Структура допустимых подкатегорий в производных категориях многообразий тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Пирожков Дмитрий Владимирович

• Пирожков Дмитрий Владимирович
• кандидат наук
• 2024

Оглавление диссертации кандидат наук Пирожков Дмитрий Владимирович

Contents

Introduction

0.1. Semiorthogonal decompositions on total spaces of tautological bundles 0.2. Rouquier dimension of some blow-ups 0.3. Stably semiorthogonally indecomposable varieties References

Appendix A. Article 1:

Pirozhkov D. Semiorthogonal Decompositions on Total Spaces of Tautological Bundles

International Mathematics Research Notices. 2022. №3. P. 2250-2273. Appendix B. Article 2:

Pirozhkov D. Rouquier dimension of some blow-ups European Journal of Mathematics. 2023. Vol. 9, art. 45. Appendix C. Article 3:

Pirozhkov D. Stably semiorthogonally indecomposable varieties Epijournal de Géométrie Algébrique. 2023. Vol. 7. Appendix D. Russian translation of the thesis

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Введение диссертации (часть автореферата) на тему «Структура допустимых подкатегорий в производных категориях многообразий»

Introduction

Algebraic geometry is, roughly speaking, the study of geometric objects defined via algebraic conditions. For example, the circle x2 + y2 = 1 in the plane R2 is an algebraic variety since its defining equation is polynomial. The key feature of polynomial equations is that we can consider the solution set not only among the real numbers, but also among the rational numbers, or complex numbers, or matrices, or in fact any ring whatsoever. This idea was codified under the name of functor of points by the Grothendieck's school of algebraic geometry and, perhaps surprisingly, is very useful even if we want to study only the geometric object formed by, say, complex solutions to a set of polynomials. More generally, associating algebraic invariants to geometric objects is a widely applied and quite powerful tool in many areas of modern mathematics.

In this thesis we study one of invariants associated to algebraic varieties, namely the derived category of coherent sheaves on it. It is a large invariant: any complex of coherent sheaves is an object in that derived category. In particular, a skyscraper sheaf at any point of the variety can be considered as an object in the derived category, so in some sense that category contains the variety, as well as all vector bundles on it. As such, understanding the structure of the derived category of coherent sheaves is very valuable: it captures a lot of information about the variety and provides a global overview of the features of that geometric object.

The study of derived categories of coherent sheaves is an active area of algebraic geometry. Besides the inherent interest in the behavior of this invariant, derived categories of coherent sheaves appear as one of two objects related via homological mirror symmetry (see, e.g., [Kon95]), they are applied in the study of moduli spaces due to the flexibility of Bridgeland's stability conditions (see, e.g., [Bri07]), and so on. In this work we analyse the derived categories using the notion of semi-orthogonal decomposition, which is a certain way to describe the category as a glueing of several smaller components, called admissible subcategories. Before we go on, let us describe the main objects of study a bit more formally.

Let X be an algebraic variety over a field k. It has an associated abelian category Coh(X) of coherent sheaves on X. We can take the derived category D(Coh(X)) of that abelian category. The key object for this object is the following:

1. Definition. The bounded derived category of coherent sheaves Dboh(X) is the full subcategory of D(Coh(X)) consisting of complexes of sheaves with only finitely many nonzero cohomology sheaves.

Remark. For technical reasons it is better to use another definition of Dcboh(X): as a full subcategory in the derived category of the abelian category of quasi-coherent sheaves, consisting of complexes with only finitely many nonzero cohomology sheaves, for which all cohomol-ogy sheaves are coherent. Since for us X is always a Noetherian scheme, this definition is equivalent to the one above [Huy06, Prop. 3.5].

The category Dcboh(X), which we will usually refer to as the derived category of the variety X, is a very large invariant of the variety X. Many more comprehensible invariants, such as algebraic K-theory or Hochschild (co)homology, can be recovered from the derived category of X. Despite the fact that the derived category of a variety is usually too large to be "computed" in a satisfactory sense, it can be productively studied, for example, by examining

its connection to the derived categories of other varieties. Examples of such results and descriptions of the methods employed can be found in the survey [B002].

With regards to the history of this field we will mention only two classical articles that have had a significant impact on its further development. In the 1978 article [Bei78], A. Beilinson studied the derived category Dcboh(Pn) of projective space P" and described it in terms of linear-algebraic objects. Using the terminology that didn't exist at that time, it can be said that Beilinson constructed an exceptional collection for P". His article served as an important step in the future study of exceptional objects, exceptional collections, and semi-orthogonal decompositions. A bit later, in 1981, the article [Muk81] by S. Mukai was published, where he proved that for any abelian variety A there exists an equivalence of derived categories DCoh(A) — DCboh(Av), where Av = Pic0 (A) is the dual abelian variety of A. This equivalence identifies degree-zero line bundles on A with the skyscraper sheaves over the corresponding points of Pic0(A). It should be noted that the varieties A and Av may not be isomorphic. This equivalence allowed Mukai to answer some questions related to Picard bundles and demonstrated that sometimes, extra symmetries arise between derived categories, and those symmetries can be highly non-trivial at the geometric level.

An important tool for studying derived categories is the concept of semi-orthogonal decomposition. It is a way to represent a category as a "gluing" of several smaller subcategories. We will need an auxiliary definition.

Список литературы диссертационного исследования кандидат наук Пирожков Дмитрий Владимирович, 2024 год

Список литературы

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