Геометрия сферических многообразий и многогранники Ньютона-Окунькова тема диссертации и автореферата по ВАК РФ 01.01.06, кандидат наук Кириченко, Валентина Алексеевна

• Кириченко, Валентина Алексеевна
• кандидат наук
• 2018

Оглавление диссертации кандидат наук Кириченко, Валентина Алексеевна

Оглавление

Введение 2-24

Приложение A. Статья 1. "Chern classes of reductive groups and an adjunction formula"

Приложение B. Статья 2. "On intersection indices of subvarieties in reductive groups"

Приложение C. Статья 3. "Gelfand-Zetlin polytopes and flag varieties"

Приложение D. Статья 4. "Schubert calculus for algebraic cobordism"

Приложение E. Статья 5. "Исчисление Шуберта и многогранники Гельфанда-Цетлина"

Приложение F. Статья 6."Equivariant cobordism of flag varieties and of symmetric varieties"

Приложение G. Статья 7. "Counting vertices in the Gelfand-Zetlin polytopes " Приложение H. Статья 8. "Geometric mitosis" Приложение I. Статья 9. "Divided difference operators on polytopes" Приложение J. Статья 10. "Newton-Okounkov polytopes of flag varieties"

Введение диссертации (часть автореферата) на тему «Геометрия сферических многообразий и многогранники Ньютона-Окунькова»

Введение

Торическая геометрия и теория многогранников Ньютона Окунькова вы явили плодотворную связь между алгебраической геометрией и выпуклой reo метрией. После доказательства теорем Кушниренко и Бернштейна Хованского в 1970 х (см. напоминание в разделе 1), Аскольд Георгиевич Хованский поста вил вопрос, как перенести эти результаты на случай, когда вместо комплексного тора рассматривается произвольная связная редуктивная группа. В частности, он привлёк внимание к задаче поиска правильных аналогов многогранников Ньютона для сферических многообразий. Последние являются естественными обобщениями торических многообразий и включают такие классические приме ры, как грассманианы, многообразия флагов и полные коники (см. напоминание в разделе 2). Понятие многогранника Ньютона было перенесено на сферические многообразия Андреем Окуньковым в 1990 х [097, 098]. Позднее его конструк ция была систематически развита в работах [KaKh, LM], и полученная теория выпуклых тел Ньютона Окунькова сейчас является активной областью алгеб рай ческой геометрии.

Хотя выпуклые тела Ньютона Окунькова можно определить для линей пых расслоений на произвольных многообразиях (без действия группы), с ни ми проще работать в случае сферических многообразий благодаря связям с теорией представлений. Например, многогранники Гельфанда Цетлина (ГЦ) и многогранники Винберга Литтельманна Фейгина Фурье (ВЛФФ) возникают естественным образом как многогранники Ньютона Окунькова многообразий флагов. Мои исследования посвящены явному описанию геометрических и то иологических инвариантов сферических многообразий через геометрические и комбинаторные инварианты их многогранников Ньютона Окунькова. Цель перенести тори ческу ю картину на более общий класс многообразий с действием редуктивной группы. Раздел 3 обзор моих результатов в этом направлении. Раздел 4 содержит точные формулировки основных результатов.

1. Выпуклые тела Ньютона Окунькова

В этом разделе мы напомним конструкцию выпуклых тел Ньютона Окунь кова для широкой математической аудитории. Начнём с определения много гранника Ньютона.

Определение 1.1. Пусть / = ^аесаха многочлен Лорапа от п переменных (здесь мультииндексное обозначение ха дл я х = (х1,..., хп) ж а = (а1,..., ап) € Ъп следует понимать как ж"1 • • • х^п)- Многогранник Нъют,она А/ С это

выпуклая оболочка всех таких а € та о са = 0.

По определению многогранник Ньютона это целочисленный многогран ник, то есть его вершины лежат в Ъп.

Пример 1.2. Для п = 2 и / = 1 + 2^1 + х2 + 3х1х2 многогранник Ньютона А! это квадрат с вершинами (0,0) (1,0) (0,1) и (1,1).

Заметим, что значения многочленов Лорана с комплексными коэффици ентами определены во всех таких точках (х1,..., хп) € Сп, та о х1,... , хп = 0. Тем самым многочлены Лорана регулярные функции на комплексном торе

(С*)п .= Сп \ = 0}.

Теорема 1.3. / ] Для данного целочисленного многогранника А С рас смотрим общий набор многочленов Лорана /1(х1,..., хп),..., /п(х1}... , хп) с многогранником Ньютона А. Тогда, система /1 = ... = /п = 0 им,еет п!Уо1ише(А) решений в комплексном торе.

Теорему Кушниренко можно воспринимать как обобщение классической теоремы Безу. Многогранники Ньютона служат уточнением понятия степени многочлена. Это позволяет применять теорему Кушниренко к наборам много членов, которые не являются общими среди всех многочленов данной степени, а только среди всех многочленов с данным многогранником Ньютона. Например, теорема Кушниренко, применённая к паре общих многочленов с многогранни

даст неправильный ответ 4 (из за двух лишних решений на бесконечности). Более геометрическая точка зрения на теорему Безу и её обобщения проис текает из исчислительной геометрии и будет обсуждаться в следующем раз деле. Теорему Кушниренко обобщили на системы многочленов Лорана с раз личными многогранниками Ньютона Давид Бернштейн и Хованский, исполь зуя смешанные объёмы многогранников [В75]. Дальнейшие обобщения включа ют явные формулы для рода и эйлеровой характеристики полных пересечений {¡г = 0} п ... П {¡т = 0} в (С*)п для т < п[ ].

Мы теперь рассмотрим немного более общую ситуацию. Зафиксируем ко нечномерное векторное пространство V С хп) рациональных функ

ций на Сп. Пусть /1,..., /п общий набор функций из V, а Х0 С Сп откры тое всюду плотное подмножество, полученное как дополнение к полюсам этих функций. Сколько решений система /1 = ... = /п = 0 имеет в Х0? Например, если V совпадает с пространством, порождённым всеми многочленами Лорана с данным многогранником Ньютона, а Х0 = (С*)те, то ответ даётся теоремой Кушниренко. Ниже простой исторический пример из теории представлений.

При,м,ер 1.4. Пусть п = 3. Рассмотрим присоединённое представление группы СЬ3(С) на пространстве Еп^С3) всех линейных операторов па С3. То есть д Е СЬ3(С) действует на операторе X Е Еп^С3) таким образом:

) : X ^ дХд

1

Пусть и С СЬ3(С) подгруппа нижнетреугольных унипотентных матриц:

Г/. 0 0 \

и~ = < Хл 10 | (Ж1, Х2, Х03) Е С3 > .

1 0 0

Х1 1 0

Х2 Х3 1

Чтобы определить подпространство V С С(х1, х2, х3)7 мы ограничим функции из двойственного пространства Еп^(С3) на и "орбиту А^^-)Е13 оператора Е13 := е1 0 е3 Е End(C3) (здесь е1} е2, е3 стандартный базис в С3). Более точно,

линейная функция / € Еп^(С3) даёт многочлен /(х\_, х2, х3) таким образом:

//.....\ л

/(жь х2, хз) := /

1 0 0 Х1 1 0

Х2 Хз 1

V

001 000 000

\ ( \—л х / 1 0 0 х

V

хл 1 0

Х2 Хз 1

/ У

Легко проверить, что пространство V порождено 8 ю многочленами: 1, х1} х2, — ж2ж3, — х1х2х3. Из следующего раздела будет видно, что

теорема Кушниренко неприменима к пространству V, то есть нормализованный объём многогранника Ньютона общего многочлена из V больше, чем число решений общей системы /1 = /2 = /3 = 0 с € V.

Чтобы связать с V выпуклое тело Ньютона Окунькова, нам понадобит ся дополнительный ингредиент. Выберем инвариантный относительно сдвигов полный порядок на решётке Ъп (например, можно взять лексикографический порядок). Рассмотрим отображение

V : С(х1,...,хп) \ {0} ^

которое ведёт себя как взятие монома минимальной степени у многочлена, а

именно: и(/ + д) > шт{г>( /),и(д)} и и(/д) = и(/) + и(д) для всех ненулевых ,

ниями. Простая конструкция нормирования приводится в примере 1.7 ниже.

Определение I.Б. Выпуклое тело Ньютона Окунькова Ау(V) это замыкание выпуклой оболочки множества

чл

к=1

ж с

и{:

—1 V

| ] €У к \ С

}

Через Vк мы обозначаем подпространство, порождённое к тыми степенями функ ций из V.

Разные нормирования дают разные выпуклые тела Ньютона Окунькова. Важное приложение тел Ньютона Окунькова это аналог теоремы Кушнирен

х3 h

Рис, 1

ко. Напомним, что через Х0 С Cn мы обозначили открытое плотное подмноже ство, на котором все функции из V регулярны (то есть не имеют полюсов).

Теорема 1.6. [ , ] Если V достаточно большое, то общая система fi = ... = fn = Ос fi е V имеет n!Volume(Aw (V)) решений в Х0.

В частности, все тела Ньютона Окунькова для V имеют один и тот же объ ём. Подробности (в частности, точный смысл понятия "достаточно большое") можно найти в [KaKh, Theorem 4.9].

Пример 1.7. Пусть V пространство из примера . Определим нормирова ние v, сопоставив каждому многочлену f е С[ж1, х3] его моном наименьшей степени относительно лексикографического порядка на мономах. Более точно,

к 1 к2 кч . 11 12

мы скажем, что xiLx2zх3Л ^ ж11 х2;хзг тогда и только тогда, когда существует такое j < та о ki = U при i < j, и kj > lj. Легко проверить, что v(V) состоит из 8 ми целых точек (О, О, О), (1, О, О), (0,1, О), (0,0,1) (1,1, О), (1, 0,1) (0,1,1), (О, 2,0). Их выпуклая оболочка изображена на рисунке 1. Это ВЛФФ много гранник FFLV (1,1) для присоединённого представления группы GL3 (в этом случае, он оказывается унимодулярно эквивалентным многограннику ГЦ). В частности, FFLV(1,1) С Av(V).

2. Сферические многообразия

В этом разделе содержится краткое введение в геометрию сферических многообразий для широкой математической аудитории. Сферические многооб разия естественным образом возникают в исчислительной геометрии. Напом ним две классические задачи исчислительной геометрии в XIX веке.

Задача 2.1 (Шуберт). Сколько прямых в трёхмерном пространстве пересекают четыре данные прямые в общем положении?

Можно отождествить прямые в СР3 с векторными плоскоетями в С4, то есть прямую можно рассматривать как точку на грассманиане С(2, 4). Уело вие, что прямая I € С(2,4) пересекает фиксированную прямую ^ определяет гиперповерхность Н1 С С(2,4). Следовательно, задача сводится к вычислению числа точек пересечения четырёх гиперповерхностей в С(2, 4). Несложно про

Н1

пиана при вложении Плюккера С(2,4) ^ Р(Л2С4) ~ СР5. Образ грассманиана будет квадрикой в СР5. Число точек пересечения квадрики в СР5 с четырьмя

2

2

Задача 2.2 (Штейнер). Сколько гладких коник касается пяти данных коник?

По аналогии с задачей Шуберта мы можем отождествить коники с точками в СР5, а именно: коника, заданная уравнением ах2+Ьху+су2+((хг+еуг+/г2 = 0 соответствует точке (а : Ь : с : ( : е : /) € СР5. Гладкие коники образуют подмножество С С СР5 (дополненне СР5 \ С это множество нулей дискрими нанта). Условие, что коника касается данной коники, задаёт гиперповерхность в СР5 степей и 6. Используя теорему Везу, можно предположить (как сам Якоб Штейнер и сделал), что ответ в задаче Штейнера 65. Однако правильный ответ гораздо меньше. Это похоже на разницу между теоремами Везу и Кушнирен ко: первая даёт лишние решения, которые не имеют исчислительного смысла.

Правильный ответ нашёл Мишель Шаль, использовав (в современной термино логии) чудесную компакт,ификацию пространства С, а именно, пространство полных коник.

Герман Шуберт развил мощный общий метод (исчисление условий) для ре шения задач исчислителыюй геометрии, таких как задачи 2.1, 2.2. В каком то смысле его метод основан на неформальной версии теории пересечений. 15 ая проблема Гильберта состояла в строгом обосновании исчисления Шуберта1. В первой половине XX столетия такие обоснования были развиты как в тополо гической (кольца когомологий), так и в алгебраической (кольца Чжоу) ситуа ции. Однако версия Шуберта теории пересечений была формализована только в 1980 ые в работах Коррадо Де Кончини и Клаудио Прочези [СР85].

Пусть G связная редуктивная группа, а Н сферическая алгебраиче екая подгруппа, то есть борелевская подгруппа В с G действует на G/Н с открытой плотной орбитой. Для сферического однородного пространства G/Н (не обязательно компактного), Де Кончини и Прочези построили кольцо уело вий, которое отвечает одновременно за все задачи исчислителыюй геометрии на G/Н. Легко проверить, что комплексный тор (C*)п, грассманиан G(2, 4) и пространство С гладких коник, рассмотренные выше, все являются сфериче скими однородными пространствами относительно редуктивных групп (C*)n, GL4(C) и GL3(C), соответственно. В частности, их кольца условий корректно определены. Элементы кольца условий это классы эквивалентности многообра зий в G/Н относительно естественной численной эквивалентности. Более точно, два подмногообразия одной размерности эквивалентны, если равны их индексы пересечения с любым подмногообразием дополнительной размерности. Транзи

G

с пересечением не трансверсальных подмногообразий. Произведение в кольце

1 Das Problem bestellt darin, diejenigen geometrischen Anzahlen strenge und unter genauer Feststellung der Grenzen ihrer Gültigkeit zu beweisen, die insbesondere Schubert auf Grund des sogenannten Princips der speziellen Lage mittelst des von ihm ausgebildeten Abzählungskalküls bestimmt hat (Гильберт).

условий соответствует пересечению подмногообразий.

В частности, многие задачи исчислителыюй геометрии (включая задачи 2.1, 2.2) сводятся к вычислению индекса самопересечения гиперповерхности в G/H. В торическом случае теорема Кушниренко даёт явную формулу для ин декса самопересечения гиперповерхности {/ = 0} где f общий многочлен Лорана с данным многогранником Ньютона. В сферическом случае, явные фор мулы были получены Борисом Казарновским (случай (G х G)/Gdmg) и Мише лем Брионом (общий случай) [Kaz, Вг89]. Хотя формула Бриона Казарновского изначально была выписана в других терминах, её можно переформулировать через многогранники Ньютона Окунькова [КаКЬ2].

Пример 2.3. Сейчас мы поместим пример 1.4 в контекст исчислителыюй reo метрии и сферических однородных пространств. Пусть X = {(V1 С V2 С C3) | dim Vг = г] многообразие полных флагов в C3. Это однородное про странство относительно действия группы GL3 (C), а имени о, X = GL3(C)/B7 где борелевская подгруппа В это подгруппа верхнетреугольных матриц. Лег ко проверить, что В действует на X с открытой всюду плотной орбитой U-В/В ~ U — в частноети, X является сферическим.

Скажем, что два флага V1 С V2 и W1 С W2 в C3 находятся не в общем положении, если либо V1 С W2, либо W1 С V2. Сколько флагов в C3 находятся не в общем положении с тремя данными флагами? С одной стороны, легко пока зать, что ответ 6, используя школьную планиметрию. С другой стороны, тот же ответ можно найти, используя простейшее проективное вложение пространства X:

р : X ^ P(C3) х Р(Л2C3) P(End(C3)); р : (V1, V2) ^ V1 х V2 ^ V1 ®K2V2

и вычислив число точек пересечения образа р(Х) с тремя общими гиперплос костями в CP8 (то есть, степень многообразия р(Х)). Ограничив отображение р ^а открытую плотную В орбиту U- С X, мы сведём последнее вычисле нне к задаче из примера 1.4. В частности, мы можем показать, что включение

РРЬУ(1,1) С (V) на самом деле равенство. Действительно, по теореме объём выпуклого тела Ау(V) умножить на 3! равен степени р(Х), то есть 6. Следовательно, объём тела Ау(V) равен 1. Поскольку объём многогранника ГГЬУ(1,1) тоже равен 1, включение ГГЬУ(1,1) С Ау (V) влечёт точное ра венство.

3. Результаты и публикации

Этот раздел содержит краткий обзор результатов докторской диссертации. Основная цель поместить эти результаты в общий контекст (не слишком вдаваясь в подробности) и дать ссылки на более новые продвижения. Точные формулировки и все необходимые определения можно найти в публикациях [1] [10] (см. список публикаций в конце этого раздела).

3.1. Эйлерова характеристика полных пересечений в редуктивных группах

В торическом случае почти все инварианты полного пересечения У = {/х = 0} П ... П {/т = 0} С (С*)те можно вычислить через многогранники Ньютона Ад,..., А/т. В редуктивном случае (для (С х формула

Бриона Казарновского для т = п (то есть для нульмерного У) довольно долго была единственной явной формулой. Заметим, что её можно интерпретировать как формулу для (топологической) эйлеровой характеристики \{у). Главный результат работ [1,2] явная формула для \{у) при всех т < п. Формула по лучена в два этапа. Во первых, определены и изучены (некомпактные версии) классов Черна редуктивных групп как элементы кольца условий [1]. Во вторых, использован алгоритм Де Кончини Прочези [СР83], чтобы вычислить индексы пересечения этих классов Черна с полными пересечениями и применить фор мулу присоединения [2].

В [2] доказано, что алгоритм Де Кончини Прочези работает для классов

и

Черна, которые как правило не лежат в подкольце условий, порождённом пол ными пересечениями. Также показано, как конвертировать этот алгоритм в яв ную формулу, используя весовой многогранник представления, связанного с У. В частности, это даёт альтернативное доказательство формулы Бриона Казар новского. Хотя формулы из [1,2] прямо не используют многогранники Ныо тона Окунькова, они имеют похожую выпукло геометрическую природу (см. точную формулу в случае т = 1 в разделе ). Недавно больше инвариантов (в частности, арифметический род) полных пересечений в сферических однород пых пространствах были найдены через многогранники Ньютона Окунькова (см. [КаКЬЗ], гда также приводится история вопроса).

3.2. Выпукло-геометрические модели для исчисления Шуберта

Кольцо условий комплексного тора порождено классами гииерповерхно стей. Это же верно для полных многообразий флагов С/В, но необязательно верно для более общих сферических однородных пространств. Таким образом, полные многообразия флагов первые кандидаты для применения выпукло геометрических методов к вычислению произведений в кольце условий. Заме тим, что поскольку С/В компактно, его кольцо условий совпадает с кольцом Чжоу, а последнее обладает естественным базисом из циклов Шуберта, кото

В

рические модели для исчисления Шуберта и находим выпукло геометрические реализации циклов Шуберта.

В [3] формула Пьери Шевалле для С/В в типе А проинтерпретирована через многогранники ГЦ. В [5] (совместная работа с Евгением Смирновым и Владленом Тимориным), мы развиваем необходимую теорию для реализации циклов Шуберта линейными комбинациями граней многогранника так, чтобы пересечение граней соответствовало произведению (в кольце Чжоу) циклов Шу А

что позволило нам представить произведение любых двух циклов Шуберта неот

рицатедыюй линейной комбинацией граней многогранника ГЦ. Этот результат был мотивирован кандидатской диссертацией Михаила Когана, который пер вым связал грани многогранников ГЦ с многообразиями Шуберта [Ко]. Мы также получили формулы для характеров Демазюра многообразий Шуберта через экспоненциальные суммы по целым точкам в гранях Когана многогран пика ГЦ.

В [8] разработан геометрический алгоритм для реализации циклов Шубер та гранями многогранников в произвольном типе. В типе А этот алгоритм сво дится к митозу Кнутсона Миллера на пайп дримах. В типах В и С, он сводится к новому комбинаторному алгоритму, который может дать явные реализации циклов Шуберта в симплектических и ортогональных многообразиях флагов гранями симплектических многогранников ГЦ (см. подробности в разделе 4.2).

В [7] (совместная работа с Павлом Гусевым и Владленом Тимориным), мы изучаем комбинаторику многогранников ГЦ, соответствующих разным частич ным многообразиям флагов (или в комбинаторной терминологии разбиениям Г12%2 ... к4). Заметим, что все многогранники ГЦ для данного разбиения име ют один и тот же комбинаторный тип. Мы определяем рекуррентное соотно шение на число вершин V(Г12п ... к4) и уравнений в частных производных на экспоненциальную производящую функцию для чисел V(Г12п ... к4). Недавно

те [АСК].

3.3. Реинкарнации операторов разделённых разностей (ОРР)

ОРР и операторы Демазюра важные инструменты в исчислении Шубер та и теории представлений. Они использовались в [BGG] и [D], чтобы выразить по индукции циклы Шуберта на полных многообразиях флагов как многочле ны от первых классов Черна линейных расслоений (как в когомологиях, так и в К теории). Ещё одно приложение это формула Демазюра для характе ров Демазюра многообразий Шуберта [D], В [4,6] мы определяем аналоги ОРР

в (эквивариантном) исчислении Шуберта для алгебраических кобордизмов. В [9] определяется выпукло геометрическая версия операторов Демазюра и ис пользуется, чтобы строить многогранники, кодирующие характеры Демазюра. В [10] многогранники Ньютона Окунькова многообразий флагов для одного из геометрических нормирований явно вычислены с помощью простой выпукло геометрической конструкции, которая также мотивирована ОРР.

Заметим, что классические ОРР и операторы Демазюра можно опреде лить единообразно, используя аддитивный (кольца Чжоу) и мультипликатив ный (К теория) формальные групповые законы, соответственно. В [4] (совмест пая работа с Иенсом Хорнбостелем) мы определяем ОРР для универсального группового закона и применяем их к изучению исчисления Шуберта в кольце алгебраических кобордизмов (определённом в работах Левина Мореля и Ле вина Пандхарипанде) полных многообразий флагов. Эта работа мотивирована результатами Бресслера Ивенса по исчислению Шуберта в комплексных кобор дизмах [ВЕ], и частично была направлена на перенос их результатов в алгебро геометрический контекст. Мы также вывели версию формулы Пьери Шевалле для кобордизмов.

К

многообразий флагов с одной стороны, и алгебраическими/комплексными ко бордизмами с другой стороны. Например, классы многообразий Шуберта да ют естественный базис в первых теориях, но не в последних, поскольку мно гообразия Шуберта не всегда гладкие. Зато многообразия Ботта Самельсона образуют естественное порождающий набор (но не базис) в кольце алгебраиче ских/комплексных кобордизмов.

В [6] (совместная работа с Амаленду Кришной) мы описываем кольцо эк инвариантных алгебраических кобордизмов многообразий флагов и чудесных компактификаций симметрических пространств минимального ранга. В част ности, пространства (С х С)/С1т& симметрические минимального ранга. В случае многообразий флагов мы использовали комплексные кобордизмы и то

дологические аргументы. Недавно чисто алгебро геометрический подход был предложен в [CZZ], В случае чудесных компактификаций мы использовали под ход Бриона Джошуа, описавших эквивариантые кольца Чжоу [BJ].

В [9] определены аналоги операторов Демазюра на выпуклых многогранни ках. В общем случае эти операторы переводят многогранник Р в многогранник или выпуклую цепь Р' на единицу большей размерности, так что экспоненци альные суммы по целым точкам в Р и Р' связаны с помощью классического оператора Демазюра. В частности, выпукло геометрические операторы Дема зюра можно использовать для индуктивного построения многогранников ГЦ А

точки. В типе С2 они использовались, чтобы построить симплектические ОРР многогранники, которые комбинаторно не эквивалентны симплектическим мно гогранникам ГЦ и ВЛФФ. Оказывается, эти многогранники совпадают с много гранниками Ньютона Окунькова симплектического многообразия флагов для естественного геометрического нормирования [8]. Недавно Наоки Фуджита вы сказал гипотезу, что несколько классов ОРР многогранников совпадают с по лиэдральными реализациями Зелевинского Накаджимы (для С2 это следует из [FN, Example 5.10]).

В [10] многогранники Ньютона Окунькова полных многообразий флагов в А

нутых подмногообразий Шуберта, которые соответствуют финальным подсло вам в разложении самого длинного элемента (s1)(s2Si)(s3s2si)(...)(sn-1... s1) (примеры , , иллюстрируют это вычисление при п = 3, см. также раздел 4.2). Удивительно, что полученные многогранники совпадают с мно гогранниками ВЛФФ, хотя последние были изначально построены из других соображений. Это совпадение мотивировало дальнейшие исследования (см. по дробности в [FaFL]).

Список публикаций

fl] V. KlRlTCHENKO. Chern classes of reductive groups and an adjunction formula, Arm. Inst. Fourier 56

(2006), no. 4, 1225 1256

[2] , On intersection indices of subvarieties in reductive groups, Moscow Math. J., 7 (2007), no. 3 (выпуск, посвящённый А.Г.Хованскому), 489 505

[3] , Gelfand-Zetlin polytopes and flag varieties, IMRN (2010), no. 13, 2512 2531

[4] J. Hornbostel, , Schubert calculus for algebraic cobordism, J. reine angew. Math. (Crelles), 2011, no. 656, 59 85

[5] Кириченко В. А., Смирнов E. Ю., Тиморин В. А., Исчисление Шуберт,а и многогранники Гельфанда-Цетлина, Успехи математических наук 67 (2012), по. 4, 89 128

[6] , A. Krishna, Equivariant Cobordism of Flag Varieties and of Symmetric Varieties, Transform. Groups, 18 (2013), no. 2, 391 413

[7] P. Gusev, , V. Timorin, Counting vertices in the Gelfand-Zetlin polytopes, J. of Comb. Theory, Series A, 120 (2013), 960 969

[8] , Geometric mitosis, Math. Res. Lett., 23 (2016), no. 4, 1069 1096

[9] , Divided difference operators on polytopes, Adv. Studies in Pure Math. 71 (2016), 161 184

[10] , Newton Okounkov polytopes of flag varieties, Transform. Groups 22 (2017), no. 2, 387 402

4. Основные результаты

В этом разделе собраны более подробные формулировки основных резуль татов докторской диссертации. Мы постарались сделать изложение как можно более самодостаточным. Однако мы предполагаем, что читатель знаком с тео рией представлений и исчислением Шуберта.

4.1. Эйлерова характеристика полных пересечений в редуктивных

группах

Пусть G связная комплексная редуктивная группа размерности п и ран га к, и пусть ж : G ^ GL(V) точное представление группы G. Назовём общим гиперплоским сечением Н^, соответствующим ж, прообраз ж-1(Н) пересечения ж(G) с общей аффинной гиперплоскостью Н С End(V). Несложно показать, что все общие гиперплоские сечения имеют одну и ту же (топологическую) эйлерову характеристику. Ниже приводится явная формула для эйлеровой ха рактеристики х(Н^). Она следует из [1, Theorem 1.1], [2, Theorem 1.3], из кото рых также следует аналогичная явная формула для эйлеровой характеристики

полного пересечения гиперплоских сечений.

Выберем максимальный тор Т С G, и обозначим через Lj его решётку ха рактеров. Выберем камеру Вейля V С Lt 3 R Обозначим через R+ множество

G

жительных корней группы G. Скалярное произведенпе (•, •) на Lj 3 R задано с помощью такой невырожденной симметрической билинейной формы на алгеб G

группы (такая форма существует, поскольку G редуктивпа). Через Р^ С Lj 3 R обозначим весовой многогранник представления ж, то есть, выпуклую оболочку весов тора Т, которые встречаются в ж.

Определим полиномиальную функцию F(х, у) на (Lj0Lj)3R по формуле:

- (х,»)=п ^ ■

aeR+ VK' '

С геометрической точки зрения этот многочлен считает индекс самопересече ния дивизоров на произведении G/B х G/B двух многообразий флагов (диви зоры соответствуют весам (Ai, А2) G Lt 0 Lt).

Теорема 4.1. [1, Theorem 1.1J, [2, Theorem 1.2] Зададим дифференциальный one pamop D на функциях на (Lt 0 Lt) 3 R формулой:

D = П (!+da)(1+ад,

aGR+

где da and da производные вдоль векторов (а, 0) and (0,а)7 соответствен но. Обозначим через [D]j однородную компоненту i той степени в D7 если рассматривать D как многочлен от переменных da и da. Тогда,

Х(Н*) = (-1)n-i (п! - (п - 1)![D]i + (п - 2)![D]2 - ■ ■ ■ + k![D]n_,) F(х, x)dx.

Ръ nv

Форм,а, объёма dx нормируется так, чтобы кообзём решётки Lt в Lt 3 R был 1

Например, для G = SL3(C) и неприводимого представления ж со старшим весом гаш1 + пш2 (через и ш2 обозначены фундаментальные веса) получается такой ответ:

х(Нп) = -3(т8 + 16т7п + 112т6п2 + 448т5п3 + 700т4п4 + 448т3п5 + 112т2п6+

16тп7 + п8 + 18(т6 + 12т5п + 50т4п2 + 80т3п3 + 50т2п4 + 12тп5 + п6) + +6(5т4 + 40т3п + 72т2п2 + 40тп3 + 5п4) + 6(т2 + 4тп + п2 )--6(т + п)(т6 + 13т5п + 71т4п2 + 139т3п3 + 71тп4 + 13тп5 + п6+

/А Ч. 0 0 Ч. А \ /О О \ \ \

+5(т + 9т п + 19т п + 9тп +п)+3(т + 5тп + п ))).

4.2. Выпукло-геометрические модели для исчисления Шуберта

В работах [3,5] многогранник Гельфанда Цетлпна используется, чтобы мо делировать исчисление Шуберта на многообразии полных флагов в Cn. При этом пересечение циклов на многообразии флагов соответствует пересечению

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Приложение A.

Статья 1.

Valentina Kiritchenko "Chern classes of reductive groups and an adjunction formula"

Annales de L'Institut Fourier, Grenoble 56, 4 (2006) 1225-1256

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Valentina KIRITCHENKO

Chern classes of reductive groups and an adjunction formula

Tome 56, no 4 (2006), p. 1225-1256.

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Ann. Inst. Fourier, Grenoble 56, 4 (2006) 1225-1256

CHERN CLASSES OF REDUCTIVE GROUPS AND AN ADJUNCTION FORMULA

by Valentina KIRITCHENKO

Abstract. — In this paper, I construct noncompact analogs of the Chern classes for equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the (topological) Euler characteristic of complete intersections in reductive groups. In the case where a complete intersection is a curve, this formula gives an explicit answer for the Euler characteristic and the genus of the curve. I also prove that the higher Chern classes vanish. The first and the last nontrivial Chern classes are described explicitly. An extension of these results to the setting of spherical homogeneous spaces is outlined.

Résumé. — Dans cet article, je construis l'analogue non compact des classes de Chern pour des fibrés vectoriel équivariants au-dessus de groupes réductifs complexes. Pour le fibré tangent, ces classes de Chern produisent une formule d'adjonction pour la caractéristique d'Euler (topologique) d'intersections complètes dans des groupes réductifs. Dans le cas d'une intersection complète qui est une courbe, cette formule donne une réponse explicite pour la caractéristique d'Euler et le genre de la courbe. Je démontre également que les classes de Chern supérieures sont nulles. La première et la dernière classe de Chern non nulle sont décrites explicitement. J'esquisse également une extension de ces résultats dans le cadre des espaces homogènes sphériques.

1. Introduction and main results

Let G be a connected complex reductive group. Consider a faithful finite-dimensional representation n : G ^ GL(V) on a complex vector space V. Let H C End(V) be a generic affine hyperplane. The hypersurface n-1 (n(G) n H) C G is called a hyperplane section corresponding to the representation n. The problem underlying this paper is how to find the

Keywords: Reductive groups, hyperplane section, Chern classes. Math. classification: 14L30, 20G05.

Euler characteristic of a hyperplane section or, more generally, of the complete intersection of several hyperplane sections corresponding to different representations.

The motivation to study such question comes from the case where the group G = (C*)n is a complex torus. In this case, D. Bernstein, A. Khovan-skii and A. Kouchnirenko found an explicit and very beautiful answer in terms of the weight polytopes of representations (see [18]). E.g. the Euler characteristic x(n) of a hyperplane section corresponding to the representation n is equal to ( — 1)n times the normalized volume of the weight polytope of n. The proof uses an explicit relation between the Euler characteristic x(n) and the degree of the affine subvariety n(G) in End(V):

(1.1) x(n) = ( —1)n-1 deg n(G).

The degree is defined as usual. Namely, the degree of an affine subvariety X C CN equals to the number of the intersection points of X with a generic affine subspace in CN of complementary dimension. For the degree deg n(G) (that can also be interpreted as the self-intersection index of a hyperplane section corresponding to the representation n) there is an explicit formula proved by Kouchnirenko. Later D. Bernstein, and Khovan-skii found an analogous formula for the intersection index of hyperplane sections corresponding to different representations.

How to extend these results to the case of arbitrary reductive groups? It turned out that the formulas for the intersection indices of several hyperplane sections can be generalized to reductive groups and, more generally, to spherical homogeneous spaces. For reductive groups, this was done by B. Kazarnovskii [17]. Later, M. Brion established an analogous result for all spherical homogeneous spaces [4]. For reductive groups, the Brion-Kazarnovskii theorem allows to compute explicitly the intersection index of n generic hyperplane sections corresponding to different representations. The precise definition of the intersection index is given in Section 2.

However, when G is an arbitrary reductive group, it is no longer true that x(n) = ( —1)n-1 deg n(G). K. Kaveh computed explicitly x(n) and deg n(G) for all representations n of SL2 (C). His computation shows that, in general, there is a discrepancy between these two numbers. Kaveh also listed some special representations of reductive groups, for which these numbers still coincide [16].

In this paper, I will present a formula that, in particular, generalizes formula (1.1) to the case of arbitrary reductive groups. To do this I will construct algebraic subvarieties Sj C G, whose degrees fill the gap between the Euler characteristic and the degree. My construction is similar to one

of the classical constructions of the Chern classes of a vector bundle in the compact setting (Subsection 3.1). The subvarieties Si can be thought of as Chern classes of the tangent bundle of G. I will also construct Chern classes of more general equivariant vector bundles over G (Subsection 3.2). These Chern classes are in many aspects similar to the usual Chern classes of compact manifolds. There is an analog of the cohomology ring for G, where the Chern classes of equivariant bundles live. This analog is the ring of conditions constructed by C. De Concini and C. Procesi [10, 8](see Section 2 for a reminder). It is useful in solving enumerative problems. In particular, the intersection product in this ring is well-defined.

I now formulate the main results. Denote by n and k the dimension and the rank of G, respectively. Recall that the rank is the dimension of a maximal torus in G. Denote by [S1],..., [Sn] the Chern classes of the tangent bundle of G as elements of the ring of conditions, and denote by S1,..., Sn subvarieties representing these classes. In the case of the tangent bundle, it turns out (see Lemma 3.8) that the the higher Chern classes [Sn-fc+i],..., [Sn] vanish. E.g. if G is a torus, then all Chern classes [Si] vanish.

Let Hi,..., Hm be a generic collection of m hyperplane sections corresponding to faithful representations n1,..., nm of the group G (for the precise meaning of "generic" see Subsection 4.3). Then the following theorem holds.

Theorem 1.1. — The Euler characteristic of the complete intersection H1 n ... n Hm is equal to the term of degree n in the expansion of the following product:

m

(1 + S1 + ... + S„-fc) n Hi(1 + Hi)-1. i=1

The product in this formula is the intersection product in the ring of conditions.

This is very similar to the classical adjunction formula in the compact setting.

In particular, the Euler characteristic of just one hyperplane section corresponding to a representation n is equal to the following alternating sum. Put S0 = G. Then

n-k

x(n) = E(-1)n-i-1 deg n(Si).

i=0

The latter formula may have applications in the theory of generalized hy-pergeometric equations. In the torus case, I. Gelfand, M. Kapranov and A. Zelevinsky showed that the Euler characteristic x(n) gives the number of integral solutions of the generalized hypergeometric system associated with the representation n [13]. A similar system can be associated with the representation n of any reductive group [15]. In the reductive case, the number of integral solutions of such a system is also likely to coincide with x(n).

The proof of Theorem 1.1 is similar to the proof by Khovanskii [18] in the torus case. Namely, Theorem 1.1 follows from the adjunction formula applied to the closure of a complete intersection in a suitable regular com-pactification of G (see Subsection 4.3). The key ingredient is a description of the tangent bundles of regular compactifications due to Ehlers [11] and Brion [5]. This description is outlined in Subsection 4.2.

The remaining problem is to describe the Chern classes [S1],..., [Sn-k] so that their intersection indices with hyperplane sections may be computed explicitly. So far there is such a description for the first and the last Chern classes (see Subsection 3.3). Namely, [S1] is the class of a generic hyperplane section corresponding to the irreducible representation with the highest weight 2p. Here p is the sum of all fundamental weights of G. This description follows from a result of A. Rittatore [25] concerning the first Chern class of reductive group compactifications. The last Chern class [Sn-k] is up to a scalar multiple the class of a maximal torus in G. There is a hope that the intersection indices of other Chern classes Sj with hyperplane sections can also be computed using a formula similar to the Brion-Kazarnovskii formula.

If a complete intersection is a curve, i.e., m = n — 1, then the formula of Theorem 1.1 involves only the first Chern class [S1 ]. In this case, the computation of [S1] together with the Brion-Kazarnovskii formula allows us to compute explicitly the Euler characteristic and the genus of a curve in G in terms of the weight polytopes of n1,..., nm (see Corollaries 4.9 and 4.10, Subsection 4.3). Note that these two numbers completely describe the topological type of a curve.

Most of the constructions and results of this paper can be extended without any change to the case of arbitrary spherical homogeneous spaces. This is discussed in Section 5.

I am very grateful to Mikhail Kapranov and Askold Khovanskii for numerous stimulating discussions and suggestions. I would like to thank Ki-umars Kaveh for useful discussions and Michel Brion for valuable remarks

on the first version of this paper. I am also grateful to the referee for many useful remarks and comments.

Part of the results of this paper were included into my PhD thesis at the University of Toronto [19].

Throughout this paper, whenever a group action is mentioned, it is always assumed that a complex algebraic group acts on a complex algebraic variety by algebraic automorphisms. In particular, by a homogeneous space for a group I will always mean the quotient of the group by some closed algebraic subgroup.

The following remarks concern notations. In this paper, the term equi-variant (e.g. equivariant compactification, bundle, etc.) will always mean equivariant under the action of the doubled group G x G, unless otherwise stated. The Lie algebra of G is denoted by g. I also fix an embedding G C GL(W) for some vector space W. Then for g G G and A G g, notation Ag and gA mean the product of linear operators in End(W).

2. Equivariant compactifications and the ring of conditions

This section contains some well-known notions and theorems, which will be used in the sequel. First, I define the notion of spherical action and describe equivariant compactifications of reductive groups following [9], [15] and [26]. Then I state Kleiman's transversality theorem [20] and recall the definition of the ring of conditions [10, 8].

Spherical action. Reductive groups are partial cases of more general spherical homogeneous spaces. They are defined as follows. Let G be a connected complex reductive group, and let M be a homogeneous space under G. The action of G on M is called spherical, if a Borel subgroup of G has an open dense orbit in M. In this case, the homogeneous space M is also called spherical. An important and very useful property, which characterizes a spherical homogeneous space M, is that any compactification of M equivariant under the action of G contains only a finite number of orbits [21].

There is a natural action of the group G x G on G by left and right multiplications. Namely, an element (g1,g2) G G x G maps an element g G G to g1gg-1. This action is spherical as follows from the Bruhat decomposition of G with respect to some Borel subgroup. Thus the group G can be considered as a spherical homogeneous space of the doubled group G x G with respect to this action. For any representation n : G ^ GL(V) this

action can be extended straightforwardly to the action of n(G) x n(G) on the whole End(V) by left and right multiplications. I will call such actions standard.

Equivariant compactifications. With any representation n one can associate the following compactification of n(G). Take the projectivization P(n(G)) of n(G) (i.e., the set of all lines in End(V) passing through a point of n(G) and the origin), and then take its closure in P(End(V)). We obtain a projective variety Xn С P(End(V)) with a natural action of Gx G coming from the standard action of n(G) x n(G) on End(V). Below I will list some important properties of this variety.

Assume that P(n(G)) is isomorphic to G. Fix a maximal torus T С G. Let LT be its character lattice. Consider all weights of the representation n, i.e., all characters of the maximal torus T occurring in n. Take their convex hull Pn in LT < R. Then it is easy to see that Pn is a polytope invariant under the action of the Weyl group of G. It is called the weight polytope of the representation n. The polytope Pn contains information about the compactification Xn.

Theorem 2.1.

1) ([26], Proposition 8) The subvariety Xn consists of a finite number of G x G-orbits. These orbits are in one-to-one correspondence with the orbits of the Weyl group acting on the faces of the polytope Pn.

2) Let a be another representation of G. The normalizations of subvarieties Xn and Xa are isomorphic if and only if the normal fans corresponding to the polytopes Xn and Xa coincide. If the first fan is a subdivision of the second, then there exists an equivariant map from the normalization of Xn to Xa, and vice versa.

The second part of Theorem 2.1 follows from the general theory of spherical varieties (see [21], Theorem 5.1) combined with the description of compactifications Xn via colored fans (see [26], Sections 7, 8).

In particular, suppose that the group G is of adjoint type, i.e., the center of G is trivial. Let n be an irreducible representation of G with a strictly dominant highest weight. It is proved in [9] that the corresponding compactification Xn of the group G is always smooth and, hence, does not depend on the choice of a highest weight. Indeed, the normal fan of the weight polytope Pn coincides with the fan of the Weyl chambers and their faces, so the second part of Theorem 2.1 applies. This compactification is called the wonderful compactification and is denoted by Xcan. It was introduced by De Concini and Procesi [9]. The boundary divisor Xcan \ G is a

divisor with normal crossings. There are k orbits O1,..., Ok of codimension one in Xcan. The other orbits are obtained as the intersections of the closures O1,..., Ok. More precisely, to any subset {¿1, ¿2,..., ¿m} C {1,..., k} there corresponds an orbit Oi1 n Oi2 n ... n Oim of codimension m. So the number of orbits equals to 2k. There is a unique closed orbit O1 n ... n Ok, which is isomorphic to the product of two flag varieties G/B x G/B. Here B is a Borel subgroup of G.

Compactifications of a reductive group arising from its representations are examples of more general equivariant compactifications of the group. A compact complex algebraic variety with an action of G x G is called an equivariant compactification of G if it satisfies the following conditions. First, it contains an open dense orbit isomorphic to G. Second, the action of G x G on this open orbit coincides with the standard action by left and right multiplications.

The ring of conditions. The following theorem gives a tool to define the intersection index on a noncompact group, or more generally, on a homogeneous space. Recall that two irreducible algebraic subvarieties Y1 and Y2 of an algebraic variety X are said to have proper intersection if either their intersection Y1 n Y2 is empty or all irreducible components of Y1 n Y2 have dimension dim Y1 + dim Y2 — dim X.

Theorem 2.2 (Kleiman's transversality theorem, [20]). — Let H be a connected algebraic group, and let M be a homogeneous space under H. Take two algebraic subvarieties X, Y C M. Denote by gX the left translate of X by an element g G H. There exists an open dense subset of H such that for all elements g from this subset the intersection gX n Y is proper. If X and Y are smooth, then gX n Y is transverse for general g G H.

In particular, if X and Y have complementary dimensions (but are not necessarily smooth), then for almost all g the translate gX intersects Y transversally at a finite number of points, and this number does not depend on g.

If X and Y have complementary dimensions, define the intersection index (X, Y) as the number #(gX n Y) of the intersection points for a generic g G H. If one is interested in solving enumerative problems, then it is natural to consider algebraic subvarieties of M up to the following equivalence. Two subvarieties X1 , X2 of the same dimension are equivalent if and only if for any subvariety Y of complementary dimension the intersection indices (X1,Y) and (X2,Y) coincide. This relation is similar to the numerical equivalence in algebraic geometry (see [12], Chapter 19). Consider all formal

linear combinations of algebraic subvarieties of M modulo this equivalence relation. Then the resulting group C*(M) is called the group of conditions of M.

One can define an intersection product of two subvarieties X, Y С M by setting X ■ Y = gX П Y, where g G G is generic. However, the intersection product sometimes is not well-defined on the group of conditions (see [10] for a counterexample). A remarkable fact is that for spherical homogeneous spaces the intersection product is well-defined, i.e., if one takes different representatives of the same classes, then the class of their product will be the same [10, 8]. The corresponding ring C*(M) is called the ring of conditions.

In particular, the group of conditions C* (G) of a reductive group is a ring. De Concini and Procesi related the ring of conditions to the cohomology rings of equivariant compactifications as follows. Consider the set S of all smooth equivariant compactifications of the group G. This set has a natural partial order. Namely, a compactification Xa is greater than Xn if Xa lies over Xn, i.e., if there exists a map Xa ^ Xn commuting with the action of G x G. Clearly, such a map is unique, and it induces a map of cohomology rings H*(Xn ) ^ H*(XCT).

Theorem 2.3 ([10, 8]). — The ring of conditions C*(G) is isomorphic to the direct limit over the set S of the cohomology rings H * (Xn ).

De Concini and Procesi proved this theorem in [10] for symmetric spaces. In [8] De Concini noted that their arguments go verbatim for arbitrary spherical homogeneous spaces.

3. Chern classes of reductive groups 3.1. Preliminaries

Reminder about the classical Chern classes. In this paragraph, I will recall one of the classical definitions of the Chern classes, which I will use in the sequel. For more details see [14].

Let M be a compact complex manifold, and let E be a vector bundle of rank d over M. Consider d global sections s1,..., sd of E that are C smooth. Define their i-th degeneracy locus as the set of all points x G M such that the vectors s1(x),..., sd_j+1(x) are linearly dependent. The homology class of the i-th degeneracy locus is the same for all generic

choices of the sections s1(x),..., sd(x) [14]. It is called the i-th Chern class of E.

In what follows, I will only consider complex vector bundles that have plenty of algebraic global sections (so that in the definition of the Chern classes, it will be possible to take only algebraic global sections instead of CTO-smooth ones).

In particular, there is the following way to choose generic global sections. Let r(E) be a finite-dimensional subspace in the space of all global C smooth sections of the bundle E. Suppose that at each point x G M the sections of r(E) span the fiber of E at the point x. Then there is an open dense subset U in r(E)d such that for any collection of global sections (s1,..., sd) C U their i-th degeneracy locus is a representative of the i-th Chern class of E.

I will also use the following classical construction that associates with the subspace r(E) a map from the variety M to a Grassmannian. Denote by N the dimension of r(E). Let G(N — d, N) be the Grassmannian of subspaces of dimension (N — d) in r(E). One can map M to G(N — d, N) by assigning to each point x G M the subspace of all sections from r(E) that vanish at x. By construction of the map the vector bundle E coincides with the pull-back of the tautological quotient vector bundle over the Grassmannian G(N—d, N). Recall that the tautological quotient vector bundle over G(N— d, N) is the quotient of two bundles. The first one is the trivial vector bundle whose fibers are isomorphic to r(E), and the second is the tautological vector bundle whose fiber at a point A G G(N — d, N) is isomorphic to the corresponding subspace A of dimension N — d in r(E).

Using the definition of the Chern classes given above, it is easy to check that the i-th Chern class of the tautological quotient vector bundle is the homology class of the following Schubert cycle. Let A1 C ... C Ad C r(E) be a partial flag such that dim Aj = j. In the sequel, by a partial flag I will always mean a partial flag of this type. The i-th Schubert cycle Ci corresponding to such a flag consists of all points A G G(N — d, N) such that the subspaces A and Ad-i+1 have nonzero intersection.

The following proposition relates the Schubert cycles Ci to the Chern classes of E.

Proposition 3.1 ([14]). — Let p : M ^ G(N — d, N) be the map constructed above, and let Ci be the i-th Schubert cycle corresponding to a generic partial flag in r(E). Then the i-th Chern class of E coincides with the homology class of the inverse image of Ci under the map p:

ci(E) = [p-1(Ci)].

In particular, this proposition allows to relate the definition of the Chern classes via degeneracy loci to other classical definitions.

In the sequel, the following statement will be used. For any algebraic subvariety X С G(N — d, N), a partial flag can be chosen in such a way that the corresponding Schubert cycle C has proper intersection with X. This follows from Kleiman's transversality theorem, since the Grassman-nian G(N—d, N) can be regarded as a homogeneous space under the natural action of the group GLN. Then any left translate of a Schubert cycle C is again a Schubert cycle of the same type.

Equivariant vector bundles. In this paragraph, I will recall the definition and some well-known properties of equivariant vector bundles.

Let E be a vector bundle of rank d over G. Denote by Vg С E the fiber of E lying over an element g G G. Assume that the standard action of G x G on G can be extended linearly to E. More precisely, there exists a homomorphism A : G x G ^ Aut(E) such that A(gi,g2) restricted to

the fiber Vg is a linear operator from Vg to V -1 .If these conditions

g * g gigg2

are satisfied, then the vector bundle E is said to be equivariant under the action of G x G.

Two equivariant vector bundles E1 and E2 are equivalent if there exists an isomorphism between E1 and E2 that is compatible with the structure of fiber bundle and with the action of G x G. The following simple and well-known proposition describes equivariant vector bundles on G up to this equivalence relation.

Proposition 3.2. — The equivalence classes of equivariant vector bundles of rank d are in one-to-one correspondence with the linear representations of G of dimension d.

Indeed, with each representation n : G ^ V one can associate a bundle E isomorphic to G x V with the following action of G x G:

A(gi,g2): (g,v) ^ (gigg-1,n(gi)v).

Then A(g, g-1) stabilizes the identity element e G G and acts on the fiber Ve = V by means of the operator n(g).

E.g. the adjoint representation of G on the Lie algebra g = T Ge corresponds to the tangent bundle T G on G. This example will be important in the sequel.

Among all algebraic global sections of an equivariant bundle E there are two distinguished subspaces, namely, the subspaces of left- and right-invariant sections. They consist of sections that are invariant under the

action of the subgroups G x e and e x G, respectively. Both spaces can be canonically identified with the vector space V. Indeed, any vector X G V defines a right-invariant section vr (g) = (g,X). Then it is easy to see that any left-invariant section v; is given by the formula v;(g) = (g,n(g)Y) for Y G V.

Denote by r(E) the space of all global sections of E that are obtained as sums of left- and right-invariant sections. Let us find the dimension of the vector space r(E). Clearly, if the representation n does not contain any trivial sub-representations, then r(E) is canonically isomorphic to the direct sum of two copies of V. Otherwise, let C C V be the maximal trivial sub-representation. Embed C to V© V diagonally, i.e., v G C goes to (v, v). It is easy to see that r(E) as a G-module is isomorphic to the quotient space (V © V)/C. Denote by c the dimension of C. Then the dimension of r(E) is equal to 2d — c.

3.2. Chern classes with values in the ring of conditions

In this subsection, I define Chern classes of equivariant vector bundles over G. These Chern classes are elements of the ring of conditions C*(G). Unlike the usual Chern classes in the compact situation, they measure the complexity of the action of G x G but not the topological complexity (topologically any G x G-equivariant vector bundle over G is trivial). While the definition of these classes does not use any compactification it turns out that they are related to the usual Chern classes of certain vector bundles over equivariant compactifications of G.

Throughout this subsection, E denotes the equivariant vector bundle over G of rank d corresponding to a representation n : G ^ GL(V). In the subsequent sections, I will only use the Chern classes of the tangent bundle.

Definition of the Chern classes. An equivariant vector bundle E has a special class Г(Е) of algebraic global sections. It consists of all global sections that can be represented as sums of left- and right-invariant sections.

Example. — If E = T G is the tangent bundle, then Г(Е) is a very natural class of global sections. It consists of all vector fields coming from the standard action of Gx G on G. Namely, with any element (X, Y) G 0®g one can associate a vector field v G Г(Е) as follows:

v(x) = dt

[etXie-iY] = Xx - xY.

t=0

This example suggests that one represent elements of r(E) not as sums but as differences of left- and right-invariant sections.

The space r(E) can be employed to define Chern classes of E as usual. Take d generic sections v1,..., vd G r(E). Then the i-th Chern class is the i-th degeneracy locus of these sections. More precisely, the i-th Chern class Sj(E) C G consists of all points g G G such that the first d — i + 1 sections v1(g),..., vd_j+1(g) taken at g are linearly dependent. This definition almost repeats one of the classical definitions of the Chern classes in the compact setting (see Subsection 3.1). The only difference is that global sections used in this definition are not generic in the space of all sections. They are generic sections of the special subspace r(E). If one drops this restriction and applies the same definition, then the result will be trivial, since the bundle E is topologically trivial. In some sense, the Chern classes will sit at infinity in this case (the precise meaning will become clear from the second part of this subsection). The purpose of my definition is to pull them back to the finite part.

Thus for each i = 1,..., d we get a family Sj(E) of algebraic subvarieties Sj(E) parameterized by collections of d — i + 1 elements from r(E). In the compact situation, all generic members of an analogous family represent the same class in the cohomology ring. The same is true here, if one uses the ring of conditions as an analog of the cohomology ring in the noncompact setting.

Lemma 3.3. — For all collections v1,..., vd_j+1 belonging to some open dense subset of (r(E))d-j+1 the class of the corresponding subvariety Sj(E) in the ring of conditions C*(G) is the same.

The lemma implies that the family Sj(E) parameterized by elements of (r(E))d-j+1 provides a well-defined class [Sj(E)] in the ring of conditions C* (G).

Definition 3.4. — The class [Sj(E)] G C*(G) defined by the family Sj(E) is called the i-th Chern class of a vector bundle E with value in the ring of conditions.

Before proving the lemma let me give another description of the Chern classes [Sj(E)].

Maps to Grassmannians. In this paragraph, I apply the classical construction discussed in Subsection 3.1 to define a map from the group G to the Grassmannian G(d — c, r(E)) of subspaces of dimension (d — c) in

the space r(E). Recall that c is the dimension of the maximal trivial sub-representation of V, and the dimension of r(E) is 2d — c (see the end of Subsection 3.1).

Note that the global sections from the subspace r(E) span the fiber of E at each point of G. Hence, one can define a map fE from G to the Grassmannian G(d — c, r(E)) as follows. A point g G G gets mapped to the subspace Ag C r(E) spanned by all global sections that vanish at g. Clearly, the dimension of Ag equals to (dim r(E) — d) = (d — c) for all g G G. We get the map

^e : G ^ G(d — c,r(E)); ^ : g ^ Ag.

The subspace Ag can be alternatively described using the graph of the operator n(g) in V©V. Namely, it is easy to check that Ag = {(X, n(g)X), X G V}/C. Then fE comes from the natural map assigning to the operator n(g) on V its graph in V © V.

Clearly, the pull-back of the tautological quotient vector bundle over G(d, r(E)) is isomorphic to E. Hence, the Chern class Si(E) constructed via elements v1 , . . . , vd is the inverse image of the Schubert cycle Ci corresponding to the partial flag (v1) C (v1, v2) C ... C (v1,..., vd) C r(E) (see Subsection 3.1). Here (v1,..., vi) denotes the subspace of r(E) spanned by the vectors v1,..., vi.

Remark 3.5. — This gives the following equivalent definition of Si(E). The Chern class Si(E) consists of all elements g G G such that the graph of the operator n(g) in V © V has a nontrivial intersection with a generic subspace of dimension d — i + 1 in V © V.

In particular, if the representation n : G ^ GL(V) corresponding to a vector bundle E has a nontrivial kernel, then the Si(E) are invariant under left and right multiplications by the elements of the kernel (since this is already true for the preimage ^>-1(A) of any point A G fE(G)). E.g. the Chern classes Si(T G) are invariant under multiplication by the elements of the center of G.

We can now relate the Chern classes Si(E) to the usual Chern classes of a vector bundle over a compact variety.

Denote by XE the closure of fE(G) in the Grassmannian G(d — c, r(E)), and denote by EX the restriction of the tautological quotient vector bundle to XE. We get a vector bundle on a compact variety. The i-th Chern class of EX is the homology class of Ci n XE for a generic Schubert cycle Ci (see Proposition 3.1). By Kleiman's transversality theorem applied to the Grassmannian G(d — c, r(E)) (see Subsection 3.1), a generic Schubert cycle

Cj has a proper intersection with the boundary divisor XE E (G). Hence, there is the following relation between the Chern classes of EX and generic members of the family Sj(E).

Proposition 3.6. — For a generic Sj(E) the homology class of the closure of fE(Sj(E)) in XE coincides with the i-th Chern class of EX.

Thus the Chern classes [Sj(E)] can be described via the usual Chern classes of the bundle EX over the compactification XE.

Let us study the variety XE in more detail. It is a G x G-equivariant compactification of the group fE (G). Indeed, the action of G x G on fE(G) can be extended to the Grassmannian G(d, r(E)) as follows. Identify r(E) with (V © V)/C (see the end of Subsection 3.1). The doubled group G x G acts on V © V by means of the representation n © n, i.e., (gi,g2)(vi, V2) = (giV!,g2V2) for gi,g2 G G, vi,V2 G V. The subspace C С V © V is invariant under this action. Hence, the group G G acts on r(E). This action provides an action of G x G on the Grassmannian G(d — c, r(E)). Clearly, the subvariety XE is invariant under this action.

Example 1 (Demazure embedding). — Let G be a group of adjoint type, and let n be its adjoint representation on the Lie algebra g. The corresponding vector bundle E coincides with the tangent bundle of G. The corresponding map fE : G ^ G(n, g © g) coincides with the embedding constructed by Demazure [9]. The Demazure map takes an element g G G to the Lie subalgebra gg = {(gXg-i,X),X G g} С g © g. Clearly, the Demazure map provides an embedding of G into G(n, g © g).

It is easy to check that the Lie subalgebra gg is the Lie algebra of the stabilizer of an element g G G under the standard action of Gx G. Thus for any A G gg the corresponding vector field vanishes at g, and the Demazure embedding coincides with fE. The compactification XE in this case is isomorphic to the wonderful compactification Xcan of the group G [9]. In particular, it is smooth.

Definition 3.7. — Let G and E be as in Example 1. The restriction of the tautological quotient vector bundle to XE ^ Xcan is called the Demazure bundle and is denoted by Vcan.

If E is the tangent bundle, then Proposition 3.6 implies that the Chern class Sj(E) is the inverse image of the usual i-th Chern class of the Demazure bundle. The Demazure bundle is considered in [5], where it is related to the tangent bundles of regular compactifications of the group G.

Example 2.

a) Let G be GL(V) and let n be its tautological representation on the

space V of dimension d. Then is an embedding of GL(V) into the Grassmannian G(d, 2d). Notice that the dimensions of both varieties are the same. Hence, the compactification XE coincides with G(d, 2d).

b) Take SL(V) instead of GL(V) in the previous example. Its compactification XE is a hypersurface in the Grassmannian G(d, 2d) which can be described as a hyperplane section of the Grassmannian in the Plucker embedding. Consider the Plucker embedding p : G(d, 2d) ^ Р(Л^(У1 ф V2)), where Vi and V2 are two copies of V. Then p(XE) is a special hyperplane section of p(G(d, 2d)). Namely, the decomposition V1 ф V2 yields a decomposition of Л^(У1 ф V2) into a direct sum. This sum contains two one-dimensional components p(V1) and p(V2) (which are considered as lines in Л^ ф V2)). In particular, for any vector in Л^(У1 ф V2) it makes sense to speak of its projections to p(V1) and p(V2). On V1 and V2 there are two special n-forms, preserved by SL(V). These forms give rise to two 1-forms 11 and 12 on p(V1) and p(V2), respectively. Consider the hyperplane H in Л^(У1 ф V2) consisting of all vectors v such that the functionals 11 and 12 take the same values on the projections of v to p(V1) and p(V2), respectively. Then it is easy to check that p(XE) = p(G(d, 2d)) П P(H).

In the next section, I will be concerned with the case when E = T G is the tangent bundle. In this case, the vector bundle EX is closely related to the tangent bundles of regular compactifications of the group G. Let us discuss this case in more detail.

Example 3. — This example is a slightly more general version of Example 1. Let g = g' ф c be the decomposition of the Lie algebra g into the direct sum of the semisimple and the central subalgebras, respectively. Denote by c the dimension of the center c. Let E = T G be the tangent bundle on G. Then maps G to the Grassmannian G(n — c, (g ф g)/c). It is easy to show that the image of the map coincides with the adjoint group of G and the image contains only subspaces that belong to (g' ф g') С (g ф g)/c. Comparing this with Example 1, one can easily see that XE is isomorphic to the wonderful compactification Xcan of the adjoint group of G.

In this case, the bundle EX is the direct sum of the Demazure bundle and the trivial vector bundle of rank c. Indeed, for any subspace Лх G XE ^ Xcan С G(n — c, r(E)) its intersection with the subspace c = {(c, —c), c G c} С r(E) is trivial. Hence, the quotient space r(E)/Лх coincides with the direct sum ((g' ф g')/Лж) ф c-.

Proof of Lemma 3.3. The proof of Lemma 3.3 relies on the following fact. Let Y1 and Y2 be two subvarieties of codimension i in the group G.

Using Kleiman's transversality theorem and continuity arguments, it is easy to show that Y1 and Y2 represent the same class in the ring of conditions C* (G) if there exists an equivariant compactification X of the group G such that the closures of Y1, Y2 in X have proper intersections with all G x G-orbits (see [10] for the proof).

In particular, to prove Lemma 3.3 it is enough to produce an equivariant compactification X such that the closure of a generic S (E) has proper intersections with all G x G-orbits in X. I claim that the compactification XE discussed in the previous paragraph (see Proposition 3.6) satisfies this condition.

Indeed, the closure of any S(E) in XE coincides with the intersection of XE with the Schubert cycle C corresponding to a partial flag in Г(Е). By Kleiman's transversality theorem applied to the Grassmannian G(d — c, Г(Е)) (see Subsection 3.1), a partial flag can be chosen in such a way that the corresponding Schubert cycle has proper intersections with all G x Gorbits in XE. All partial flags with such property form an open dense subset in the space of all partial flags. Hence, for generic flags the corresponding subvarieties S represent the same class in the ring of conditions.

In the sequel, Sj(E) will denote any subvariety of the family Sj(E) whose class in the ring of conditions coincides with the Chern class [Sj(E)].

Remark. — Recall that the ring of conditions C*(G) can be identified with the direct limit of cohomology rings of equivariant compactifications of G (see Theorem 2.3). It follows that under this identification the Chern class [Sj(E)] G C*(G) corresponds to an element in the cohomology ring of the compactification XE. In particular for an adjoint group G, the Chern class [Sj (T X)] of the tangent bundle corresponds to some cohomology class of the wonderful compactification of G.

Properties of the Chern classes of reductive groups. The next lemma computes the dimensions of the Chern classes. It also shows that if G acts on V without an open dense orbit, then the higher Chern classes automatically vanish.

For any representation n : G ^ GL(V), there exists an open dense G-invariant subset in V such that the stabilizers of any two elements from this subset are conjugate subgroups of G (see [24]). In particular, all elements from this subset have isomorphic G-orbits. Such orbits are called principal. Denote by d(n) the dimension of a principal orbit of G in V. If G has an open dense orbit in V, then d(n) = d. In my main example, when n is the adjoint representation, d(n) = n — k.

Lemma 3.8. — If i > d(n), then Sj(E) is empty, and if i < d(n) then the dimension of Sj(E) is equal to n — i.

Proof. — Recall that Sj(E) is the inverse image of Cj under the map fE : G ^ G(d — c, r(E)). Here Cj is the i-th Schubert cycle corresponding to a generic partial flag in r(E). The codimension of Cj in the Grass-mannian G(d — c, r(E)) is equal to i. Hence, by Kleiman's transversality theorem applied to G(d — c, r(E)) (see Subsection 3.1), the intersection Cj n fE (G) is either empty or proper and has codimension i in fE (G). Then Sj(E) = ^is1(Cj n fE (G)) is either empty or has codimension i in G, because all fibers of the map f E are isomorphic to each other (each of them is isomorphic to the kernel of n). It remains to find out all i for which Sj(E) is empty.

By Remark 3.5, the Chern class Sj(E) consists of all elements g G G such that the graph rg = {(v, n(g)v), v G V} C V © V of n(g) has a nontrivial intersection with a generic subspace Ad-j+1 of dimension d — i + 1 in V © V. For all g G Sj(E) \ Sj+1(E) the intersection rg n Ad-j+1 has dimension 1. Indeed, if dim(rg n Ad-j+1) > 2, then dim(rg n Ad-j) > 1 (since the subspace Ad-j C Ad-j+1 has codimension one in Ad-j+1), and g belongs to Sj+1(E). Hence, there is a well-defined map

p : Sj(E) \ Sj+1(E) ^ P(D n Ad-j+1); p : g ^ P(rg n Ad-j+1).

Here D C V © V is the union of all graphs rg for g G G. In particular, the Chern class Sj(E) is nonempty if and only if P(D n Ad-j+1) is nonempty.

We now estimate the dimension of D n Ad-j+1. Since D is not a variety, it is more convenient to take its Zariski closure D. The subvariety D is the closure of the image of the following morphism:

F : G x V ^ V x V; F :(g,v) ^ (v,n(g)v).

The source space G x V is an irreducible variety of dimension n+d, and the general fibers of F are isomorphic to the principal stabilizers, of dimension n — d(n). Hence dim D = d + d(n), that is, D has codimension d — d(n).

Next, observe that D is a constructible set, invariant under scalar multiplication. Hence it contains a dense open subset (also invariant under scalar multiplication) of the irreducible variety D. Thus a general vector space Ad-j+1 satisfies dim(D n Ad-j+1) = d(n) — i + 1, if i < d(n), and D n Ad-j+1 is dense in this intersection. In particular, if i = d(n), then D n Ad-j+1 consists of several lines whose number is equal to the degree of D. If i > d(n), then D n Ad-j+1 contains only the origin. It follows that if i > d(n), then Sj(E) is empty.

□

This proof also implies the following corollary. Denote by H C G the stabilizer of an element in a principal orbit of G in V. The subgroup H is defined up to conjugation so its class in the ring of conditions is well-defined.

Corollary 3.9. — An open dense subset of the subvariety Sj(E) admits almost a fibration whose fibers are translates of H. Here almost means that the intersection of different fibers always lies in Sj+1(E) C Sj(E). In particular, the last Chern class Sd(n)(E) admits a true fibration and coincides with the disjoint union of several translates of H. Their number equals to the degree of a generic principal orbit of G in V.

The last statement follows from the fact that the degree of D in V © V (see the proof of Lemma 3.8) is equal to the degree of a generic principal orbit of G in V.

In particular, let E be the tangent bundle. Then the stabilizer of a generic element in g is a maximal torus in G. Hence, the last Chern class Sn-k(T G) is the union of several translates of a maximal torus. The number of translates is the cardinality of the Weyl group (the degree of a general orbit in the adjoint representation).

3.3. The first and the last Chern classes

Throughout the rest of the paper, I will only consider the Chern classes Sj = Sj(T G) of the tangent bundle unless otherwise stated. Theorem 1.1 expresses the Euler characteristic of a complete intersection via the intersection indices of the Chern classes Sj with generic hyperplane sections. The question is how to compute these indices. If [Sj] is a linear combination of complete intersections of generic hyperplane sections corresponding to some representations of G, then the answer to this question is given by the Brion-Kazarnovskii formula. A hyperplane section corresponding to the representation n is called generic if its closure in the compactification Xn has proper intersections with all G x G-orbits in Xn.

In this subsection, I describe S1 as a generic hyperplane section. The description follows from a result of Rittatore [25]. One can also compute the intersection indices with the last Chern class Sn-k, because Sn-k is the union of translates of a maximal torus (see Corollary 3.9). However, it seems that in general the Chern class Sj, for i = 1, is not a sum of complete intersections. E.g. I can show that for G = SL3(C) the Chern class [S3] does not lie in the subring of C*(G) generated by the classes of hypersurfaces.

Description of Si. The result of Rittatore for the first Chern class of regular compactifications (see [25], Proposition 4) implies that the class [S1] in the ring of conditions can be represented by the doubled sum of the closures of all codimension one Bruhat cells in G. Below I will deduce this description directly from the definition of S1.

It is easy to show that S1 С G is given by the equation det(Ad(g) — A) = 0 for a generic A G End(g). Indeed, the first Chern class S1(E) of any equivariant vector bundle E over G consists of elements g G G such that the graph of the operator n(g) in V ф V has a nontrivial intersection with a generic subspace of dimension n in V ф V (see Remark 3.5). As a generic subspace, one can take the graph of a generic operator A on V. Then the graphs of operators n(g) and A have a nonzero intersection if and only if the kernel of the operator n(g) — A is nonzero.

The function det(Ad(g) — A) is a linear combination of matrix coefficients corresponding to all exterior powers of the adjoint representation. Hence, the equation of S1 is the equation of a hyperplane section corresponding to the sum of all exterior powers of the adjoint representation. Denote this representation by a. It is easy to check that the weight polytope coincides with the weight polytope of the irreducible representation в with the highest weight 2p (here p is the half sum of all positive roots, or equivalently the sum of all fundamental weights). It remains to prove that S1 is generic, which means that the closure of S1 in intersects all G x G-orbits along subvarieties of codimension one. The proof of Lemma 3.3 implies that this is true for the wonderful compactification, and the normalization of is the wonderful compactification by Theorem 2.1 (since Pg = ).

It is now easy to show that the doubled sum of the closures of all codi-mension one Bruhat cells in G is equivalent to S1. This is because the closures of codimension one Bruhat cells are generic hyperplane sections corresponding to the irreducible representations with fundamental highest weights.

Description of Sn-k. By Corollary 3.9 the last Chern class Sn-k is the disjoint union of translates of a maximal torus. Their number is equal to the degree of a generic adjoint orbit in g. The latter is equal to the order of the Weyl group W. Denote by [T] the class of a maximal torus in the ring of conditions C*(G). Then the following identity holds in C*(G):

[Sn-k] = |W |[T].

The degree of n(T) can be computed using the formula of D.Bernstein, Khovanskii and Koushnirenko [18].

3.4. Examples

G = SL2(C). Consider the tautological embedding of G, namely, G = {(a, b, c, d) G C4 : ad—bc = 1}. Since the dimension of G is 3 and the rank is 1, by Lemma 3.8 we get that there are only two nontrivial Chern classes: S1 and S2. Let us apply the results of the preceding subsection to find them. The first Chern class S1 is a generic hyperplane section corresponding to the second symmetric power of the tautological representation, i.e., to the representation 0 : SL2(C) ^ SO3(C). In other words, it is the intersection of SL2 (C) with a generic quadric in C4. The second Chern class S2 (which is also the last one in this case) is the union of two translates of a maximal torus (or the intersection of S1 with a hyperplane in C4).

Let n be a faithful representation of SL2(C). It is a direct sum of irreducible representations. Any irreducible representation of SL2(C) is iso-morphic to the i-th symmetric power of the tautological representation for some i. Its weight polytope is the line segment [—i, i]. Hence the weight polytope of n is the line segment [—n, n] where n is the greatest exponent of symmetric powers occurring in n. Then the matrix coefficients of n are polynomials in a, b, c, d of degree n. In this case, it is easy to compute the degrees of subvarieties n(G), n(S1) and n(S2) by the Bezout theorem. Then deg n(G) = 2n3, deg n(S1) = 4n2, deg n(S2) = 4n. Also, if one takes another faithful representation a with the weight polytope [—m, m], then the intersection index of S1 with two generic hyperplane sections corresponding to n and a, equals to 4mn.

Since by Theorem 1.1 the Euler characteristic x(n) of a generic hyperplane section is equal to deg n(G) — deg n(S1) + deg n(S2), we get

x(n) = 2n3 — 4n2 + 4n.

This answer was first obtained by Kaveh who used different methods [16].

If n is not faithful, i.e., n(SL2(C)) = SO3(C), consider n as a representation of SO3(C). Then x(n) is two times smaller and equals to n3 — 2n2 + 2n.

Apply Theorem 1.1 to a curve C that is the complete intersection of two generic hyperplane sections corresponding to the representations n and a. Then

x(C) = Hn • Hff • H0 — Hn • Hff • (Hn + Hff) = —2mn(m + n — 2).

G = (C* )n is a complex torus. In this case, all left-invariant vector fields are also right-invariant since the group is commutative. Hence, they are linearly independent at any point of G = (C*)n as long as their values at the identity are linearly independent. It follows that all subvarieties Sj

are empty, and all the Chern classes vanish. Then Theorem 1 coincides with a theorem of D.Bernstein and Khovanskii [18].

4. Chern classes of regular compactifications and proof of Theorem 1.1

4.1. Preliminaries

Chern classes of the tangent bundle. In this paragraph, I explain a method from [11], which in some cases allows to find the Chern classes of smooth varieties.

Let X be a smooth complex variety of dimension n, and let D C X be a divisor. Suppose that D is the union of l smooth irreducible hypersurfaces D1,..., D; with normal crossings. One can relate the tangent bundle T X of X to the logarithmic tangent bundle, consisting of those vector fields that preserve the divisor D.

Let LX (D1),..., LX (D;) be the line bundles over X associated with the hypersurfaces D1,..., D;, respectively. i.e., the first Chern class of the bundle LX (Dj) is the homology class of Dj. One can also associate with D the logarithmic tangent bundle VX(D). It is a holomorphic vector bundle over X of rank n that is uniquely defined by the following property. The holomorphic sections of VX (D) over an open subset U C X consist of all holomorphic vector fields v(x) on U such that v(x) restricted to U n Dj is tangent to the hypersurface Dj for any i. The precise definition is as follows. Cover X by local charts. If a chart intersects the divisors Dj1,..., Djfc choose local coordinates x1,... ,xn such that the equation of Dj. in these coordinates is xj = 0. Then VX is given by the collection of trivial vector bundles spanned by the vector fields x^-^-,..., xk ,-r,——,..., over each chart with the natural transition operators.

For a vector bundle E, denote by O(E) the sheaf of its holomorphic sections.

Proposition 4.1 ([11]). — There is an exact sequence of coherent sheaves

i

0 ^ O(Vx(D)) ^ O(TX) ^ 0 O(Lx(Dj)) ^ 0.

j=1

In particular, the tangent bundle T X has the same Chern classes as the direct sum of the bundle VX(D) with LX(D1),..., LX(D;).

Proposition 4.1 gives the answer for the Chern classes of X, when the Chern classes of VX (D) are known. In particular, this is the case when X is a smooth toric variety, and D = X \ (C*)n is the divisor at infinity. In this case, the vector bundle VX (D) is trivial, and the Chern classes of T X can be found explicitly. This was done by Ehlers [11]. A more general class of examples is given by regular compactifications of reductive groups (see the next paragraph for the definition) and, more generally, of arbitrary spherical homogeneous spaces (see Section 5). In this case, the vector bundle VX (D) is no longer trivial but still has a nice description, which is due to Brion [5]. I recall his result in Subsection 4.2 and use it to prove Theorem 1.1.

Regular compactifications. In this paragraph, I will define the notion of regular compactifications of reductive groups following [6]. Let X be a smooth G x G-equivariant compactification of a connected reductive group G of dimension n. Denote by Oi,..., O; the orbits of codimension one in X. Then the complement X \ G to the open orbit is the union of the closures O1,..., O; of codimension one orbits.

Definition 4.2. — A smooth G x G-equivariant compactification X is called regular if the following three conditions are satisfied.

(1)The hypersurfaces O1,..., O; are smooth and intersect each other transversally.

(2)The closure of any G x G-orbit in X \ G coincides with the intersection of those hypersurfaces O1,..., O; that contain it.

(3) For any point x G X and its G x G-orbit Ox С X, the stabilizer (G x G)x С G x G acts with a dense orbit on the normal space TxX/TxOx to the orbit.

This definition was introduced by E. Bifet, De Concini and Procesi in a more general setting ([2], see also Section 5).

If G is a complex torus, then the regularity of X is just equivalent to the smoothness. However, for other reductive groups, there exist compact-ifications that are smooth but not regular. In particular, it follows from Proposition 4.3 below that the compactification Xn associated with a representation n : G ^ GL(V) (see Section 2) is regular if and only it is smooth and none of the vertices of the weight polytope of n lies on the walls of the Weyl chambers.

Regular compactifications of reductive groups generalize smooth toric varieties and retain many nice properties of the latter. E.g. any regular compactification X can be covered by affine charts Xa ~ Cn in such a

way that only k hypersurfaces O^.., Oik intersect Xa, and intersections Ojj П Xa,..., Oik П Xa are k coordinate hyperplanes in Xa [9, 6]. Here k denotes the rank of G. In particular, all G x G-orbits in X have codimension at most k, and all closed orbits have codimension k.

If G is of adjoint type, then it has the wonderful compactification Xcan, which is regular. This example is crucial for the study of the other regular compactifications.

For arbitrary reductive group G, denote by Xcan the wonderful com-pactification of the adjoint group of G. There is the following criterion of regularity.

Proposition 4.3 ([6]). — Let X be a smooth G x G-equivariant com-pactißcation of G. Then the condition that X is regular is equivalent to the existence of a G x G-equivariant map from X to Xcan.

E.g. if G is a complex torus, then the latter condition is always satisfied because Xcan is a point in this case.

Thus the set of regular compactifications of G consists of all smooth G x G-equivariant compactifications lying over Xcan. In particular, for reductive groups of adjoint type the wonderful compactification is the minimal regular compactification.

4.2. Demazure bundle and the Chern classes of regular compactifications

In this subsection, I state a formula for the Chern classes of regular compactifications of reductive groups. It follows from a more general result proved for arbitrary toroidal spherical varieties by Brion [5]. This formula gives a description of the Chern classes in terms of two different collections of subvarieties. The first collection is given by the Chern classes of G, which are independent of a compactification, and the second is given by the closures of codimension one orbits, which are easy to deal with (in particular, all their intersection indices with other divisors can be computed via the Brion-Kazarnovskii theorem).

Let X be a regular compactification of G, and let Oi,..., O; be the closures of the G x G-orbits of codimension one in X. Then the tangent bundle T X of X can be described using the Demazure vector bundle Vcan over the wonderful compactification Xcan (see Example 1 from Subsection 3.2) and the line bundles corresponding to the hypersurfaces Oj.

Let L(O1),..., L(O;) be the line bundles over X associated with the hypersurfaces O1,..., O;, respectively. Let p : X ^ Xcan be the canonical map from Proposition 4.3, and let p*(Vcan) be the pull-back of the De-mazure vector bundle to X. It turns out that p*(Vcan) coincides up to a trivial summand with the logarithmic tangent bundle corresponding to the boundary divisor X \ G.

Theorem 4.4 ([5]). — The tangent bundle TX has the same Chern classes as the direct sum of the pull-back p*(Vcan) with the line bundles

L(01),...,L("O).

In the case when G is a complex torus, Theorem 4.4 was proved by Ehlers [11]. For arbitrary reductive groups, Theorem 4.4 follows from a more general result by Brion ([5], 1.6 Corollary 1).

This theorem implies the following formula for the Chern classes c1(X),..., cn(X) of the tangent bundle of X. Let Sj = S,(TG) C G for i = 1,..., n — k be the Chern classes of the tangent bundle of G defined in the previous section (see Definition 3.4). Denote by Sj the closure of Sj in X. Note that Sj has proper intersections with all G x G-orbits in X (since this is already true for the wonderful compactification Xcan, and X lies over Xcan).

Corollary 4.5. — The total Chern class c(X) = 1+c1(X) + .. .+cn(X) coincides with the following product:

i

c(X) = (1+ S1 + ... + Sn_k) n(1+ Oi).

i=1

The product in this formula is the intersection product in the (co)homology ring of X.

Below I sketch the proof of Theorem 4.4 following mostly the proofs by Ehlers and Brion. The goal is to explain the main idea of their proofs, which is very transparent, and motivate the definition of the Chern classes Sj. In the torus case, this idea can be extended to a complete elementary proof. For more details see [11] and [5].

Take n generic vector fields v1,..., vn coming from the action of G x G. It is not hard to show that v1,...,vn are generic in the space of all C^-smooth vector fields on X (it is enough to prove it for each affine chart on X). Hence, their degeneracy loci give Chern classes of X. Note that these fields are not only C^-smooth but also algebraic so their degeneracy loci are algebraic subvarieties in X.

The picture is especially simple in the torus case, because in this case vi,..., v„_j+i are linearly dependent precisely on all orbits of codimension greater than or equal to i (since they all belong to the tangent bundle of the orbit) and independent on the other orbits. Hence, the i-th Chern class of X consists of all orbits of codimension at least i.

In the reductive case, the situation is more complicated because the degeneracy loci of v1,..., vn have nontrivial intersections with the open orbit G С X. These intersections are exactly the Chern classes Si,..., Sn-k of G. So it seems more convenient to use the method described in Subsection 4.1 (see Proposition 4.1). Namely, consider the logarithmic tangent bundle VX = VX (X \ G) corresponding to the boundary divisor X \ G = O1 U ... UO|. Recall that c denotes the dimension of the center of G.

Proposition 4.6. — The vector bundle VX is isomorphic to the direct sum of the pull-back p*(Vcan) with the trivial vector bundle Ec of rank c.

Proof. — The vector fields coming from the action of G x G on X are global sections of the bundle VX, since they are tangent to all G x G-orbits in X. It follows easily from condition (3) in the definition of regular compactifications that these global sections span the fiber of VX at any point of X. Hence, the map : G ^ G(n — c, (g ф g)/c) considered in Example 3 extends to a map p : X ^ G(n — c, (g фg)/c). The rest follows from Example 3 □

Remark 4.7. — There is also another construction of the map p : X ^ Xcan by Brion (see [4]).

4.3. Applications

In this subsection, I prove Theorem 1.1 using the formula for the Chern classes of regular compactifications (Corollary 4.5). Then I apply it to compute the Euler characteristic and the genus of a curve in G.

Proof of Theorem 1.1. First, define the notion of generic collection of hyperplane sections used in the formulation of Theorem 1.1. A collection of m hyperplane sections H1,..., Hm corresponding to representations n1,..., nm, respectively, is called generic, if there exists a regular compactification X of G such that the closure H of any hyperplane section H is smooth, and all possible intersections of H1,..., Hm with the closures of G x G-orbits in X are transverse. E.g. one can take the compactification Xn corresponding to the tensor product n of the representations n1,..., nm,

where is any irreducible representation with a strictly dominant highest weight. Then it is not hard to show that the set of all generic collections (with respect to the compactification Xn) is an open dense subset in the space of all collections.

So the closure Y = C of C = Hi n ... fl Hm in X is the transverse intersection of smooth hypersurfaces. In particular, Y is smooth, and its normal bundle NY in X is the direct sum of m line bundles corresponding to the hypersurfaces Hj. The analogous statement is true for any subvariety of the form Y f 0/, where I = {¿i,..., ip} is a subset of {1,..., 1} and 0/ = 0^ f • • • f 0ip. Let us find the Euler characteristic of Y n 0/ using the classical adjunction formula. Denote by J = {1,..., 1} \ I the complement to the subset I. We get that x(Y f 0/) is the term of degree n in the decomposition of the following intersection product in X:

m

(1 + Si + • • • + S„-fc) n Hs (1 + Hs)-i n Oi n (1 + 0j). (*)

s=i ie/ jeJ

On the other hand, since the Euler characteristic is additive, and C = Y \ (0i U • • • U 0;), one can express the Euler characteristic x(C) in terms of the Euler characteristics x(Y f 0/) over all subsets I C {1,..., 1}:

X(C)= £ ( —1)|/|x(Y f0/). (**)

/c{i,...,;}

Combining formulas (*) and (**), we get the formula of Theorem 1.1. Indeed, we have that x(C) is the term of degree n in the decomposition of the following intersection product in X:

m /

(1+Si+- • •+£„-*HS(1+HS)-M £ (-i)|J|nOi n(1+Oj

s=1 Vuj={i,...,i} ie/ je J

The sum in the parentheses is equal to 1, since for any commuting variables xi, x2, ..., x; we have the identity:

i

1 = n((1+xj) - xi) = E (-1)|/|nxj n(1+xj).

j=i /uj={i,...,;} ie/ jeJ

Computation for a curve. Apply Theorem 1.1 and the formula for the first Chern class Si to a curve in G. We get that if C = Hi f • • • f Hn-i is a complete intersection of n - 1 generic hyperplane sections, then

n-i

x(C) = (Si - Hi-----Hn-i) n Hi.

Since S1 is also a generic hyperplane section, the computation of x(C) reduces to the computation of the intersection indices of hyperplane sections.

Recall the Brion-Kazarnovskii formula for such intersection indices. Denote by R+ the set of all positive roots of G. Recall that p denotes the half of the sum of all positive roots of G and LT denotes the character lattice of a maximal torus T C G. Since G is reductive, we can assume that g is embedded into gl(W) so that the trace form tr(A, B) = tr(AB) for A, B G gl(W) is nondegenerate on g. Then the inner product (•, •) on LT <R used in Theorem 4.8 is given by the trace form on g. Choose a Weyl chamber D C LT < R.

Theorem 4.8 ([4, 17]). — If Hn is a hyperplane section corresponding to a representation n with the weight polytope Pn C LT < R , then the self-intersection index of Hn in the ring of conditions is equal to

The measure dx on LT < R is normalized so that the covolume of LT is 1.

This theorem in particular implies that the self-intersection index Hn depends not on a representation but only on its weight polytope. Note also that the integrand is invariant under the action of the Weyl group.

Let H1,..., Hn be n generic hyperplane sections corresponding to different representations n1,..., nn. To compute their intersection index one needs to take the polarization of H^. Namely, the formula of Theorem 4.8 gives a polynomial function D(P) of degree n on the space of all virtual polytopes P C LT < R (the addition in this space is the Minkowski sum). The polarization Dpol is the unique symmetric n-linear form on this space such that Dpol (Pn ,...,Pn) = D(Pn). Then Dpoi(Pni,..., Pnn) is the intersection index H1 • ... • Hn. For instance, it can be found by applying the differential operator n dtf at to the function F(t1,..., tn) = D(t1Pni + • • • + tnPn„). E.g. if Pn2 = • • • = Pnn, then the computation of Dp0i(Pni,... ,Pn„) = ndt |t=0 D(tPni + P^2) reduces to the integration over the facets of Pn2.

Thus we get the following answer for x(C). For simplicity, the answer is given in the case when n1 = • • • = n„_1 = n. Then its polarization provides the answer in the general case. Denote by P2p the weight polytope of the irreducible representation of G with the highest weight 2p.

Corollary 4.9. — Let C be a curve obtained as the transverse intersection of a generic collection of n — 1 hyperplane sections corresponding

to the representation n. Then

x(C) = Dpoi (P2p, Pn, . . . ,Pn ) - (n - 1)D(Pn )

A similar answer can be obtained for the genus of C since it is equal to the genus of the compactified curve C C Xn. Hence, g(C) = g(C) = 1-x(C)/2. To compute the Euler characteristic of C we need to sum up x(C) and the number of points in C \ C. The latter is the intersection index of Hn-i with the codimension one orbits in Xn and can be again computed by the Brion-Kazarnovskii formula. Choose 1 facets Fi,..., of Pn so that they parameterize the codimension one orbits in Xn. This means that each orbit of the Weyl group acting on the facets of Pn contains exactly one F (see Theorem 2.1).

Corollary 4.10. — The genus g(C) of C is given by the following formula:

g(C)=1 - 2 (x(c)+(n -1)! E / n (xSdx)

The measure dx on a facet Fi is normalized as follows. Let H C L < R be the hyperplane containing Fi. Then the covolume of the sublattice L f H in H is equal to 1.

In the above answer, one can rewrite the polarization Dpo;(P2p,Pn,..., Pn) in terms of the integrals over the facets of Pn. E.g. in the case when n is the irreducible representation with a strictly dominant highest weight A, the answer takes the following form. Let 2p = ^k=i aiai be the decomposition of 2p in the basis of simple roots ai,..., ak.

x(C> = n'(n E1- / 11+ ^(n - ') I n ^dX)

i=i F,nD aeR+ PnnD aeR+

g(C) = 1 - n £ [(ai + 1) / n (xS dX

2 Vn i=i F-nv aeR+ (P'a)

- <» - «y iit

nv «eI+

5. The case of regular spherical varieties

The results of this paper concerning the Chern classes of the tangent bundle can be generalized straightforwardly to the case of arbitrary spherical homogeneous space. In this section, I briefly outline how this can be done.

Let G be a connected complex reductive group of dimension r, and let H be a closed algebraic subgroup of G . Suppose that the homogeneous space G/H is spherical, i.e., the action of G on the homogeneous space G/H by left multiplication is spherical. In the preceding sections, we considered a particular case of such homogeneous spaces, namely, the space (G x G)/G ~ G.

The definition of the Chern classes Sj of the tangent bundle T(G/H) can be repeated verbatim for G/H. Denote the dimension of G/H by n. There is a space of vector fields on G/H coming from the action of G. Take n arbitrary vector fields v1,..., vn of this type. Define the subvariety Sj С G/H as the set of all points x G G/H such that the vectors v1(x),..., v„_j+1(x) are linearly dependent.

Denote by (j С g the Lie algebra of the subgroup H. Again, there is the Demazure map p : G/H ^ G(r — n, g), which takes g G G/H to the Lie subalgebra gjg-1. Denote by Xcan the closure of p(X) in the Grassmannian G(r — n, g) . This is a compactification of a spherical homogeneous space G/N(j), where N(j) С G is the normalizer of j. Brion conjectured that if H coincides with N(H), then the compactification Xcan is smooth, and hence, regular [5]. F. Knop proved that under the same assumption the normalization of Xcan is smooth [22]. The conjecture has been proved for semisimple Lie algebras of type A by D. Luna [23], and in type D by P. Bravi and G. Pezzini [3]. In the general case, one can still define the Demazure bundle over Xcan as the restriction of the tautological quotient vector bundle over G(r — n, g).

Since we have not used the regularity of Xcan in the proof of Lemma 3.3 the same arguments imply two facts. First, for a generic choice of vector fields v1,..., vn, the resulting subvariety Sj belongs to a fixed class [Sj] in the ring of conditions. Second, for any compactification X of G/H lying over Xcan the closure of a generic Sj in X intersects properly any orbit of X. Repeating the proof of Lemma 3.8 one can also show that Sj is empty unless i ^ n — k. Here k is the difference between the ranks of G and of H. Therefore, we have n — k well-defined classes [S1],..., [Sn-k] in the ring of conditions C*(G/H). Recently, M. Brion and I. Kausz proved that

the G-equivariant Chern classes of the Demazure bundle also vanish for i > n — k [7].

To extend Theorem 1.1 to an arbitrary spherical homogeneous space one can use the same description of the Chern classes of its regular compactifi-cations. The definition of regular compactifications repeats Definition 4.2.

Theorem 5.1. — Let X be a regular compactification of G/H. Then the total Chern class of X equals to

(1 + Si + ••• + S n-k ) + Oi)

n-

i=1

This description also follows from Subsection 4.1. The proof uses the methods mentioned in Subsection 4.2. In fact, regular compactifications of spherical homogenous spaces arise naturally when one try to apply these methods to a wider class of varieties with a group action. Namely, suppose that a connected complex affine group G of dimension r acts on a compact smooth irreducible complex variety X with a finite number of orbits. Then there is a unique open orbit in X isomorphic to G/H for some subgroup H C G, so X can be regarded as a compactification of G/H. Denote by O1,..., O; the orbits of codimension one in X. Then one can describe the tangent bundle of X exactly by the methods mentioned in Subsection 4.2 if the following conditions hold. First, the hypersurfaces O1,..., O; are smooth and intersect each other transversally (this allows to apply Ehlers' method to the divisor X \ (G/H) = O1 U • • • U O;). Second, the vector bundle VX (defined as in Subsection 4.2) is generated by its global sections v1,..., vr, where v1,..., vr are infinitesimal generators of the action of G on X (this allows to give a uniform description of VX for all compactifications of G/H satisfying these conditions). It is not hard to check that these two conditions are equivalent to the definition of regular compactifications.

It turns out that a homogeneous space G/H admits a regular compactification if and only if G/H is spherical [1]. Regular compactifications of arbitrary spherical homogeneous spaces are exactly their smooth toroidal compactifications [1]. A compactification X of the spherical homogeneous space G/H is called toroidal if for any codimension one orbit of a Borel subgroup of G acting on G/H, its closure in X does not contain any G-orbit in X.

The proof of Theorem 1.1 goes without any change for complete intersections in arbitrary spherical homogeneous space G/H. Let H1,..., Hm

be smooth hypersurfaces in some regular compactification of G/H. Suppose that all possible intersections of Hi with the closures of G-orbits are transverse.

Theorem 5.2. — Let Hi,..., Hm C G/H be the hypersurfaces Hi f (G/H), and let C = Hi f • • • f Hm be their intersection. Then the Euler characteristic of C equals to the term of degree n in the decomposition of

m

(1 + Si + • • • + Sn-k) nHi(1 + Hi)-i.

i=i

The products are taken in the ring of conditions C* (G/H).

For instance, if G/H is compact, then the Si become the usual Chern classes and the above formula coincides with the classical adjunction formula. However, if G/H is noncompact then the Chern classes in the usual sense (as degeneracy loci of generic vector fields on G/H) do not usually yield the adjunction formula (although they do for G = (C*)n). Indeed, when the homogeneous space is a noncommutative reductive group, all usual Chern classes are trivial but as we have seen x(H) = (-1)nHn even for one smooth hypersurface H. Theorem 5.2 shows that the adjunction formula still holds for noncompact spherical homogeneous spaces, if one replaces the usual Chern classes with the refined Chern classes Si that are defined as the degeneracy loci of the vector fields coming from the action of G.

BIBLIOGRAPHY

[1] F. Bien & M. Brion, "Automorphisms and local rigidity of regular varieties", Compositio Math. 104 (1996), no. 1, p. 1-26.

[2] E. Bifet, C. de Concini & C. Procesi, "Cohomology of regular embeddings", Adv. in Math. 82 (1990), no. 1, p. 1-34.

[3] P. Bravi & G. Pezzini, "Wonderful varieties of type D", arXiv.org/math.AG/ 0410472.

[4] M. Brion, "Groupe de Picard et nombres caractéristiques des variétés sphériques", Duke Math J. 58 (1989), no. 2, p. 397-424.

[5 ]--, "Vers une généralisation des espaces symétriques", J. Algebra 134 (1990),

no. 1, p. 115-143.

[6 ]--, "The behaviour at infinity of the Bruhat decomposition", Comment. Math.

Helv. 73 (1998), no. 1, p. 137-174.

[7] M. Brion & I. Kausz, "Vanishing of top equivariant Chern classes of regular em-beddings", preprint arxiv.org/math.AG/0503196.

[8] C. de Concini, "Equivariant embeddings of homogeneous spaces", in Proceedings of the International Congress of Mathematicians (Berkeley, California, USA) (Providence, RI), vol. 1,2, Amer. Math. Soc., 1986, p. 369-377.

[9] C. de Concini & C. Procesi, "Complete symmetric varieties I", in Invariant theory (Montecatini, 1982) (Berlin), Lect. Notes in Math., vol. 996, Springer, 1983, p. 1-44.

[10 ]--, "Complete symmetric varieties II Intersection theory", in Algebraic groups

and related topics (Kyoto/Nagoya, 1983) (Amsterdam), Adv. Stud. Pure Math., vol. 6, North-Holland, 1985, p. 481-513.

[11] F. Ehlers, "Eine Klasse komplexer Mannigfaltigkeiten und die Auflösung einiger isolierter Singularitäten", Math. Ann. 218 (1975), no. 2, p. 127-157.

[12] W. Fulton, Intersection theory, Springer, Berlin, 1984.

[13] I. M. Gelfand, M. M. Kapranov & A. V. Zelevinsky, "Generalized Euler integrals and A-hypergeometric functions", Adv. Math. 84 (1990), no. 2, p. 255-271.

[14] P. Griffiths & J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978.

[15] M. Kapranov, "Hypergeometric functions on reductive groups", in Integrable systems and algebraic geometry (Kobe/Kyoto, 1997) (River Edge, NJ), World Sci. Publishing, 1998, p. 236-281.

[16] K. Kaveh, "Morse theory and Euler characteristic of sections of spherical varieties", Transformation Groups 9 (2004), no. 1, p. 47-63.

[17] B. Y. Kazarnovskii, "Newton polyhedra and the Bezout formula for matrix-valued functions of finite-dimensional representations", Funct. Anal. Appl. 21 (1987), no. 4, p. 319-321.

[18] A. G. Khovanskii, "Newton polyhedra, and the genus of complete intersections", Funct. Anal. Appl. 12 (1978), no. 1, p. 38-46.

[19] V. Kiritchenko, "A Gauss-Bonnet theorem, Chern classes and an adjunction formula for reductive groups", PhD Thesis, University of Toronto, Toronto, Ontario, 2004.

[20] S. L. Kleiman, "The transversality of a general translate", Compositio Mathemat-ica 28 (1974), no. 3, p. 287-297.

[21] F. Knop, "The Luna-Vust theory of spherical embeddings", in Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) (Madras), Manoj Prakashan, 1991, p. 225-249.

[22 ]--, "Automorphisms, root systems, and compactifications of homogeneous varieties", J. Amer. Math. Soc. 9 (1996), no. 1, p. 153-174.

[23] D. Luna, "Sur les plongements de Demazure", J. Algebra 258 (2002), no. 1, p. 205215.

[24] R. W. Richardson, "Principal orbit types for algebraic transformation spaces in characteristic zero", Invent. Math. 16 (1972), p. 6-14.

[25] A. Rittatore, "Reductive embeddings are Cohen-Macaulay", Proc. Amer. Math. Soc. 131 (2003), no. 3, p. 675-684.

[26] D. Timashev, "Equivariant compactifications of reductive groups", Sb. Math. 194 (2003), no. 3-4, p. 589-616.

Manuscrit reçu le 21 avril 2004, révisé le 10 octobre 2005, accepté le 11 novembre 2005.

Valentina KIRITCHENKO

State University of New York at Stony Brook

Dept. of Mathematics

[e-mail]

Приложение B.

Статья 2.

Valentina Kiritchenko "On intersection indices of subvarieties in reductive groups"

Moscow Mathematical Journal, Volume 7, Number 3, July-September 2007, Pages 489-505

Разрешение на копирование: Согласно https://www.ams.org/distribution/mmj/ копирование не требует получения разрешения, если копии используются в образовательных и научных целях. При копировании требуется указать источник.

MOSCOW MATHEMATICAL JOURNAL

Volume 7, Number 3, July-September 2007, Pages 489-505

ON INTERSECTION INDICES OF SUBVARIETIES IN REDUCTIVE GROUPS

VALENTINA KIRITCHENKO

To my Teacher Askold Khovanskii

Abstract. In this paper, I give an explicit formula for the intersection indices of the Chern classes (defined earlier by the author) of an arbitrary reductive group with hypersurfaces. This formula has the following applications. First, it allows to compute explicitly the Euler characteristic of complete intersections in reductive groups thus extending the beautiful result by D. Bernstein and Khovanskii, which holds for a complex torus. Second, for any regular compactification of a reductive group, it computes the intersection indices of the Chern classes of the compactification with hypersurfaces. The formula is similar to the Brion-Kazarnovskii formula for the intersection indices of hypersur-faces in reductive groups. The proof uses an algorithm of De Concini and Procesi for computing such intersection indices. In particular, it is shown that this algorithm produces the Brion-Kazarnovskii formula.

2000 Math. Subj. Class. 14L30.

KEY words AND phrases. Reductive groups, Chern classes, Euler characteristic of hyperplane sections.

1. Introduction

Let G be a connected complex reductive group of dimension n, and let n: G ^ GL(V) be a faithful representation of G. A generic hyperplane section Hn corresponding to n is the preimage n-1(H) of the intersection of n(G) with a generic affine hyperplane H C End(V). There is a nice explicit formula for the self-intersection index Hn of Hn in G, and more generally, for the intersection index of n generic hyperplane sections corresponding to different representations (see Theorem 1.1 below) in terms of the weight polytopes of the representations [3], [9]. In this paper, I give a similar formula for the intersection indices of the Chern classes of G (defined in [11]) with generic hyperplane sections (see Theorem 1.2).

The Chern classes of G can be defined using the Chern classes of the logarithmic tangent bundle over a regular compactification of G (see Section 3 for a precise definition). They were introduced in [11] as main ingredients in a formula for the Euler characteristic of a generic hyperplane section and of complete intersections

Received June 11, 2006.

©2007 Independent University of Moscow

of several hyperplane sections. In the case where a reductive group is a complex torus (C*)n, there are beautiful explicit formulas for the Euler characteristic due to D. Bernstein and A. Khovanskii [10]. The result of the present paper combined with [11] provides analogous formulas in the case of an arbitrary reductive group.

Denote by k the rank of G, i. e. the dimension of a maximal torus in G. Only the first (n-k) Chern classes are not trivial [11]. These Chern classes are elements of the ring of conditions of G, which was introduced by C. De Concini and C. Procesi [7] (see also Section 2.4 for a brief reminder). They can be represented by subvarieties Si, ..., Sn-k C G, where Sj has codimension i. All enumerative problems for G, such as the computation of the intersection index S^Hn-1, make sense in the ring of conditions.

First, I recall the usual Brion-Kazarnovskii formula for the intersection indices of hyperplane sections. Choose a maximal torus T C G, and denote by LT its character lattice. Choose also a Weyl chamber D C LT < R. Denote by R+ the set of all positive roots of G and denote by p the half of the sum of all positive roots of G. The inner product (•, •) on LT < R is given by a nondegenerate symmetric bilinear form on the Lie algebra of G that is invariant under the adjoint action of G (such a form exists since G is reductive).

Theorem 1.1 [3], [9]. If Hn is a hyperplane section corresponding to a representation n with the weight polytope Pn C Lt < R, then the self-intersection index Hn of Hn is equal to

n! J II (p^dx

p/nD '

The measure dx on Lt < R is normalized so that the covolume of Lt is 1.

This theorem was first proved by B. Kazarnovskii [9]. Later, M. Brion proved an analogous formula for arbitrary spherical varieties using a different method [3].

The integrand in this formula has the following interpretation. The direct sum LT © LT can be identified with the Picard group of the product G/B x G/B of two flag varieties. Here B is a Borel subgroup of G. Hence, to each lattice point (A1, A2) £ LT © LT one can assign the self-intersection index of the corresponding divisor in G/B x G/B. The resulting function extends to the polynomial function (n - k)! F on (LT © LT) < R, where

Note that the integrand is the restriction of F onto the diagonal {(x, x): x £ LT < R}.

This interpretation leads to another proof of the Brion-Kazarnovskii formula (different from those of Kazarnovskii and Brion). Namely, take any regular com-pactification X of G that lies over the compactification Xn corresponding to the representation n (see Section 2.2). Then reduce the computation of Hn to the computation of the intersection indices of divisors in the closed orbits of X (see Section 4). All closed orbits are isomorphic to the product of two flag varieties. The precise algorithm for doing this was given by De Concini and Procesi [6] in the

2

case, where X is a wonderful compactification of a symmetric space. Then E. Bifet extended this algorithm to all regular compactifications of symmetric spaces [2]. I will show that in the case, where a symmetric space is a reductive group, this algorithm actually produces the Brion-Kazarnovskii formula if one uses the weight polytope of n to keep track of all transformations.

Moreover, the De Concini-Procesi algorithm works not only for divisors. It can also be carried over to the Chern classes of G (which are, in general, not linear combinations of complete intersections). In particular, there is the following explicit formula for the intersection indices of the Chern classes of G with hyperplane sections. Assign to each lattice point (A1? A2) € LT © LT the intersection index of the i-th Chern class of the tangent bundle over G/B x G/B with the divisor D(A1, A2) corresponding to (A1; A2), that is, the number cj(G/B x G/B)Dn-k-i(A1, A2). Extend this function to the polynomial function on (LT © LT) < R. Since the Chern classes of G/B are known the resulting function can be easily computed (see Section 4). The final formula is as follows.

Let D be the differential operator (on functions on (LT © LT) < R) given by the formula

D = n (1+ da)(1+ da),

a£fl+

where da and da are directional derivatives along the vectors (a, 0) and (0, a), respectively. Denote by [D]j the i-th degree term in D.

Theorem 1.2. If Hn is a generic hyperplane section corresponding to a representation n with the weight polytope Pn C Lt < R, then the intersection index SiHn-1 of the i-th Chern class of G with Hn-j is equal to

(n - i)! J [DjjF(x,x)dx.

Pn nD

The measure dx on Lt < R is normalized so that the covolume of Lt is 1.

Of course, this formula also allows one to compute the intersection index Sj x Hni ... Hnn-i for any n — i generic hyperplane sections corresponding to different representations n1, ..., nn-i.

Since, in general, the Chern classes of G are not complete intersections, this formula extends computation of the intersection indices to a bigger part of the ring of conditions of G. Theorem 1.2 also completes some results of [11]. Namely, the Chern classes S1? ..., Sn-k were used there in the following adjunction formula for the topological Euler characteristic of complete intersections of hyperplane sections in G.

Theorem 1.3 [11]. Let H1, .. ., Hm be generic hyperplane sections corresponding to m (possibly different) representations of G. The Euler characteristic of the complete intersection H1 fl- • •O Hm is equal to the term of degree n in the expansion of the following product:

(1 + S1 + • • • + Sn-k) • HHi(1 + Hi)-1.

i=1

Theorem 1.2 in the present paper allows to make this formula explicit, since it allows to compute all terms of the form SjH-f1 ... Hm, where k1 + ... + km = n — i. For example, if a complete intersection is just one hyperplane section Hn, then

x(Hn) = ( — 1)n-i / (n! — (n — 1)![D]i + (n — 2)![D]2 — ... + k![D]n-k )F (x, x) dx.

Pn nD

There is also a formula for the Chern classes c.j(X) of the tangent bundle over any regular compactification X of G in terms of S1, ..., Sn-k (see Corollary 4.4 in [11]). Theorem 1.2 allows to compute explicitly the intersection index of c.j(X) with a complete intersection of complementary dimension in X.