Исключительные наборы на изотропных грассманианах тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Новиков Александр Андреевич

• Новиков Александр Андреевич
• кандидат наук
• 2025

Оглавление диссертации кандидат наук Новиков Александр Андреевич

Contents

Introduction

1 Preliminaries

1.1 Lefschetz exceptional collections

1.2 Equivariant bundles

1.3 Borel-Bott-Weil theorem and its applications

1.4 Computations with tensor products

2 Semiorthogonality lemmas

2.1 General semiorthogonality lemmas

2.2 Semiorthogonality lemmas for k =

3 Some exceptionalities

3.1 Exceptionalities for B

3.2 Some exceptionalities for H

4 Secondary staircase complexes in general case

4.1 Notation

4.2 Grassmannians of isotropic lines

4.2.1 Vector bundles Ea'b

4.2.2 Secondary staircase complexes on IGr(2, V)

4.2.3 Cohomology of secondary staircase complexes on IGr(2, V)

4.3 General isotropic Grassmannians

4.3.1 Fourier-Mukai transforms

4.3.2 General secondary staircase complexes

4.4 Bicomplexes

5 Complexes on IGr(3,2n) and properties of H

5.1 Staircase complexes and their truncations

5.2 Secondary staircase complexes

5.3 Secondary staircase complexes are mutations

5.4 Semiorthogonality to H

6 Proof of exceptionality

7 Relations and notation

7.1 Staircase complexes and strips

7.2 Secondary staircase complexes and triangular relations

8 Proof of fullness

8.1 Notation

8.2 Generation of additional bundles

Conclusion

References

Appendix: Russian translation of the thesis

Введение диссертации (часть автореферата) на тему «Исключительные наборы на изотропных грассманианах»

Introduction

This work is dedicated to the study of bounded derived categories of coherent sheaves over isotropic symplectic Grassmannians IGr(k, V). More specifically, we introduce and study a new class of objects in these categories. Using these objects, we construct minimal Lefschetz exceptional collections on Grassmannians of isotropic 3-subspaces for any symplectic vector space V over an algebraically closed field k of characteristic 0, generalizing already existing results for spaces of small dimension.

Derived categories of coherent sheaves and exceptional collections

The bounded derived category of coherent sheaves Db(X) is an important invariant of an algebraic variety X, but its structure can be complicated. These categories often admit decompositions into several admissible semiorthogonal subcategories. In the simplest case each of those subcategories is generated by an exceptional object and, thus, is equivalent to the derived category of vector spaces. Such ordered sets of objects (E\,E2,..., Em) are called exceptional collections. Then each object of Db(X) admits a unique filtration, with the i-th subquotient being a direct sum of shifts of the objects Ei. Therefore, an exceptional collection serves as a kind of non-commutative basis for Db(X). A long-standing conjecture predicts the existence of a full exceptional collection on all homogeneous varieties of reductive algebraic groups.

Reductive algebraic groups

Each connected linear group G over an algebraically closed field k of characteristic zero has a maximal unipotent radical R C G — a largest smooth connected unipotent normal subgroup. If R is trivial, then G is called reductive. Reductive groups are also characterized by complete reducibility of their rational finite-dimensional representations, i.e., every short exact sequence of their finite-dimensional rational representations splits. If the group G is reductive, then its derived subgroup G' = [G, G] is semi-simple. Then G can be reconstructed as an almost-direct product of G' and connected component of the center of G that is a torus, i.e., it is a factor of the direct product by a finite central subgroup. Furthermore, each semi-simple algebraic group is an almost-direct product of simple groups.

Finally, there is a well-known classification of simple algebraic groups over k into 4 classical infinite series of equivalence classes that correspond to Dynkin diagrams An for n > 1, Bn for n > 2, Cn for n > 3, and Dn for n > 4: representatives of these classes are SL(n + 1), SO(2n + 1), Sp(2n), and SO(2n), respectively, and 5 classes of exceptional groups corresponding to E6, E7, Eg, F4, and G2. In each of these classes there is a distinguished group that is connected and simply-connected (i.e., its algebraic fundamental group is trivial) that is the simply-connected cover of all other groups in the class. Its representation theory is the simplest and can be fully described in terms of combinatorial data of the corresponding Dynkin diagram.

Reductive algebraic groups and classification of homogeneous spaces

Homogeneous spaces of reductive groups are a ubiquitous and well-studied class of algebraic varieties. All homogeneous varieties of algebraic groups are conjectured to have full exceptional collections. Grassmannians constitute the simplest class homogeneous varieties, that serve as the building blocks for other homogeneous spaces. In particular, this allows to automatically construct full exceptional collections on all homogeneous spaces, once the case of Grassmannians is closed. Grassmannians are parameterized by pairs of a Dynkin diagram and its simple root. The Grassmannian is a quotient of the simply connected simple algebraic group by the maximal parabolic subgroup corresponding to the distinguished root.

Exceptional collections on homogeneous spaces

Projective space Pn for integral n > 1 always has a structure of a homogeneous space as Pn = Gr(1,n + 1). Moreover, P2n-1 can also be represented as an isotropic Grassmannian of 1-subspaces of a symplectic 2n-dimensional space: P2n-1 = IGr(1,2n). The first example of a full exceptional collection was constructed by Beilinson, who showed in [1] that the twists of the structure sheaf O, O(1),..., O(n) form such a collection on the projective space Pn.

Afterward, Kapranov [8] (1984) constructed full exceptional collections on the Grassmannians and all flag varieties of groups GLn and on smooth quadrics OGr(1, V) in both cases: for odd and even-dimensional V. Later, Fonarev (2013) in [3] suggested a more powerful framework for constructing and studying exceptional collections on the classical Grassmannians, connected to mutations and quantum cohomology, and suggested a new proof of fullness based on a consistent application of the so-called staircase complexes.

In this text we study the derived categories of symplectic isotropic Grassmannians IGr(k, 2n), where the progress has been significantly slower. As the case IGr(1,2n) = P2n-1 is covered by the result of Beilinson, the first new series is the isotropic Grassmannian of lines IGr(2,2n); in this case a full exceptional collection was constructed in [12]. The main idea of the Kuznetsov's proof consists in representing IGr(2,2n) as a codimension 1 subvariety of Gr(2,2n), where all the required homological algebra and representation theory is greatly simplified. Our computation in the proof of fullness for k = 3 also follows this method to a significant degree. However, the codimension of IGr(k, 2n) c Gr(k, 2n) is k(k2 1), so the complexity of computations quickly rises.

A general construction of full exceptional collections over all homogeneous spaces over an arbitrary ring was recently proposed by Samokhin and van der Kallen in [20] (2024). However, their construction is implicit and the properties of the objects are not understood yet. Another, more explicit, set of exceptional collections that are potentially full were constructed for all classical groups by Kuznetsov and Polischchuk in [13] (2016), where they extended the ideas from [19] (2001) and [18] (2011), where the cases of IGr(3,6), and IGr(4,8) and IGr(5,10) are discussed. Their fullness was proved by Fonarev over classical Grassmannians Gr(k,n) in [4] (2015) and in the case of Lagrangian Grassmannians IGr(n, 2n) in [5] (2019) and, recently, a general proof for symplectic isotropic case was suggested by L. Guseva and the author in [7] (2025). In particular, the latter paper heavily relies on a further generalization of secondary staircase complexes that are introduced as a part of the present work.

Lefschetz exceptional collections

There is an important class of exceptional collections introduced by Kuznetsov in [10], which are crucial for many applications, including homological projective duality, mirror symmetry, and connections with quantum cohomology. Some powerful theorems that describe structure of derived categories explicitly demand exceptional collections with Lefschetz structure. More specifically, only minimal Lefschetz collections allow one to make the full use of these results. However, most of the collections described above do not have Lefschetz structure.

Goals and Objectives

To construct minimal Lefschetz collections on isotropic Grassmannians IGr(k, V) for k = 3 and arbitrary n.

Degree of development of the topic

Even in the case of classical Grassmannians exceptional collections that were constructed by Fonarev in [3] (2013), that have Lefschetz structure, are not minimal Lefschetz in general. For isotropic symplectic Grassmannians minimal Lefschetz exceptional collections were known in the following cases only:

• IGr(1,2n) = P2n-1 — A. Beilinson (1978) [1]

• IGr(3,6) — A. Samokhin (2001) [19]

• IGr(2,2n) — A. Kuznetsov (2008) [12]

• IGr(4,8) and IGr(5,10) — A. Polishchuk, A. Samokhin (2011) [18], corrected by A. Fonarev (2019) [5]

• IGr(3,8) — L. Guseva (2018) [6]

• IGr(3,10) — the author (2020) [14]

Equivariant bundles and complexes

Apart from providing an invaluable source of examples of exceptional collections and tools of working with them, the above papers conveyed an important message: to construct sufficiently long exceptional collections one has to consider equivariant vector bundles which are not necessarily irreducible, and to prove fullness of the constructed collections, one has to find exact sequences relating these bundles, similar to the staircase complexes that work so well in the case of GLn.

Some complexes of this sort already appeared in the papers mentioned above. For instance, the key step in the proof [12] of fullness of the exceptional collection on IGr(2,2n) was a construction of a certain bicomplex in [12, Proposition 5.3], that can be considered as a complex consisting of the objects represented by its rows; one of our results is another construction of such a complex fixing an inaccuracy made in [12], see Remarks 4.7, 4.13 and Corollary 4.16. Similarly, the proof of fullness of an exceptional collection on IGr(3,8) and IGr(3,10) uses bicomplexes described in [6, Section 5.2] and [14, Section 3.2], respectively.

Provisions for defense

• Construction of minimal Lefschetz exceptional collections on isotropic Grassmannians IGr(3,2n) over an algebraically closed field of characteristic zero for arbitrary n > 5.

• Construction of a class of Sp2n-equivariant complexes on IGr(k, 2n) (called secondary staircase complexes) and computation of their cohomologies. Interpretation of the corresponding objects in Db(IGr(2,2n)) as total complexes of certain bicomplexes.

Publications

• Alexander Novikov. "Minimal Lefschetz Collections on Isotropic Grassmannians IGr(3,2n)". In: Russian Mathematical Surveys 80.3(483) (2025), pp. 187-188. URL: https ://www. mathnet.ru/rus/rm10250

• Alexander Novikov. "Secondary staircase complexes on isotropic Grassmannians". In: Sbornik: Mathematics 216.7 (2025), pp. 78-95. DOI: 10.4213/sm10210. arXiv: 2412.02914 [math.AG]. URL: https://www.mathnet.ru/rus/sm10210

Relevance of the topic

Derived categories are one of the most important invariants of spaces and are a central object of study in modern algebraic geometry and representation theory. However, their complexity makes them difficult to study. Complete exceptional collections, particularly minimal Lefschetz collections, allow for a complete description of the structure of the derived category in combinatorial terms and are one of the most powerful tools for studying derived categories. Homogeneous spaces, such as isotropic Grassmannians, are a commonly encountered class of varieties whose

study is of interest far beyond algebraic geometry and serves as an important step towards the study of more general spaces.

Validity of scientific propositions

All results of this dissertation have been mathematically rigorously substantiated and proven. The dissertation is executed in accordance with all scientific standards, including all definitions, statements, and theorems necessary within the scope of this research are contained in the dissertation. All proofs are carried out at a high scientific level, leaving no doubt about the validity of the scientific propositions of this dissertation.

Degree of reliability of the results

All scientific results presented for the defense of the dissertation by the candidate are formulated and proven as statements and theorems in his works. Most of the statements are accompanied by illustrative examples. Due to the clarity and rigor of the presentation of the research, the results presented for the defense of the dissertation are beyond doubt.

Scientific novelty of the work

In the dissertation research, complete minimal Lefschetz exceptional collections were constructed on isotropic Grassmannians of three-dimensional subspaces of symplectic spaces of any dimension. Previously, similar results were known only for dimensions 6, 8, and 10. Additionally, new equivariant complexes were constructed on all isotropic Grassmannians, and their cohomologies were computed.

Theoretical significance

A new method for proving the completeness of exceptional collections on isotropic Grassmannians is presented. A new class of objects in the derived categories of isotropic Grassmannians is studied, for which various resolutions are constructed, and interpretations as mutations in the derived category are proven. The methods presented in the work are original and can be used to study the derived categories and geometry of homogeneous spaces and other varieties, including odd symplectic Grassmannians.

Practical significance of the results

The results obtained by the candidate are of practical interest due to potential applications in geometry and representation theory, as well as other areas of mathematics and physics, particularly in problems of homological mirror symmetry and Gromov-Witten invariant theory.

Presentation of the work

The results of the dissertation were presented in the talk "Derived Categories of Isotropic Grassmannians" at the seminar of the Laboratory of Algebraic Geometry and Its Applications, HSE, Moscow, 05.10.2023.

Potential applications

First, as we have already mentioned, secondary staircase complexes, were successfully applied in [7] as one of the main tools in the proof of fullness of Kuznetsov-Polishchuk exceptional collections on isotropic Grassmannians IGr(k, 2n). In particular, there is an alternative proof for the Lagrangian case that only uses secondary staircase complexes. Therefore, they constitute a fundamental class of complexes on IGr(k, 2n).

Applications of different exceptional collections as computational tools are also numerous. Especially well suited for computations are minimal Lefschetz collections, since they possess a high degree of symmetry, and essentially decompose the category into several identical semiorthogonal pieces generated by exceptional collections. For example, only minimal non-rectangular Lefschetz exceptional collections allow to compute and study residual categories that are conjectured to be closely connected to quantum cohomology, theory of Gromov-Witten invariants, and Dubrovin conjectures.

Заключение диссертации по теме «Другие cпециальности», Новиков Александр Андреевич

Conclusion

There are still many open questions left that can be tackled using the same straightforward methods. For example, the objects H in the constructed collections are not pure in general. There are potential candidates for collections that are pure that use Kuznetsov-Polishchuk objects. Moreover, these collections are easier to generalize to other cases like IGr(4,2n).

It is still possible to construct more Lefschetz exceptional collections for bigger isotropic Grassmannians, however some of the computations would unwind. As we have already mentioned, the computations in cases of all known series fundamentally rely on their closeness to classical Grassmannians GL(k,V). This seem to be a dead end, however the developed methods like secondary staircase complexes proved to be invaluable for progress in general.

As the next goal in the study of exceptional collections on isotropic Grassmannians and their categories in general, one should aim for better understanding of the connection with quantum cohomology. In particular, it would be useful to construct and better understand an action similar to the one given by staircase complexes for classical and Lagrangian staircase complexes for Lagrangian Grassmannians.

Furthermore, the constructions developed for symplectic groups usually have direct analogues for orthogonal groups. This is another even less-studied area that could bring better understanding of geometry and combinatorics of homogeneous spaces.

Список литературы диссертационного исследования кандидат наук Новиков Александр Андреевич, 2025 год

References

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[14] Alexander Novikov. "Lefschetz Exceptional Collections on Isotropic Grassmannians". Master Thesis. HSE, 2020. URL: https://www.hse.ru/en/edu/vkr/369850047.

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[16] Alexander Novikov. "Secondary staircase complexes on isotropic Grassmannians". In: Sbornik: Mathematics 216.7 (2025), pp. 78-95. DOI: 10.4213/sm10210. arXiv: 2412.02914 [math.AG]. URL: https://www.mathnet.ru/rus/sm10210.

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[20] Alexander Samokhin and Wilberd van der Kallen. Highest weight category structures on Rep(B) and full exceptional collections on generalized flag varieties over Z. 2024. arXiv: 2407.13653 [math.AG]. URL: https://arxiv.org/abs/2407.13653.