Stable Maps and Branch Divisors

We construct a natural branch divisor for equidimensional projective morphisms where the domain has lci singularities and the target is nonsingular. The method involves generalizing a divisor construction of Mumford from sheaves to complexes. The construction is valid in flat families. The generalized branch divisor of a stable map to a nonsingular curve X yields a canonical morphism from the space of stable maps to a symmetric product of X. This branch morphism (together with virtual localization) is used to compute the Hurwitz numbers of covers of the projective line for all genera and degrees in terms of Hodge integrals.     

The Hodge conjecture for certain moduli varieties

For smooth projective varietiesX over ℂ, the Hodge Conjecture states that every rational Cohomology class of type (p, p) comes from an algebraic cycle. In this paper, we prove the Hodge conjecture for some moduli spaces of vector bundles on compact Riemann surfaces of genus 2 and 3.     

Hodge polynomials of the moduli spaces of rank 3 pairs

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple on X consists of two holomorphic vector bundles E ₁ and E ₂ over X and a holomorphic map . There is a concept of stability for triples which depends on a real parameter σ. In this paper, we determine the Hodge polynomials of the moduli spaces of σ-stable triples with rk(E ₁) = 3, rk(E ₂) = 1, using the theory of mixed Hodge structures. This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we recover the Hodge polynomial of the moduli space of odd degree rank 3 stable vector bundles.      

On K1 and K2 of Algebraic Surfaces

In this paper we study higher Chow groups of smooth, projective surfaces over a field k of characteristic zero, using some new Hodge theoretic methods which we develop for this purpose. In particular we investigate the subgroup of CH ^r+1 (X,r) with r = 1,2 consisting of cycles that are supported over a normal crossing divisor Z on X. In this case, the Hodge theory of the complement forms an interesting variation of mixed Hodge structures in any geometric deformation of the situation. Our main result is a structure theorem in the case where X is a very general hypersurface of degree d in projective 3-space for d sufficiently large and Z is a union of very general hypersurface sections of X. In this case we show that the subgroup of CH ^r+1 (X,r) we consider is generated by obvious cycles only arising from rational functions on X with poles along Z. This can be seen as a generalization of the Noether–Lefschetz theorem for r = 0. In the case r = 1 there is a similar generalization by Müller-Stach, but our result is more precise than it, since it is geometric and not only cohomological. The case r = 2 is entirely new and original in this paper. For small d, we construct some explicit examples for r = 1 and 2 where the corresponding higher Chow groups are indecomposable, i.e. not the image of certain products of lower order groups. In an appendix Alberto Collino constructs even more indecomposable examples in CH ³ (X,2) which move in a one-dimensional family on the surface X.Contribution to appendix.     

Half twists and the cohomology of hypersurfaces

A Hodge structure V of weight k on which a CM field acts defines, under certain conditions, a Hodge structure of weight , its half twist. In this paper we consider hypersurfaces in projective space with a cyclic automorphism which defines an action of a cyclotomic field on a Hodge substructure in the cohomology. We determine when the half twist exists and relate it to the geometry and moduli of the hypersurfaces. We use our results to prove the existence of a Kuga-Satake correspondence for certain cubic 4-folds. Received: 25 August 2000; in final form: 8 January 2001 / Published online: 18 January 2002     

A Serre-Swan Theorem for Bundles of Topological Modules

Let R be a unital topological ring whose set of invertible elements is open and inversion is continuous, and let X be a compact Hausdorff space admitting continuous R-valued partitions of unity. Considering bundles over X of fibre type a projective finitely generated R-module, we prove a Serre-Swan type theorem: namely, the category of these bundles is equivalent to the category of projective finitely generated modules over the ring of continuous R-valued functions on X.     

Torelli Theorem for the Moduli Spaces of Rank 2 Quadratic Pairs

Let X be a smooth projective complex curve. We prove that a Torelli type theorem holds, under certain conditions, for the moduli space of α-polystable quadratic pairs on X of rank 2.     

The 2-factoriality of the O’Grady moduli spaces

The aim of this work is to show that the moduli space M ₁₀ introduced by O’Grady is a 2-factorial variety. Namely, M ₁₀ is the moduli space of semistable sheaves with Mukai vector v: = (2, 0, −2) in H^ev(X,\mathbbZ){H^{ev}(X,\mathbb{Z})} on a projective K3 surface X. As a corollary to our construction, we show that the Donaldson morphism gives a Hodge isometry between v^{^}{v^{\perp}} (sublattice of the Mukai lattice of X) and its image in H² ([(M)\tilde]₁₀, \mathbbZ){H^{2} (\widetilde{M}_{10}, \mathbb{Z})}, lattice with respect to the Beauville form of the 10-dimensional irreducible symplectic manifold [(M)\tilde]₁₀{\widetilde{M}_{10}}, obtained as symplectic resolution of M ₁₀. Similar results are shown for the moduli space M ₆ introduced by O’Grady to produce its 6-dimensional example of irreducible symplectic variety.     

Existence of a non‐reflexive embedding with birational Gauss map for a projective variety

We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in ℙ^N to re‐embed into some projective space ℙ^M so as to be non‐reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in ℙ^N is identically zero; hence the projective varietyX re‐embedded in ℙ^M yields a negative answer to Kleiman–Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in ℙ^N with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙ^N satisfying the assumption. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

SU X (r, L) is separably unirational

We show that the moduli space of SU _X (r, L) of rank r bundles of fixed determinant L on a smooth projective curve X is separably unirational.      

Motivation for Hodge cycles

Given two smooth projective varieties X and Y, X is defined to motivate Y if the motive of Y is contained in the tensor category generated by X. Some techniques are given for checking this condition. It is shown that in a number of cases moduli spaces of sheaves over curves or surfaces are motivated by the underlying curve or surface. This is used to check the Hodge and related conjectures for some of these examples.     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998     

CR Structures on the 3‐Sphere and the Blow‐Up of the Projective Plane

Motivated by a problem of characterizing CR‐structures on the 3‐sphere, we give a geometric construction of formal deformations of a complex surface, which is the complement of a ball in the projective plane. They are described by cohomology groups of the blow‐up X of the projective plane. Moreover it will be shown that the space of these formal deformations is an infinite dimensional space with a natural stratification by finite dimensional subspaces. This stratification re ects algebro‐geometric properties of X. It is expected that our construction will clarify the complex geometric nature of the space of non‐embeddable CR‐structures on the 3‐sphere.     

Stringy E-functions of varieties with A-D-E singularities

The stringy E-function for normal irreducible complex varieties with at worst log terminal singularities was introduced by Batyrev. It is defined by data from a log resolution. If the variety is projective and Gorenstein and the stringy E-function is a polynomial, Batyrev also defined the stringy Hodge numbers as a generalization of the Hodge numbers of nonsingular projective varieties, and conjectured that they are nonnegative. We compute explicit formulae for the contribution of an A-D-E singularity to the stringy E-function in arbitrary dimension. With these results we can say when the stringy E-function of a variety with such singularities is a polynomial and in that case we prove that the stringy Hodge numbers are nonnegative. Research Assistant of the Fund for Scientific Research - Flanders (Belgium) (F.W.O.),     

Fréchet Spaces of Holomorphic Functions without Copies of l1

Let X be a Banach space. Let H_w*(X*) the Fréchet space whose elements are the holomorphic functions defined on X* whose restrictions to each multiple mB(X*), m = 1,2, …, of the closed unit ball B(X*) of X* are continuous for the weak-star topology. A fundamental system of norms for this space is the supremum of the absolute value of each element of H_w*(X*) in mB(X*), m = 1,2,…. In this paper we construct the bidual of l₁ when this space contains no copy of l¹. We also show that if X is an Asplund space, then H_w*(X*) can be represented as the projective limit of a sequence of Banach spaces that are Asplund.     

On images of weak Fano manifolds

We consider a smooth projective morphism between smooth complex projective varieties. If the source space is a weak Fano (or Fano) manifold, then so is the target space. Our proof is Hodge theoretic. We do not need mod p reduction arguments. We also discuss related topics and questions.     

Standard P3-bundles of exponent two on algebraic surfaces

Let V_K be a twisted form of P³ over the function field of an algebraic surface with Pic V_K generated by the half of the canonical line bundle. We construct an algebraic fibre space V → X projective flat over a smooth projective surface X with the generic fibre V _K → Spac K satisfying some properties.     

Projective resolutions associated to projections

In this paper we will describe projective resolutions of d dimensional Cohen–Macaulay spaces X by means of a projection of X to a hypersurface in d+1-dimensional space. We will show that for a certain class of projections, the resulting resolution is minimal. Received: 22 February 1999     

An extension theorem for holomorphic functions of slow growth on covering spaces of projective manifolds

LetX be a projective manifold of dimension n ≥ 2 andY →X an infinite covering space. EmbedX into projective space by sections of a sufficiently ample line bundle. We prove that any holomorphic function of sufficiently slow growth on the preimage of a transverse intersection ofX by a linear subspace of codimension <n extends toY. The proof uses a Hausdorff duality theorem for L₂ cohomology. We also show that every projective manifold has a finite branched covering whose universal covering space is Stein.