Vafa–Witten Invariants from Exceptional Collections

It is proved that a triangulated category generated by a strong exceptional collection is equivalent to the derived category of modules over an algebra of homomorphisms of this collection. For the category of coherent sheaves on a Fano variety, the functor of tightening to a canonical class is described by means of mutations of an exceptional collection generating the category. The connection between mutability of strong exceptional collections and the Koszul property is studied. It is proved that in the geometric situation mutations of exceptional sheaves consist of present sheaves and the corresponding algebra of homomorphisms is Koszul and selfconsistent. Bibliography: 18 titles.

In this paper we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant $c>\frac34$ there exist infinitely many toric Fano varieties $Y$ with Picard number three, such that the maximal length of exceptional collection of line bundles on $Y$ is strictly less than $c\rk K_0(Y).$ To obtain varieties without exceptional collections of line bundles, it suffices to put $c=1.$ On the other hand, we prove that for any toric nef-Fano DM stack $Y$ with Picard number three, there exists a strong exceptional collection of line bundles on $Y$ of length at least $\frac34 \rk K_0(Y).$ The constant $\frac34$ is thus maximal with this property.

We consider the bounded derived category D_S_k(X^n_k) of S_k-equivariant coherent sheaves on (P^n)^k. One of the ways to describe derived category is via an exceptional collection. We consider a special class of exceptional collections - Lefschetz exceptional collections. The goal of the dissertation is to construct in the category D_S_k(X^n_k) a rectangular Lefschetz exceptional collection when this is possible, or a minimal Lefschetz exceptional collection when a rectangular one does not exist.The main results of the dissertation include the construction of a rectangular Lefschetz exceptional collection in the case k= 3 and in the case n= 1 when gcd(n+1,k) = 1. We also construct minimal Lefschetz exceptional collection for n= 1 and even k, and for n= 2 and k= 3.--Author's abstract

L’espace de modules M=M(0,3) des faisceaux semi-stables de rang 2, de classes de Chern (0,3) sur le plan projectif ℙ 2 est une variété projective irréductible et lisse de dimension 9. Dans M, les points qui ne proviennent pas d’un faisceau localement libre constituent une hypersurface ∂M. Dans cet article, nous montrons que toute surface complété de M rencontre la frontière ∂M, autrement dit qu’il n’existe pas de famille de fibrés vectoriels paramétrée par une surface complète de M. La démonstration repose sur la construction d’un morphisme birationnel φ:M→ Gr de M dans la grassmannienne Gr des réseaux de coniques de ℙ 2 , ce qui nous permet d’identifier M avec l’éclaté d’une surface de Gr. Ceci conduit à une description précise de l’algèbre de cohomologie de M, description utilisée pour déterminer quelle pourrait être la classe fondamentale des surfaces complètes de M ne rencontrant pas ∂M et aboutir à une contradiction.

In the previous paper by Tarasov and Varchenko the equivariant quantum differential equation ( q D E qDE ) for a projective space was considered and a compatible system of difference q K Z qKZ equations was introduced; the space of solutions to the joint system of the q D E qDE and q K Z qKZ equations was identified with the space of the equivariant K K -theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant K K -theory algebra. This paper is a continuation of the paper by Tarasov and Varchenko. We describe the relation between solutions to the joint system of the q D E qDE and q K Z qKZ equations and the topological-enumerative solution to the q D E qDE only, definitionned as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant K K -theory algebra, the equivariant Gamma class of the projective space, and the equivariant first Chern class of the tangent bundle of the projective space. We consider a Stokes basis, the associated exceptional basis in the equivariant K-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincaré pairing wrt to the basis, which is the left dual to the associated exceptional basis. We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space. These statements are equivariant analogs of results of G. Cotti, B. Dubro-vin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani.

We investigate exceptional sheaves on the Hirzebruch surface $\mathbb{F}_2$, as the first attempt toward the classification of exceptional objects on weak del Pezzo surfaces.

We construct a full exceptional collection of vector bundles in the derived categories of coherent sheaves on the Grassmannian of isotropic two-dimensional subspaces in a symplectic vector space of dimension 2n and in an orthogonal vector space of dimension 2n + 1 for all n.

We study Le Potier's strange duality conjecture on a rational surface. We focus on the case involving the moduli space of rank 2 sheaves with trivial first Chern class and second Chern class 2, and the moduli space of 1-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show the conjecture for this case is true under some suitable conditions on $L$, which applies to $L$ ample on any Hirzebruch surface $Σ_e:=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(e))$ except for $e=1$. When $e=1$, our result applies to $L=aG+bF$ with $b\geq a+[a/2]$, where $F$ is the fiber class, $G$ is the section class with $G^2=-1$ and $[a/2]$ is the integral part of $a/2$.

Admissible subcategories are building blocks of semiorthogonal decompositions. Many examples of them are known, but few general properties have been proved, even for admissible subcategories in the derived categories of coherent sheaves on basic varieties such as projective spaces. We use a relation between admissible subcategories and anticanonical divisors to study admissible subcategories of del Pezzo surfaces. We show that any admissible subcategory of the projective plane has a full exceptional collection, and since all exceptional objects and collections for the projective plane are known, this provides a classification result for admissible subcategories. We also show that del Pezzo surfaces of degree at least three do not contain so-called phantom subcategories. These are the first examples of varieties of dimension larger than one that have some nontrivial admissible subcategories, but provably do not contain phantoms.

In this article, we construct Chern classes in rational Deligne cohomology for coherent sheaves on a smooth complex compact manifold. We prove that these classes satisfy the functoriality property under pullbacks, the Whitney formula and the Grothendieck–Riemann–Roch theorem for projective morphisms between smooth complex compact manifolds.

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is M. Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim $X\geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

D-branes, exceptional sheaves and quivers on Calabi–Yau manifolds: from Mukai to McKay

Recently, a universal formula for a non-holomorphic modular completion of the generating functions of refined BPS indices in various theories with $N=2$ supersymmetry has been suggested. It expresses the completion through the holomorphic generating functions of lower ranks. Here we show that for $U(N)$ Vafa-Witten theory on Hirzebruch and del Pezzo surfaces this formula can be used to extract the holomorphic functions themselves, thereby providing the Betti numbers of instanton moduli spaces on such surfaces. As a result, we derive a closed formula for the generating functions and their completions for all $N$. Besides, our construction reveals in a simple way instances of fiber-base duality, which can be used to derive new non-trivial identities for generalized Appell functions. It also suggests the existence of new invariants, whose meaning however remains obscure.

We describe various properties of Hirzebruch surfaces and related constructions: degenerations, braid monodromy, Galois covers and their Chern classes.

In this article, we investigate an axiomatic approach introduced by Grivaux for the study of rational Bott-Chern cohomology, and use it in that context to define Chern classes of coherent sheaves. This method also allows us to derive a Riemann-Roch-Grothendieck formula for a projective morphism between smooth complex compact manifolds. In the general case of complex spaces, the Poincar\'e and Dolbeault-Grothendieck lemmas are not always valid. For this reason, and to simplify the exposition, we only consider non singular complex spaces. The appendix presents a calculation of integral Bott-Chern cohomology in top degree for a connected compact manifold.