A great number of theoretical results are known about log Gromov-Witten invariants, but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov-Witten invariants. The first such calculation by Gross-Pandharipande-Siebert deals with multiple covers over rigid curves in the log Calabi-Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi-Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of "logarithmic" 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.
We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincare polynomials of the moduli spaces to Donaldson-Thomas invariants of the surface. In support of this result, we compute explicitly intersection Poincare polynomials for sheaves with rank two and three on ruled surfaces.
We consider generalized complete intersection manifolds in the product space of projective spaces, and work out useful aspects pertaining to the cohomology of sheaves over them. First, we present and prove a vanishing theorem on the cohomology groups of sheaves for subvarieties of the ambient product space of projective spaces. We then prove an equivalence between configuration matrices of complete intersection Calabi-Yau manifolds. We also present a formula of the genus of curves in generalized complete intersection manifolds. Some of these curves arise as the fixed point locus of certain sym…
The theory of relative logarithmic jet spaces is developed for log schemes. With this theory the existence of bounds of intersection multiplicities of curves and divisors on certain log schemes is established. This result extends those of Noguchi and Winkelmann by replacing the semi-abelian condition by a differential one.
We study intersection theoretic problems in the setting of Chow-Witt groups with coefficients in a fixed Milnor-Witt cycle algebra over a perfect field. We prove that the product maps on such groups satisfy the following property: given two points in a regular local scheme with supports which do not intersect properly, their product vanishes. This gives an analogue of Serre's vanishing result for intersection multiplicities.
We establish a criterion for sheaves on an adically complete DG scheme to be coherent. We deduce a description of coherent sheaves on an adically complete lci singularity in terms of modules for a DG Lie algebra.
We show by explicit example that local intersection multiplicities in holomorphic dynamical systems can grow arbitrarily fast, answering a question of V. I. Arnold. On the other hand, we provide results showing that such behavior is exceptional, and that typically local intersection multiplicities grow subexponentially.
In this short note we prove a formula for local heights on elliptic curves over number fields in terms of intersection theory on a regular model over the ring of integers.
Intersection sheaves, i.e., the Deligne pairing, were first introduced by Deligne in the setting of Poincare duality for etale cohomology, and later in his work on the determinant of cohomology. Intersection sheaves were generalized from smooth schemes to Cohen-Macaulay schemes by Elkik, and then beyond Cohen-Macaulay schemes by Munoz-Garcia. To define the Abel maps arising in rational simple connectedness on the natural parameter spaces of rational sections, we need a variant of the construction of Munoz-Garcia that has the good properties of the construction using det and Div. In addition, w…
Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces
The School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
Let K be an algebraic number field and O_K the ring of integers of K. Let f : X --> Spec(O_K) be a stable arithmetic surface over O_K of genus g >= 2. In this short note, we will prove that if f has a reducible geometric fiber, then the self intersection of dualizing sheaf of X with Arakelov metric is greater than or equal to log(2)/6(g-1).
We study intersections of complex Lagrangian in complex symplectic manifolds, proving two main results. First, we construct global canonical perverse sheaves on complex Lagrangian intersections in complex symplectic manifolds for any pair of {\it oriented} Lagrangian submanifolds in the complex analytic topology. Our method uses classical results in complex symplectic geometry and some results from arXiv:1304.4508. The algebraic version of our result has already been obtained by the author et al. in arXiv:1211.3259 using different methods, where we used, in particular, the recent new theory of…
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 $\mathbf{M}_n$ 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.
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…
The method of intersection spaces associates rational Poincaré complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. Moreover, the intersection space complex …
We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration, generalizing the Kapranov's description of the derived category of a single quadric. As an application we verify that the noncommutative algebraic variety $(\PP(S^2W^*),\CB_0)$, where $\CB_0$ is the universal sheaf of even parts of Clifford alg…
In this paper, we consider a problem of counting multiplicities. We fix a counting function of multiplicity of rational points in a hypersurface of a projective space over a finite field, and we give an upper bound for the sum with respect to this counting function in terms of the degree of the hypersurface, the dimension and the cardinality of the finite field. This upper bound gives a description of the complexity of the singular locus of this hypersurface. In order to obtain this upper bound, we introduce a notion called intersection tree by intersection theory. We construct a sequence of i…
In two articles by Barthel, Brasselet, Fieseler and Kaup, and, Bressler and Lunts, a combinatorial theory of intersection cohomology and perverse sheaves has been developed on fans. In the first one, one tried to present everything on an elementary level,using only some commutative algebra and no derived categories. There remained two major gaps: First of all the Hard Lefschetz Theorem was only conjectured and secondly the intersection product seemed to depend on some non-canonical choices. Meanwhile the Hard Lefschetz theorem has been proved by Karu. The proof relies heavily on the intersecti…
For an irreducible subvariety Z in an algebraic group G we define a nonnegative integer gdeg(Z) as the degree, in a certain sense, of the Gauss map of Z. It can be regarded as a substitution for the intersection index of the conormal bundle to Z with the zero section of T^*G, even though G may be non-compact. For G a semiabelian variety (in particular, an algebraic torus (C^*)^n) we prove a Riemann-Roch-type formula for constructible sheaves on G, which involves our substitutions for the intersection indices. As a corollary, we get that a perverse sheaf on such a G has nonnegative Euler charac…
Given pure-dimensional (generalized) cycles $μ_1$ and $μ_2$ on a complex manifold $Y$ we introduce a product $μ_1\diamond_{Y} μ_2$ that is a generalized cycle whose multiplicities at each point are the local intersection numbers at the point. % If $Y$ is projective, then given a very ample line bundle $L\to Y$ we define a product $μ_1\bl μ_2$ whose multiplicities at each point also coincide with the local intersection numbers. In addition, provided that $μ_1$ and $μ_2$ are effective, this product satisfies a Bézout inequality. If $i\colon Y\to \Pk^N$ is an embedding such that $i^*\Ok(1)=L$, th…
In this paper, we investigate the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in term of the special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.
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 prove a characteristic p analogue of a result of Massey which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic cycle of a perverse sheaf in characteristic p by Saito. We apply this to prove a conjecture of Shende and Tsimerman on the Betti numbers of the intersections of two translates of theta loci in a hyperelliptic Jacobian. This implies a function field analogue of the Michel-Venkatesh mixing conjecture about the equidistribution of CM points…
We describe the E-infinity algebra structure on the complex of singular cochains of a topological space, in the context of sheaf theory. As a first application, for any algebraic variety we define a weight filtration compatible with its E-infinity structure. This naturally extends the theory of mixed Hodge structures in rational homotopy to p-adic homotopy theory. The spectral sequence associated to the weight filtration gives a new family of multiplicative algebraic invariants of the varieties for any coefficient ring, carrying Steenrod operations. As a second application, we promote Deligne'…
Starting from certain perverse sheaves on an abelian variety, including the intersection cohomology sheaves of curves and smooth ample divisors, we construct a semisimple super-Tannakian category.
We compute Betti numbers of both the components of the moduli space of rank 2 semi-stable torsion-free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.
As a continuation of the work of Freiermuth and Trautmann, we study the geometry of the moduli space of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $4m+1$. The moduli space has three irreducible components whose generic elements are, respectively, sheaves supported on rational quartic curves, on elliptic quartic curves, or on planar quartic curves. The main idea of the proof is to relate the moduli space with the Hilbert scheme of curves by wall crossing. We present all stable sheaves contained in the intersections of the three irreducible components. We also classify stable sheav…
By generalizing the Landau-Ginzburg/Calabi-Yau correspondence for hypersurfaces, we can relate a Calabi-Yau complete intersection to a hybrid Landau-Ginzburg model: a family of isolated singularities fibered over a projective line. In recent years Fan, Jarvis, and Ruan have defined quantum invariants for singularities of this type, and Clader and Clader-Ross have provided a equivalence between these invariants and Gromov-Witten invariants of complete intersections. For Calabi-Yau complete intersections of two cubics, we show that this equivalence is directly related - via Chen character - to t…
We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The decisive step is an exact description of the intersection of two Schubert varieties, from which the multiplicities (which are powers of 2) in the Pieri-type formula are immediately obvious.
Let Y,Z be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety X. We show that any data of first order deformation of the structure sheaf O_X to a sheaf of noncommutative algebras and of the sheaves O_Y and O_Z to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkoviski algebra structure on the Tor-sheaf Tor^{O_X}_*(O_Y, O_Z). The induced Gerstenhaber bracket on the Tor-sheaf turns out to be canonically defined; it is independent of the choices of deformations involved. There are similar results for Ext-sheaves as…
In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f_1,...,f_n are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f_1,...,f_n. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the…
We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme $\mathcal{M}(k), ~ k \geq 3$ of coherent semistable rank 2 sheaves with Chern classes $c_1=0,~ c_2=k,~ c_3=0$ on $\mathbb{P}^3$ whose general points are sheaves with singularities of mixed dimension. These sheaves are constructed by elementary transformations of stable and properly $μ$-semistable reflexive sheaves along disjoint union of collections of points and smooth irreducible curves which are rational or complete intersection curves. As a special member of this series we obtain a new compo…
We investigate the geometry of the Simpson moduli space M of stable sheaves on P_3 with Hilbert polynomial H(m)=3m+1 and describe explicitly the two smooth, rational components, their 11-dimensional smooth, transversal intersection and the relation between M and the Hilbert scheme Hilb_H(P_3) of cubic space curves. We also compute the Betti numbers of the Chow groups of the moduli space M.
The aim of this paper is to study the modified diagonal cycle in the triple product of a curve over a global field defined by Gross and Schoen. Our main result is an identity between the height of this cycle and the self-intersection of the relative dualising sheaf. We have some applications to the following problems in number theory and algebraic geometry.
We derive two consequences of the multivariate fundamental theorem of algebra (MFTA). The first one is the Bezout theorem for $n$ polynomials. Notably the intersection multiplicities, as in MFTA, are characterized just by means of partial differential operators given by polynomials from $D$-invariant linear spaces. The second consequence provides a solution to the ideal membership problem, based on the above characterization of intersection multiplicities. Let us mention that one readily gets Nullstellensatz from here.
This paper studies mixed multiplicities of an arbitrary standard bigraded algebra and mixed multiplicities of two ideals I, J in a local ring (A,m), where I is an m-primary ideal and J an arbitrary ideal. The main results are criteria for their positivity which can be used to compute them effectively. We also show that the range of positive mixed multiplicities of a bigraded algebra is rigid if the algebra satisfies the first chain condition and is connected in codimension one and that this range is always rigid for mixed multiplicities of ideals. These results can be used to study the mu-inva…
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 establish a close connection between intersection multiplicities of special cycles on arithmetic models of the Shimura variety for GU(1,2) and Fourier coefficients of derivatives of certain incoherent Eisenstein series, confirming a conjecture of Kudla and Rapoport. The paper is divided into two parts. The first part is the bulk of the paper and investigates intersections of local special cycles which are defined on a certain moduli space of p-divisible groups. This moduli space of p-divisible groups can be used for the uniformization of the completion of the Shimura variety along the super…
On a smooth projective threefold, we show that there are only two isomorphism types for the moduli of stable objects with respect to Bayer's standard polynomial Bridgeland stability - the moduli of Gieseker-stable sheaves and the moduli of PT-stable objects - under the following assumptions: no two of the stability vectors are collinear, and the degree and rank of the objects are relatively prime. We also describe a close relation between the intersection of the moduli spaces of PT-stable and dual-PT-stable objects, and the moduli of reflexive sheaves.
We prove that if X, X' are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C of Trop(X) \cap Trop(X') lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a suitable toric variety X(Δ) and its associated extended tropicalization N_R(Δ); the algebraic intersection points lifting the stable tropical intersection will have tropicalization somewhere in the …
Let $\mathbf{M}_d$ be the moduli space of stable sheaves on $\mathbb{P}^2$ with Hilbert polynomial $dm+1$. In this paper, we determine the effective and the nef cone of the space $\mathbf{M}_d$ by natural geometric divisors. Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the Grothendieck-Riemann-Roch theorem. We also present the stable base locus decomposition of the space $\mathbf{M}_6$. As a byproduct, we obtain the Betti numbers of the moduli spaces, which confirm the prediction i…
The present paper is an extension of a previous paper written in collaboration with Markus Reineke dealing with quiver representations. The aim of the paper is to generalize the theory and to provide a comprehensive theory of Donaldson-Thomas invariants for abelian categories of homological dimension one (without potential) satisfying some technical conditions. The theory will apply for instance to representations of quivers, coherent sheaves on smooth projective curves, and some coherent sheaves on smooth projective surfaces. We show that the (motivic) Donaldson-Thomas invariants satisfy the …
Let $X$ be an elliptic surface over ${\bf P}^1$ with $κ(X)=1$, and $M=M(c_2)$ be the moduli scheme of rank-two stable sheaves $E$ on $X$ with $(c_1(E),c_2(E))=(0,c_2)$ in $\operatorname{Pic}(X)\times\mathbb{Z}$. We look into defining equations of $M$ at its singularity $E$, partly because if $M$ admits only canonical singularities, then the Kodaira dimension $κ(M)$ can be calculated. We show the following. (A) $E$ is at worst canonical singularity of $M$ if the restriction of $E_η$ to the generic fiber of $X$ has no rank-one subsheaf, and if the number of multiple fibers of $X$ is a few. (B) W…
Let Y be a Calabi-Yau complete intersection in a weighted projective space. We show that the space of quadratic invariants of the hypergeometric group associated with the twisted I-function is one-dimensional, and spanned by the Gram matrix of a split-generator of the derived category of coherent sheaves on Y with respect to the Euler form.
We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The decisive step is an explicit description of the intersection of two Schubert varieties, from which the multiplicities (which are powers of 2) in the Pieri-type formula are deduced.
