Some remarks on tautological sheaves on Hilbert schemes of points on a surface

We study tautological sheaves on the Hilbert scheme of points on a smooth quasi-projective algebraic surface by means of the Bridgeland–King–Reid transform. We obtain Brion–Danila’s Formulas for the derived direct image of tautological sheaves or their double tensor product for the Hilbert–Chow morphism; as an application we compute the cohomology of the Hilbert scheme with values in tautological sheaves or in their double tensor product, thus generalizing results previously obtained for tautological bundles.      

Local Gromov-Witten invariants and tautological sheaves on Hilbert schemes

Abstract We study the local Gromov-Witten invariants of O(k)⊕O(-k-2) → P1 by localization techniques and the Marino-Vafa formula, using suitable circle actions. They are identified with the equivariant Riemann-Roch indices of some power of the determinant of the tautological sheaves on the Hilbert schemes of points on the affine plane. We also compute the corresponding Gopakumar-Vafa invariants and make some conjectures about them.     

Chern classes of reflexive sheaves on normal surfaces

We define Chern classes of reflexive sheaves using Wahl's relative local Chern classes of vector bundles. The main result of the paper bounds contributions of singularities of a sheaf to the Riemann–Roch formula. Using it we are able to prove inequality in Wahl's conjecture on relative asymptotic RR formula for rank 2 vector bundles. Moreover, we prove that if Wahl's conjecture is true for a singularity then it is true for any its quotient. This implies Wahl's conjecture for quotient singularities and for quotients of cones over elliptic curves. Received March 2, 1998; in final form March 24, 1999 / Published online September 14, 2000     

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     

Generating series of classes of Hilbert schemes of points on orbifolds

The notion of power structure over the Grothendieck ring of complex quasi-projective varieties is used for describing generating series of classes of Hilbert schemes of zero-dimensional subschemes (“fat points”) on complex orbifolds.     

Chern classes of bundles on blown-up surfaces

Consider the blow upπ [Xtilde]→ Xof a complex surface Xat a point. Let [Etilde] be a holomorphic bundle on [Xtilde]whose restriction to the exceptional divisor is O(j)n ? O(-j), and define E+(π_* E*)^vv. Friedman and Morgan gave the following bounds for the second Chern classes j≤ c ₂([Etilde])- c ₂(E) ≤ j ²We show that these bounds are sharp.     

Chern classes in Deligne cohomology for coherent analytic sheaves

In this article, we construct Chern classes in rational Deligne cohomology for coherent sheaves on a smooth complex compact manifold. We prove that these classes satisfy the functoriality property under pullbacks, the Whitney formula and the Grothendieck–Riemann–Roch theorem for projective morphisms between smooth complex compact manifolds.     

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     

Segments and Hilbert schemes of points

Using results obtained from a study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to a degreverse term order (as, for example, the degrevlex order) in the Hilbert scheme of points in Pⁿ. In this context, we look into the properties of several types of “segment” ideals that we define and compare. This study also leads us to focus on the connections between the shape of generators of Borel ideals and the related Hilbert polynomial, thus providing an algorithm for computing all saturated Borel ideals with a given Hilbert polynomial.     

Flag Hilbert schemes and moduli spaces of torsion plane sheaves

We find certain relations between flag Hilbert schemes of points on plane curves and moduli spaces of one-dimensional plane sheaves. We show that some of these moduli spaces are stably rational.     

Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P².The generic noncommutative plane corresponds to the Sklyanin algebra S=Skl(E,σ) constructed from an automorphism σ of infinite order on an elliptic curve E⊂P². In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1−n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P²?E.     

Stable bundles with torsion Chern classes on non-Kählerian elliptic surfaces

If π:X→B is a non-Kählerian elliptic surface with generic fibreF, the moduli space of stable holomorphic vector bundles with torsion Chern classes onX has an induced fibred structure with base Pic^o(F) and the moduli space of stable parabolic bundles onB ^orb as fibre. This is specific to the non-Kähler case.     

Integral operators and integral cohomology classes of Hilbert schemes

The methods of integral operators on the cohomology of Hilbert schemes of points on surfaces are developed. They are used to establish integral bases for the cohomology groups of Hilbert schemes of points on a class of surfaces (and conjecturally, for all simply connected surfaces).Mathematics Subject Classification (2000): 14C05, 14F43, 17B69Partially supported by an NSF grant.     

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.     

The Chow ring of punctual Hilbert schemes on toric surfaces

Let X be a smooth projective toric surface, and the Hilbert scheme parametrizing the length d zero-dimensional subschemes of X. We compute the rational Chow ring . More precisely, if is the two-dimensional torus contained in X, we compute the rational equivariant Chow ring and the usual Chow ring is an explicit quotient of the equivariant Chow ring. The case of some quasi-projective toric surfaces such as the affine plane are described by our method too.     

On vector bundles over surfaces and Hilbert schemes

Let X be an irreducible smooth projective surface over ${{\mathbb{C}}}$ and Hilb ^d (X) the Hilbert scheme parametrizing the zero-dimensional subschemes of X of length d. Given a vector bundle E on X, there is a naturally associated vector bundle ${{\mathcal{F}}_d(E)}$ over Hilb ^d (X). If E and V are semistable vector bundles on X such that ${{\mathcal{F}}_d(E)}$ and ${{\mathcal{F}}_d(V)}$ are isomorphic, we prove that E is isomorphic to V. A key input in the proof is provided by Biswas and Nagaraj (see [1]).