Quiver varieties and Frenkel–Kac construction

An affine Lie algebra acts on cohomology groups of quiver varieties of affine type. A Heisenberg algebra acts on cohomology groups of Hilbert schemes of points on a minimal resolution of a Kleinian singularity. We show that in the case of type A the former is obtained by Frenkel–Kac construction from the latter.     

The Hilbert scheme of points for supersingular abelian surfaces

We study the geometry of Hilbert schemes of points on abelian surfaces and Beauville’s generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of two points to the abelian surface is a quasifibration such that all fibers are nonsmooth. In particular, the corresponding generalized Kummer surface is nonsmooth, and minimally elliptic singularities occur in the supersingular case. We unravel the structure of the singularities in dependence of p-rank and a-number of the abelian surface. To do so, we establish a McKay Correspondence for Artin’s wild involutions on surfaces. Along the line, we find examples of canonical singularities that are not rational singularities.     

Hilbert schemes of surfaces are dense in moduli

We prove that the locus of Hilbert schemes of n points on a projective K 3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds is proven as well. Along the way we prove an integral constraint on the monodromy group of generalized Kummer manifolds.     

On the Cohomology of Configuration Spaces on Surfaces

The integral cohomology rings of the configuration spaces ofn-tuples of distinct points on arbitrary surfaces (not necessarilyorientable, not necessarily compact and possibly with boundary)are studied. It is shown that for punctured surfaces the cohomologyrings stabilize as the number of points tends to infinity, similarlyto the case of configuration spaces on the plane studied byArnold, and the Goryunov splitting formula relating the cohomologygroups of configuration spaces on the plane and punctured planeto arbitrary punctured surfaces is generalized. Moreover, onthe basis of explicit cellular decompositions generalizing theconstruction of Fuchs and Vainshtein, the first cohomology groupsfor surfaces of low genus are given.     

Monomial multiple structures

In this paper we study Cohen–Macaulay monomial multiple structures (non-reduced schemes) on a linear subspace of codimension two in projective space. We show that these structures determine smooth points in their respective Hilbert schemes, with (smooth) neighbourhoods of two such points intersecting if their Hilbert functions are equal. We generalize a construction for multiple structures on points in the plane to this setting, giving a kind of product of monomial multiple structures. For points, this construction can be found in Nakajima’s book (Lectures on Hilbert schemes of points on surfaces, Univ Lecture Ser AMS, vol 18, 1999). The tools we use for studying multiple structures are developed in Vatne (Math Nachr 281(3):434–441, 2008; Comm Algebra 37(11):3861–3873, 2009) (see also Vatne in Towards a classification of multiple structures, PhD thesis, University of Bergen, 2001).     

Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces

Using vertex algebra techniques, we determine a set of generators for the cohomology ring of the Hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers. Received: 28 November 2000 / Published online: 23 May 2002     

Non-special scrolls with general moduli

Abstract  In this paper we study smooth, non-special scrolls S of degree d, genus g ≥ 0, with general moduli. In particular, we study the scheme of unisecant curves of a given degree on S. Our approach is mostly based on degeneration techniques. Keywords Ruled surfaces, Hilbert schemes of scrolls, Moduli, Embedded degenerations Mathematics Subject Classification (2000) 14J26, 14D06, 14C20, (Secondary) 14H60, 14N10     

Non degenerate projective curves with very degenerate hyperplane section

In this paper, we study the Hilbert scheme of non degenerate locally Cohen- Macaulay projective curves with general hyperplane section spanning a linear space of dimension 2 and minimal Hilbert function. The main result is that those curves are almost always the general element of a generically smooth component H_n,d,g of the corresponding Hilbert scheme. Moreover, we show that the curves with maximal cohomology almost always correspond to smooth points of H_n,d,g.All the authors were partially supported by Acción Integrada Italia-España, HI2000-0091, and by the Italian counterpart of the project.     

Graph theoretic techniques in algebraic geometry II: construction of singular complex surfaces of the rational cohomology type of CP2

Methods of graph theory are used to obtain rational projective surfaces with only rational double points as singularities and with rational cohomology rings isomorphic to that of the complex projective plane. Uniqueness results for such cohomologyCP ²'s and for rational and integral homologyCP ²'s are given in terms of the typesA _k,D _k, orE _k of singularities allowed by the construction. Supported in part by National Science Foundation grant no. MCS 77-03540.     

Vanishing of Chern Classes of the De Rham Bundles for Some Families of Moduli Spaces

Given a family of nonsingular complex projective surfaces, there is a corresponding family of Hilbert schemes of zero dimensional subschemes. We prove that the Chern classes, with values in the rational Chow groups, of the de Rham bundles for such a family of Hilbert schemes vanish. A similar result is proved for any relative moduli space of rank one sheaves with trivial integral first Chern class over any family of complex projective surfaces.     

Quantum cohomology of the Hilbert scheme of points in the plane

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of ℂ². The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. A relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves is proven.     

Frobenius splitting of Hilbert schemes of points on surfaces

Let X be a quasiprojective smooth surface defined over an algebraically closed field of positive characteristic. In this note we show that if X is Frobenius split then so is the Hilbert scheme Hilb of n points inX. In particular, we get the higher cohomology vanishing for ample line bundles on Hilb when X is projective and Frobenius split. Received November 2, 1999 / Published online March 12, 2001     

Galois representations modulo p and cohomology of Hilbert modular varieties

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let us mention:

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control of the image of Galois representations modulo p,

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Hida's congruence criterion outside an explicit set of primes,

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freeness of the integral cohomology of a Hilbert modular variety over certain local components of the Hecke algebra and Gorenstein property of these local algebras.

We study the arithmetic properties of Hilbert modular forms by studying their modulo p Galois representations and our main tool is the action of inertia groups at primes above p. In order to determine this action, we compute the Hodge-Tate (resp. Fontaine-Laffaille) weights of the p-adic (resp. modulo p) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of Siegel modular varieties and builds upon geometric constructions of Tilouine and the author.     

Critical values of automorphic L-functions for GL(<Emphasis Type="Italic">r</Emphasis>)×GL(<Emphasis Type="Italic">r</Emphasis>)

 We compute, up to an element of a fixed number field, the critical values of the L-function of a pair of automorphic, cuspidal, cohomological representations of any GL(r). The result is expressed as a product of cohomological periods divided by an archimedean integral. The main tool used is the rationality of the cohomology of the three representations involved in the Rankin–Selberg integral. As an intermediate step, we also obtained the rationality of the Eisenstein cohomology. Received: 14 February 2002 / Revised version: 17 September 2002 Published online: 14 February 2003     

Techniques,computations, and conjectures for semi-topological <Emphasis Type="Italic">K</Emphasis>-theory

We establish the existence of an Atiyah-Hirzebruch-like spectral sequence relating the morphic cohomology groups of a smooth, quasi-projective complex variety to its semi-topological K-groups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that relates the motivic cohomology and algebraic K-theory of varieties, and it is also compatible with the classical Atiyah-Hirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the Borel-Moore (singular) homology of complex varieties introduced by H. Gillet and C. Soulé – to compute the semi-topological K-theory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational three-folds, and related varieties, the semi-topological K-groups and topological K-groups are isomorphic in all degrees permitted by cohomological considerations. We also formulate integral conjectures relating semi-topological K-theory to topological K-theory analogous to more familiar conjectures (namely, the Quillen-Lichtenbaum and Beilinson-Lichtenbaum Conjectures) concerning mod-n algebraic K-theory and motivic cohomology. In particular, we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by A. Suslin. Our computations verify that these conjectures hold for the list of varieties above.Mathematics Subject Classification (2000): 19E20, 19E15, 14F43The first author was partially supported by the NSF and the NSAThe second author was supported by the Helen M. Galvin Fellowship of Northwestern UniversityThe third author was partially supported by the NSF and the NSA     

Un contre-exemple sur les families de courbes gauches

In this paper we give an example of two composants X and X,_o of the Hilbert scheme of space curves (i.e. components of subschemes with constant cohomology) satisfying the condition of semi-continuity (the cohomology of curves in X is less than the cohomology of curves in X,_o) but not the property X ₀?[Xbar]≠Ø.     

De Rham and Infinitesimal Cohomology in Kapranov's Model for Non commutative Algebraic Geometry

The title refers to the nilcommutative or NC-schemes introduced by M. Kapranov in Noncommutative Geometry Based on Commutator Expansions, J. Reine Angew. Math 505 (1998) 73–118. The latter are noncommutative nilpotent thickenings of commutative schemes. We also consider the parallel theory of nil-Poisson or NP-schemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for NC- and NP-schemes. The variants include nilcommutative and nil-Poisson versions of the de Rham complex as well as of the cohomology of the infinitesimal site introduced by Grothendieck in Crystals and the de Rham Cohomology of Schemes, Dix exposés sur la cohomologie des schémas, Masson, Paris (1968), pp. 306–358. It turns out that each of these noncommutative variants admits a kind of Hodge decomposition which allows one to express the cohomology groups of a noncommutative scheme Y as a sum of copies of the usual (de Rham, infinitesimal) cohomology groups of the underlying commutative scheme X (Theorems 6.1, 6.4, 6.7). As a byproduct we obtain new proofs for classical results of Grothendieck (Corollary 6.2) and of Feigin and Tsygan (Corollary 6.8) on the relation between de Rham and infinitesimal cohomology and between the latter and periodic cyclic homology.     

Approximating fixed points of holomorphic mappings in the Hilbert ball

We approximate fixed points of holomorphic and ρ-nonexpansive self-mappings of the Hilbert ball using both continuous and discrete schemes.