We construct an obstruction theory for relative Hilbert schemes in the sense of [BF] and compute it explicitly for relative Hilbert schemes of divisors on smooth projective varieties. In the special case of curves on a surface V , our obstruction theory determines a virtual fundamental class [[Hilb m V ]] ∈ A m(m−k)

No abstract

We study the solutions of the Seiberg-Witten equations on complex surfaces. We show that for a large class of parameters, the gauge equivalence classes of irreducible solutions of the twisted Seiberg-Witten equations correspond to stable Witten triples. We prove that on Kähler surfaces this correspondence is the set-theoretical support of an isomorphism of real-analytic spaces. This makes it possible to take multiplicities into account and generalizes and unifies results previously obtained by Witten. IntroductionWe study the Seiberg-Witten equations and the moduli spaces of monopoles on complex surfaces. The main objective of this article is to state a Kobayashi-Hitchin correspondence.In his article [28], Witten showed that on a Kähler surface, for certain perturbations the moduli space of irreducible monopoles corresponds set-theoretically to decompositions of a canonical divisor into two effective divisors. If one wants to compute the Seiberg-Witten invariants, such an information does not suffice. One also has to compute the multiplicities of the solutions. In order to endow the set of decompositions with the structure of a complex-analytic space, we introduce the concept of a Witten triple and study the moduli space of stable Witten triples. Our version of the Kobayashi-Hitchin correspondence then compares the moduli space of irreducible Seiberg-Witten monopoles with the analytic space of stable Witten triples, and thus makes more precise, generalizes and unifies the previously known versions of this correspondence.In the literature, Kobayashi-Hitchin correspondences for Seiberg-Witten moduli spaces have been considered for two kinds of perturbations:(a) Perturbing the term F + A by a pure imaginary self-dual form 2πiβ + , where β is a real 2-form of type (1, 1); (b) Perturbing the term F + A by a pure imaginary self-dual formη − η, where η is a holomorphic 2-form. In the first case, the moduli spaces of irreducible monopoles can be identified with Douady spaces of divisors with bounded volume (see [28], [22], [24], [20]). In the second case, one can identify the moduli spaces of irreducible monopoles as a set with the set of decompositions of the canonical divisor (η) into two effective divisors (see [28] Our goal is twofold: we want to approach both kinds of Kobayashi-Hitchin correspondences in a uniform way, i.e. we want to describe the moduli space of irreducible monopoles, where we perturb the term F + A by a pure imaginary self-dual form 2πiβ, where β is a real 2-form of type (1, 1) and η is a holomorphic 2-form. Secondly, we want to endow the set of decompositions of the canonical divisor (η) into two effective divisors with the structure of a complex-analytic space.On a complex surface, any Hermitian metric gives rise to a canonical Spin c (4)-structure τ can . Moreover, any other Spin c (4)-structure is of the form τ M , where M is a Hermitian line bundle. If the Hermitian metric g is Kähler, the Seiberg-Witten equations take a specially simple form. By perturbing the Dirac operator by the term γ(θ...

In [DKO] we constructed virtual fundamental classes [[Hilb m V ]] for Hilbert schemes of divisors of topological type m on a surface V , and used these classes to define the Poincaré invariant of V :We conjecture that this invariant coincides with the full Seiberg-Witten invariant computed with respect to the canonical orientation data.In this note we prove that the existence of an integral curve C ⊂ V induces relations between some of these virtual fundamental classes [[Hilb m V ]]. The corresponding relations for the Poincaré invariant can be considered as algebraic analoga of the fundamental relations obtained in [OS].