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)
IntroductionRecently, Seiberg and Witten [W] introduced new invariants of 4-manifolds, which are defined by counting solutions of a certain non-linear differential equation.The new invariants are expected to be equivalent to Donaldson's polynomial-invariants-at least for manifolds of simple type [KM 1]-and they have already found important applications, like e.g. in the proof of the Thom conjecture by Kronheimer and Mrowka [KM 2].Nevertheless, the equations themselves remain somewhat mysterious, especially from a mathematical point of view.The present paper contains our attempt to understand and to generalize the Seiberg-Witten equations by coupling them to connections in unitary vector bundles, and to relate their solutions to more familiar objects, namely stable pairs. Fix a Spin c -structure on a Riemannian 4-manifold X, and denote by Σ ± the associated spinor bundles. The equations which we will study are: * Partially supported by: AGE-Algebraic Geometry in Europe, contract No ER-BCHRXCT940557 (BBW 93.0187), and by SNF, Proposition 1.4 Let J be a g-orthogonal almost complex structure on X, compatible with the orientation. i) The spinor bundles Σ ± J ofP J are:is given by Λ 20 J ⊕Λ 02 J ⊕Λ 00 ω g ∋ (λ 20 , λ 02 , ω g ) Γ −→ 2 −i − * (λ 20 ∧ ·) λ 02 ∧ · i ∈ End 0 (Λ 00 ⊕Λ 02 ).
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