Abstract. 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.
Abstract. 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).
Given a closed complex manifold X of even dimension, we develop a systematic (vertex) algebraic approach to study the rational orbifold cohomology rings H * orb (X n /Sn) of the symmetric products. We present constructions and establish results on the rings H * orb (X n /Sn) including two sets of ring generators, universality and stability, as well as connections with vertex operators and W algebras. These are independent of but parallel to the main results on the cohomology rings of the Hilbert schemes of points on surfaces as developed in our earlier works joint with W.-P. Li. We introduce a deformation of the orbifold cup product and explain how it is reflected in terms of modification of vertex operators in the symmetric product case. As a corollary, we obtain a new proof of the isomorphism between the rational cohomology ring of Hilbert schemes X [n] and the ring H * orb (X n /Sn) (after some modification of signs), when X is a projective surface with a numerically trivial canonical class; we show that no sign modification is needed if both cohomology rings use C-coefficients.
In this paper, we apply the technique of chamber structures of stability polarizations to construct the full moduli space of rank-2 stable sheaves with certain Chern classes on Calabi–Yau manifolds which are anti-canonical divisor of ℙ1×ℙn or a double cover of ℙ1×ℙn. These moduli spaces are isomorphic to projective spaces. As an application, we compute the holomorphic Casson invariants defined by Donaldson and Thomas.
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