The real intersection cohomology of a toric variety is described in a purely combinatorial way using methods of elementary commutative algebra only. We define, for arbitrary fans, the notion of a ``minimal extension sheaf'' on the fan as an axiomatic characterization of the equivariant intersection cohomology sheaf. This provides a purely algebraic interpretation of Stanley's generalized f- and g-vector of an arbitrary polytope or complete fan under a natural vanishing condition. -- The results presented in this note originate from joint work with G.Barthel, J.-P.Brasselet and L.Kaup, continuing earlier research (see math.AG/9904159). A detailed exposition will appear elsewhere (see math.AG/0002181).
We continue the approach toward a purely combinatorial "virtual" intersection cohomology for possibly non-rational fans, based on our investigation of equivariant intersection cohomology for toric varieties (see math.AG/9904159). Fundamental objects of study are "minimal extension sheaves" on "fan spaces". These are flabby sheaves of graded modules over a sheaf of polynomial rings, satisfying three relatively simple axioms that characterize the properties of the equivariant intersection cohomology sheaf on a toric variety, endowed with the finite topology given by open invariant subsets. These…
We review the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for non-rational fans, which do not define a toric variety. As a result, a number of interesting results on the toric $g$ and $h$ polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two com…
We construct an intersection product on tropical cycles contained in the Bergman fan of a matroid. To do this we first establish a connection between the operations of deletion and restriction in matroid theory and tropical modifications as defined by Mikhalkin. This product generalises the product of Allermann and Rau, and Allermann and also provides an alternative procedure for intersecting cycles which is not based on intersecting with Cartier divisors. Also, we simplify the definition in the case of one dimensional fan cycles in two dimensional matroidal fans and given an application of th…
We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in R^n and then for "abstract" cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in R^n.
For any conical symplectic resolution, we give a conjecture relating the intersection cohomology of the singular cone to the quantum cohomology of its resolution. We prove this conjecture for hypertoric varieties, recovering the ring structure on hypertoric intersection cohomology that was originally constructed by Braden and the second author.
We say that a complex analytic space, $X$, is an intersection cohomology manifold if and only if the shifted constant sheaf on $X$ is isomorphic to intersection cohomology; this is quickly seen to be equivalent to $X$ being a homology manifold. Given an analytic function $f$ on an intersection cohomology manifold, we describe a simple relation between $V(f)$ being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of $f$. We then describe how the Sebastiani-Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Fin…
We describe the relationship between intersection cohomology with twisted coefficients and the perverse sheaves which play the role of the eigenspaces for the Milnor monodromy of an affine hypersurface.
We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on previous result from Mellit for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincaré polynomial specialization of a conjecture from Letellier.
We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks.
We introduce the semi-infinite category of sheaves on the affine Grassmannian, and construct a particular object in it, which we call the the semi-infinite intersection cohomology sheaf. We relate it to several other entities naturally appearing in the geometric Langlands theory.
We construct Landau-Ginzburg models for numerically effective complete intersections in toric manifolds as partial compactifications of families of Laurent polynomials. We show a mirror statement saying that the quantum D-module of the ambient part of the cohomology of the submanifold is isomorphic to an intersection cohomology D-module defined from this partial compactification and we deduce Hodge properties of these differential systems.
The eigenvalues of Frobenius acting on $\ell$-adic cohomology of a complete intersection over a finite field have the divisibility predicted by the theorem of Ax and Katz. We have corrected some unforgivable typos.
We completely determine the intersection cohomology of the Satake compactifications of the moduli space of principally polarized abelian varieties in genera 2,3,4, except for the degree 10 intersection cohomology in genus 4. We also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the Satake compactification in these genera. As a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved.
This paper applies the decomposition theorem in intersection cohomology to geometric invariant theory quotients, relating the intersection cohomology of the quotient to that of the semistable points for the action. Suppose a connected reductive complex algebraic group $G$ acts linearly on a complex projective variety $X$. We prove that if $1 \to N \to G \to H \to 1$ is a short exact sequence of connected reductive groups, and $X^{ss}$ the set of semistable points for the action of $N$ on $X$, then the $H$-equivariant intersection cohomology of the geometric invariant theory quotient $X^{ss}//N…
We improve the known Hodge type bound for the exotic cohomology of complete intersections. In the revised version, we included a simplification of our original argument due to Pierre Deligne. The note appears in the C. R. de l'Aca. des Sc. de Paris.
We present several approaches to equivariant intersection cohomology. We show that for a complete algebraic variety acted by a connected algebraic group $G$ it is a free module over $H^*(BG)$. The result follows from the decomposition theorem or from a weight argument.
For a normal variety X defined over an algebraically closed field with an action of the multiplicative group G_m, we consider the ``hyperbolic localization'' functor from D^b(X) to D^b(X^T), which localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that the hyperbolic localization of the intersection cohomology sheaf is a direct sum of intersection cohomology sheaves.
Let $X^{2n}\subseteq \mathbb{P} ^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}π{_*}\mathbb{Q}$, where $π$ denotes the projection from the universal hyperplane family of $X^{2n}$ to ${(\mathbb{P} ^N)}^{\vee}$. We investigate the cohomology of the intersection cohomology complex $IC(R^{2n-1}π{_*}\mathbb{Q})$ over the points of a Severi's variety, parametrizing nodal hypersurfaces, whose nodes impose independent conditions on the very ample linear system giving the embedding in $\mathbb{P} ^N$.
The goal of this article is to give a Lefschetz type decomposition for the cd-index of a complete fan. To a complete simplicial fan one can associate a toric variety X, the even Betti numbers h_i of X and the numbers g_i = h_i-h_{i-1}. If the fan is projective, then non-negativity of g_i follows from the Lefschetz decomposition of the cohomology. In the case of a nonsiplicial complete fan one can analogously compute the flag h-numbers h_S and, by a change of variable formula, the cd-index. We give an analogue of the Lefschetz operation for the cd-index. This gives another proof of the non-nega…
We study certain foliated complex manifolds that behave similarly to complete nonsingular toric varieties. We classify them by combinatorial objects that we call marked fans. We describe the basic cohomology algebras of them in terms of corresponding marked fans. We also study the basic Dolbeault cohomology algebras of them.
We describe the homology intersection form associated to regular holonomic GKZ systems in terms of the combinatorics of regular triangulations. Combining this result with the twisted period relation, we obtain a formula of cohomology intersection numbers in terms of a Laurent series. We show that the cohomology intersection number depends rationally on the parameters. We also prove a conjecture of F. Beukers and C. Verschoor on the signature of the monodromy invariant hermitian form. This is a continuation of the previous work arXiv:1904.00565.
The purpose of this article is to investigate the intersection cohomology for algebraic varieties with torus action. Given an algebraic torus $\mathbb{T}$, one of our result determines the intersection cohomology Betti numbers of any normal projective $\mathbb{T}$-variety admitting an algebraic curve as global quotient. The calculation is expressed in terms of a combinatorial description involving a divisorial fan which is the analogous of the defining fan of a toric variety. Our main tool to obtain this computation is a description of the decomposition theorem in this context.
We study a Pfaffian of the system of differential equations annihilating Appell's hypergeometric series $F_4(a,b,c;x)$ by twisted cohomology groups associated with integrals representing solutions to this system. We simplify its connection matrix by the pull-back under a double cover of the complement of the singular locus. We express the simplified connection matrix in terms of the intersection form of the twisted cohomology groups.
We consider a hypersurface in $\mathbb{C}^n$ with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular we express the dimension of each graded piece of the weight filtration of this mixed Hodge structure in terms of the numbers of the Jordan blocks in the Milnor monodromy.
This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. It is based on lectures given at Montréal is Summer 1997. Our main aim is to obtain explicit descriptions of cohomology or Chow rings of certain manifolds with group actions which arise in representation theory, e.g. homogeneous spaces and their compactifications. As another appplication of equivariant intersection theory, we obtain simple versions of criteria for smoothness or rati…
In this paper, we show that the Galois representations provided by $\ell$-adic cohomology of proper smooth varieties, and more generally by $\ell$-adic intersection cohomology of proper varieties, over any field, are orthogonal or symplectic according to the degree. We deduce this from a preservation result of orthogonal and symplectic pure perverse sheaves by proper direct image. We show moreover that the subgroup of the Grothendieck group generated by orthogonal pure perverse sheaves of even weights and symplectic pure perverse sheaves of odd weights are preserved by Grothendieck's six opera…
For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf on Proj(N), showing that the Chern classes depend on subtle geometry of P and giving criteria for the splitting of the sheaf as a sum of line bundles. For certain fans associated to the reflection arrangement A_n, we describe a connection between C^0(P) and logarithmic vector fields tangent to …
Let $X$ be a smooth complete curve, $G$ be a reductive group and $P\subset G$ a parabolic. Following Drinfeld, one defines a compactification $\widetilde{\on{Bun}}_P$ of the moduli stack of $P$-bundles on $X$. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of $\widetilde{\on{Bun}}_P$. The description is given in terms of the combinatorics of the Langlands dual Lie algebra $\check{\mathfrak g}$.
This paper is a sequel to [Ga1]. We study the semi-infinite category on the Ran version of the affine Grassmannian, and study a particular object in it that we call the semi-infinite intersection cohomology sheaf. Unlike the situation of [Ga1], this version of the semi-infinite intersection IC sheaf is defined as the middle of extension of the constant (more precisely, dualizing) sheaf on the basic stratum, in a certain t-structure. We give several explicit description and characterizations of our semi-infinite intersection IC sheaf: we describe its !- and *- stalks; we present it explicitly a…
Let $\mathbf{M}_n$ be the Simpson compactification of twisted ideal sheaves $\mathcal{I}_{L,Q}(1)$ where $Q$ is a rank $4$ quardric hypersurface in $\mathbb{P}^n$ and $L$ is a linear subspace of dimension $n-2$. This paper calculates the intersection Poincaré polynomial of $\mathbf{M}_n$ using Kirwan's desingularization method. We obtain the intersection Poincaré polynomial of the moduli space for one-dimensional sheaves on del Pezzo surfaces of degree $\geq 8$ by considering wall-crossings of stable pairs and complexes.
This article studies the relationship between tropical Severi varieties and secondary fans. In the case when tropical Severi varieties are hypersurfaces this relationship is very well known; specifically, in this case, a tropical Severi variety of codimension 1 is a subfan of the corresponding secondary fan. It was expected for some time that this continues to hold more generally, but Katz found a counterexample in codimension 2, showing that this relationship is more subtle. The two main results in this paper are as follows. The first theorem finds a simple condition under which a tropical Se…
We generalize Givental's Theorem for complete intersections in smooth toric varieties in the Fano case. In particular, we find Gromov--Witten invariants of Fano varieties of dimension $\geq 3$, which are complete intersections in weighted projective spaces and singular toric varieties. We generalize the Riemann-Roch equations to weighted projective spaces. We compute counting matrices of smooth Fano threefolds with Picard group Z and anticanonical degrees 2, 8, and 16.
In this paper, we provide two different resolutions of structural sheaves of projectivized tangent bundles of smooth complete intersections. These resolutions allow in particular to obtain convenient (and completely explicit) descriptions of cohomology of twisted symmetric powers of cotangent bundles of complete intersections, which are easily implemented on computer. We then provide several applications. First, we recover the known vanishing theorems on the subject, and show that they are optimal via some non-vanishing theorems. Then, we study the symmetric algebra of global sections of symme…
We show that the etale cohomology (with compact supports) of an algebraic variety $X$ over an algebraically closed field has the canonical weight filtration $W$, and prove that the middle weight part of the cohomology with compact supports of $X$ is a subspace of the intersection cohomology of a compactification $X'$ of X, or equivalently, the middle weight part of the (so-called) Borel-Moore homology of $X$ is a quotient of the intersection cohomology of $X'$. We are informed that this has been shown by A. Weber in the case $X$ is proper (and $k=\bC$) using a theorem of G. Barthel, J.-P. Bras…
The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko-Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of CP^1 by CP^1 with prescribed ramification points and ramification types is related to the degree of the Lyashko--Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko-Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations …
Viewing a fan as a partially ordered set (of cones) we consider a category of sheaves on the fan which corresponds to a category of equivariant sheaves on the corresponding toric variety if the fan is rational. In this category we define an object which corresponds to the equivariant intersection cohomology complex. Our first main result is the ``elementary'' decomposition theorem for the direct image under subdivision of fans We also develop the Borel-Moore- Verdier duality in the derived category of sheaves on the fan.
Let $X$ be any subanalytic compact pseudomanifold. We show a De Rham theorem for $L^\infty$ forms. We prove that the cohomology of $L^\infty$ forms is isomorphic to intersection cohomology in the maximal perversity.
We investigate the equivariant intersection cohomology of a toric variety. Considering the defining fan of the variety as a finite topological space with the subfans being the open sets (that corresponds to the "toric" topology given by the invariant open subsets), equivariant intersection cohomology provides a sheaf (of graded modules over a sheaf of graded rings) on that "fan space". We prove that this sheaf is a "minimal extension sheaf", i.e., that it satisfies three relatively simple axioms which are known to characterize such a sheaf up to isomorphism. In the verification of the second o…
We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complex $IH_T^*(X)\otimes H^*(T)$. We also describe the weight filtration in $IH^*(X)$.
The purpose of this paper is to provide a way to compute the intersection cohomology of the GIT quotient of a nonsingular projective variety. We show that the middle perversity intersection cohomology of the GIT quotient $M//G$ is naturally embedded into the $G$ equivariant cohomology of $M^{ss}$ under an assumption satisfied by many interesting spaces. This embedding enables us to compute the intersection pairing in terms of the cup product structure of the equivariant cohomology ring. We also show that the image is a subspace defined by truncation. Overall, we can compute the intersection co…
We revisit the localization formulas of cohomology intersection numbers associated to a logarithmic connection. The main contribution of this paper is threefold: we prove the localization formula of the cohomology intersection number of logarithmic forms in terms of residue of a connection; we prove that the leading term of the Laurent expansion of the cohomology intersection number is Grothendieck residue when the connection is hypergeometric; and we prove that the leading term of stringy integral discussed by Arkani-Hamed, He and Lam is nothing but the self-cohomology intersection number of …
Given a projective morphism of compact, complex, algebraic varieties and a relatively ample line bundle on the domain we prove that a suitable choice, dictated by the line bundle, of the decomposition isomorphism of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber, yields isomorphisms of pure Hodge structures. The proof is based on a new cohomological characterization of the decomposition isomorphism associated with the line bundle. We prove some corollaries concerning the intersection form in intersection cohomology, the natural map from cohomology to intersection cohomol…
Under a certain condition A we give a construction to calculate the intersection cohomology of a rank one local system on the complement to a hyperplane-like divisor
Let $\Bbbk$ be a field of characteristic 0. Let $X$ be a smooth complete intersection over $k$ of dimension $n-k$ in the projective space $\mathbf{P}^n_{k}$, for given positive integers $n$ and $k$. When $k=\mathbb{C}$, Terasoma (\cite{Ter90}) and Konno (\cite{Ko91}) provided an explicit representative (in terms of differential forms) of a basis for the primitive middle-dimensional algebraic de Rham cohomology $H_{dR,\operatorname{prim}}^{n-k}(X;\mathbb{C})$. Later Dimca constructed another explicit representative of a basis of $H_{dR,\operatorname{prim}}^{n-k}(X;\mathbb{C})$ in \cite{Dim95}. …
