Integral operators and integral cohomology classes of Hilbert schemes

Abstract

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).

Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces

“…We claim, that φ × φ pulls back the class (27) to the class (36). The proposition follows from this claim, since the homomorphism (35) is injective.…”

“…The proposition follows from this claim, since the homomorphism (35) is injective. The claim follows from the equivalence (38) and the invariance of the class (36), under replacement of E by E ⊗ F −1 . The invariance follows from Eq.…”

“…and comparing both pullbacks to a third class (36). P is the projectivization of a global topological vector bundle, by Lemma …”

“…Zhenbo Qin and Weiqiang Wang have recently and independently obtained bases for the integral cohomology groups (modulo torsion) of Hilbert schemes of points on a projective surface X with vanishing H 1 (X, O X ) and H 2 (X, O X ) [36].…”

A transition matrix for two bases of the integral cohomology of the Hilbert scheme of points in the projective plane

“…In 2004, using Heisenberg operators, Qin and Wang ([22]) were able to find an integral basis for the integral cohomology of X [n] whenever X was a projective 1 surface such that H 1 (X; O X ) = H 2 (X; O X ) = 0 ( [22]). In 2008, Li and Qin ([16]) improved upon this result, now only requiring that X have vanishing odd Betti numbers ( [16]).…”