We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent of the Euler characteristic of the sheaves. We also prove an analogous result for the moduli space of semistable Higgs bundles with respect to an effective divisor $D$ of degree $\mathrm{deg}(D)>2g-2$. Our results confirm the cohomological $χ$-independence conjecture by Bousseau for $\mathbb{P}^2$, and verify Toda's conjecture for Gopakumar-Vafa invariants for certain local curves and local surfaces.
For the proof, we combine a generalized version of Ngô's support theorem, a dimension estimate for the stacky Hilbert-Chow morphism, and a splitting theorem for the morphism from the moduli stack to the good GIT quotient.
We give bounds for the cohomology of constructible sheaves on the moduli stacks M_{g,n} over the complex field. This enables us recover Harer's bound for the virtual cohomological dimension of the associated mapping class groups as well the theorem of Diaz on complete subvarieties of M_g. We also obtain such bounds for any open subset of the Deligne-Mumford compactification of M_{g,n} that is a union of strata. Our proof yields a template for obtaining similar bounds for the cohomological dimension for quasicoherent sheaves on M_{g,n}.
Let $alb_X: X \rightarrow A$ be the Albanese map of a smooth projective variety and $f: X \rightarrow Y$ the fibration from the Stein factorization of $alb_X$. For a positive integer $m$, if $f$ and $m$ satisfy the assumptions AS(1,2), then the translates through the origin of all components of cohomological locus $V^0(ω_X^m, alb_X)$ generates $I^*Pic^0(S)$ where $I: X \rightarrow S$ denotes the Iitaka fibration. This result applies to studying pluricanonical maps. We also considered the problem about whether a fibration is isotrivial and isogenous to a product.
We define the cohomological Hall algebra ${AHA}_{Higgs(X)}$ of the ($2$-dimensional) Calabi-Yau category of Higgs sheaves on a smooth projective curve $X$, as well as its nilpotent and semistable variants, in the context of an arbitrary oriented Borel-Moore homology theory. In the case of usual Borel-Moore homology, ${AHA}_{Higgs(X)}$ is a module over the (universal) cohomology ring $\mathbb{H}$ of the stacks of coherent sheaves on $X$ . We show that it is a torsion-free $\mathbb{H}$-module, and we provide an explicit collection of generators (the collection of fundamental classes $[Coh_{r,d}]…
This paper investigates the cohomological property of vector bundles on biprojective space. We will give a criterion for a vector bundle to be isomorphic to the tensor product of pullbacks of exterior products of differential sheaves.
We construct a functorial decomposition of de Rham cohomology sheaves, called weight decomposition, for smooth analytic spaces over non-Archimedean fields embeddable into $\mathbf{C}_p$, which generalizes a construction of Berkovich and solves a question raised by him. We then investigate complexes of real tropical differential forms and currents introduced by Chambert-Loir and Ducros, by establishing a relation with the weight decomposition and defining tropical cycle maps with values in the corresponding Dolbeault cohomology. As an application, we show that algebraic cycles that are cohomolo…
For an Abel-Prym curve contained in a Prym variety, we determine the cohomological support loci of its twisted ideal sheaves and the dimension of its theta-dual.
For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall multiplication. This multiplication involves data (virtual pullbacks) governing the derived moduli stack, i.e., the perfect obstruction theory naturally existing on the non-derived stack. By restricting to sheaves with support of given dimension, we obtain several types of Hecke operators. In particular, we study R(S), the Hecke algebra of 0-dimensional sheaves. For t…
This paper is devoted to the study of the gluing construction for perverse sheaves on $G/U$ introduced by Kazhdan and Laumon ($G$ is a semisimple gourp, $U$ is the unipotent radical of a Borel subgroup in $G$). Kazhdan and Laumon conjectured that all Ext-groups in the glued category are finite-dimensional and that global cohomological dimension is finite. We prove the first part of this conjecture. In the appendix we show that the simple object in the glued category corresponding to the constant sheaf has infinite cohomological dimension, thus disproving the second part of the above conjecture…
We introduce a notion of regularity for coherent sheaves on Grassmannians of lines. We use this notion to prove some extension of Evans-Griffith criterion to characterize direct sums of line bundles. We also give a cohomological characterization of exterior and symmetric powers of the universal bundles of the Grassmannian.
We propose a conjecture on the relative twist formula of $\ell$-adic sheaves, which can be viewed as a generalization of Kato-Saito's conjecture. We verify this conjecture under some transversal assumptions. We also define a relative cohomological characteristic class and prove that its formation is compatible with proper push-forward. A conjectural relation is also given between the relative twist formula and the relative cohomological characteristic class.
Recently, the Riemann-Hilbert correspondence was generalized in the context of general holonomic D-modules by A. D'Agnolo and M. Kashiwara. Namely, they proved that their enhanced de Rham functor gives a fully faithfully embedding of the derived category of cohomologically holonomic complexes of D-modules into the derived category of complexes of real constructible enhanced ind-sheaves. In this paper, we study a condition when a complex of real constructible enhanced ind-sheaves $K$ is induced by a cohomologically holonomic complex of D-modules. We characterize such $K$ in terms of the restric…
We study a class of torsion-free sheaves on complex projective spaces which generalize the much studied mathematical instanton bundles. Instanton sheaves can be obtained as cohomologies of linear monads and are shown to be semistable if its rank is not too large, while semistable torsion-free sheaves satisfying certain cohomological conditions are instanton. We also study a few examples of moduli spaces of instanton sheaves.
This review gives an introduction to cohomological Donaldson-Thomas theory: the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau threefolds, and of complexes in 3-Calabi-Yau categories, categorifying their numerical DT invariant. Local and global aspects of the theory are both covered, including representations of quivers with potential. We will discuss the construction of the DT sheaf, a nontrivial topological coefficient system on such a moduli space, along with some cohomology computations. The Cohomological Hall Algebra, an algebra structure on cohomological DT spaces…
Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of $\mathbb Q$-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincaré transform, which has several consequences. In many concrete geometric contexts these functions provide useful invariants. We illustrate this with two different applications, the first one to GV-subschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.
Given a Calabi-Yau manifold and considering the $B$-branes on it as objects in the derived category of coherent sheaves, we identify the vertex operators for strings between two branes with elements of the cohomology groups of Ext sheaves. We define the correlation functions for these general vertex operators. Strings stretching between two coherent sheaves are studied as homological extensions of the corresponding branes. In this context, we relate strings between different pairs of branes when there are maps between these branes. We also interpret some strings with ghost number $k$ as obstru…
We prove that the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite, cohomological functors on the perfect derived category of a quasi-projective scheme over a field. We introduce the notions of pseudo-adjoints and Rouquier functors and study them. As an application of these ideas and results, we extend the reconstruction result of Bondal and Orlov to Gorenstein projective varieties.
We give sufficient cohomological criteria for the classes of given varieties over a field $k$ to be algebraically independent in the Grothendieck ring of varieties over $k$ and construct some examples.
University of Southern California（南加州大学）；Institut de Recherche Mathématique Avancée（高级数学研究所）；Institut Universitaire de France (IUF)（法国高等研究院）；Institut de Mathématiques de Jussieu（儒勒·让数学研究所）
The moduli stack of representations of a quiver, or coherent sheaves on a proper curve, carries two structures on its cohomology: a Hall algebra and braided vertex coalgebra. We show that they are compatible, by developing a formulation of abelian localisation which works in the derived/singular setting. An application of these techniques gives an explicit formula for the cohomological Hall algebra products.
In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by A.Vaintrob and two of the authors. Hopefully our result clarifies to some extent the constructions of the above work.
In this note, we prove a result on the independence of $\ell$ for the supports of irreducible perverse sheaves occurring in the Decomposition Theorem, as well as for the family of local systems on each support. It generalizes Gabber's result on the independence of $\ell$ of intersection cohomology to the relative case.
A constructible sheaf corresponding to Gel'fand Zelevinski hypergeometric functions on a torus is called hypergeometric sheaf. We consider Hodge and Tate conjectrue for hypergeomtric sheaves. Hodge conjecture is formulated in terms of variation of Hodge strucure and Tate conjecture is done for l-adic sheaves on an open set of torus. We prove Hodge and Tate conjecture up to Hodge and Tate cycle of Fermat motifes. We use cohomological Mellin transform to get the main theorem. This is the final revision for preprint.
For oriented $-1$-shifted symplectic derived Artin stacks, Ben-Bassat-Brav-Bussi-Joyce introduced certain perverse sheaves on them which can be regarded as sheaf theoretic categorifications of the Donaldson-Thomas invariants. In this paper, we prove that the hypercohomology of the above perverse sheaf on the $-1$-shifted cotangent stack over a quasi-smooth derived Artin stack is isomorphic to the Borel-Moore homology of the base stack up to a certain shift of degree. This is a global version of the dimensional reduction theorem due to Davison. We give two applications of our main theorem. Firs…
We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions, and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing, and reflexive differentials associated to Z.
Let X be a projective complex K3 surface. Beauville and Voisin singled out a 0-cycle c_X on X of degree 1: it is represented by any point lying on a rational curve in X. Huybrechts proved that the second Chern class of a rigid simple vector-bundle on X is a multiple of the Beauville-Voisin class c_X if certain hypotheses hold and he conjectured that the additional hypotheses are unnecessary. We believe that the following generalization of Huybrechts' conjecture holds. Let M and N be moduli spaces of stable pure sheaves on X (with fixed cohomological Chern characters) and suppose that they have…
We study twisted vector bundles of infinite rank on gerbes, giving a new spin on Grothendieck's famous problem on the equality of the Brauer group and cohomological Brauer group. We show that the relaxed version of the question has an affirmative answer in many, but not all, cases, including for any algebraic space with the resolution property and any algebraic space obtained by pinching two closed subschemes of a projective scheme. We also discuss some possible theories of infinite rank Azumaya algebras, consider a new class of "very positive" infinite rank vector bundles on projective variet…
Donaldson-Thomas theory on a Calabi-Yau can be described in terms of a certain six-dimensional cohomological gauge theory. We introduce a certain class of defects in this gauge theory which generalize surface defects in four dimensions. These defects are associated with divisors and are defined by prescribing certain boundary conditions for the gauge fields. We discuss generalized instanton moduli spaces when the theory is defined with a defect and propose a generalization of Donaldson-Thomas invariants. These invariants arise by studying torsion free coherent sheaves on Calabi-Yau varieties w…
Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians
Rutgers, The State University Of New Jersey（新泽西州立罗格斯大学）；The University of Tokyo（东京大学）；Simion Stoilow Institute of Mathematics（西蒙·斯托伊洛数学研究所）；Université de Paris（巴黎大学）
We categorify the inclusion-exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In charac…
Let $X$ be a finite connected simplicial complex, and let $δ$ be a perversity (i.e., some function from integers to integers). One can consider two categories: (1) the category of perverse sheaves cohomologically constructible with respect to the triangulation, and (2) the category of sheaves constant along the perverse simplices ($δ$-sheaves). We interpret the categories (1) and (2) as categories of modules over certain quadratic (and even Koszul) algebras $A(X,δ)$ and $B(X,δ)$ respectively, and we prove that $A(X,δ)$ and $B(X,δ)$ are Koszul dual to each other. We define the $δ$-perverse topo…
We describe derived moduli functors for a range of problems involving schemes and quasi-coherent sheaves, and give cohomological conditions for them to be representable by derived geometric n-stacks. Examples of problems represented by derived geometric 1-stacks are derived moduli of polarised projective varieties, derived moduli of vector bundles, and derived moduli of abelian varieties.
In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category $\mathsf{Coh}^{\mathsf{b}}(\mathbb{R}\mathsf{M})$ of complexes of sheaves with bounded coherent cohomology on a derived moduli stack $\mathbb{R}\mathsf{M}$. In the surface case, $\mathbb{R}\mathsf{M}$ is a suitable derived enhancement of the moduli stack $\mathsf{M}$ of coherent sheaves on the surface. This construction categorifies the K-theoretical and cohomological Hall algebras of…
Hecke operators on symplectic surfaces and $χ$-independence
University of Edinburgh（爱丁堡大学）；Université de Picardie Jules Verne（约尔·凡尔纳皮卡第大学）；Kyoto University（京都大学）；Université Paris-Saclay（巴黎萨克雷大学）；Simion Stoilow Institute of Mathematics（西米翁·斯托伊洛瓦数学研究所）；Université Paris-Cité（巴黎城市大学）
Let $X$ be a smooth quasi-projective algebraic surface and let $Δ_n$ the big diagonal in the product variety $X^n$. We study cohomological properties of the ideal sheaves $\mathcal{I}^k_{Δ_n}$ and their invariants $(\mathcal{I}^k_{Δ_n})^{\mathfrak{S}_n}$ by the symmetric group, seen as ideal sheaves over the symmetric variety $S^nX$. In particular we obtain resolutions of the sheaves of invariants $(\mathcal{I}_{Δ_n})^{\mathfrak{S}_n}$ for $n = 3,4$ in terms of invariants of sheaves over $X^n$ whose cohomology is easy to calculate. Moreover, we relate, via the Bridgeland-King-Reid equivalence,…
Rutgers, The State University Of New Jersey（新泽西州立罗格斯大学）；The University of Tokyo（东京大学）；Simion Stoilow Institute of Mathematics（罗马尼亚科学院西米恩·斯托伊洛数学研究所）；Université de Paris（巴黎大学）
We study the relation between Donaldson-Thomas theory of Calabi-Yau threefolds and a six-dimensional topological Yang-Mills theory. Our main example is the topological U(N) gauge theory on flat space in its Coulomb branch. To evaluate its partition function we use equivariant localization techniques on its noncommutative deformation. As a result the gauge theory localizes on noncommutative instantons which can be classified in terms of N-coloured three-dimensional Young diagrams. We give to these noncommutative instantons a geometrical description in terms of certain stable framed coherent she…
We give a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity W and a subgroup G of the diagonal group of symmetries of W. This theory can be viewed as an analogue of the Gromov-Witten theory for an orbifoldized Landau-Ginzburg model for W/G. The main geometric ingredient for our construction is provided by the moduli of curves with W-structures introduced by Fan, Jarvis and Ruan. We construct certain matrix factorizations on the products of these moduli stacks with affine spaces which play a role similar to that of…
