Gromov-Witten invariants of blow-ups along submanifolds with convex normal bundles

Given a submanifold Z inside X, let Y be the blow-up of X along Z. When the normal bundle of Z in X is convex with a minor assumption, we prove that genus-zero GW-invariants of Y with cohomology insertions from X, are identical to GW-invariants of X. Under the same hypothesis, a vanishing theorem is also proved. An example to which these two theorems apply is when the normal bundle is generated by global sections. These two main theorems do not hold for arbitrary blow-ups, and counter-examples are included.Comment: 34 page

Curves on K3 surfaces and modular forms

We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equivalences relating the reduced Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable pairs moduli spaces are proven. As a consequence, we prove the Katz-Klemm-Vafa conjecture evaluating $\lambda_g$ integrals (in all genera) in terms of explicit modular forms. Indeed, all K3 invariants in primitive classes are shown to be governed by modular forms. The method of proof is by degeneration to elliptically fibered rational surfaces. New formulas relating reduced virtual classes on K3 surfaces to standard virtual classes after degeneration are needed for both maps and sheaves. We also prove a Gromov-Witten/Pairs correspondence for toric 3-folds. Our approach uses a result of Kiem and Li to produce reduced classes. In Appendix A, we answer a number of questions about the relationship between the Kiem-Li approach, traditional virtual cycles, and symmetric obstruction theories. The interplay between the boundary geometry of the moduli spaces of curves, K3 surfaces, and modular forms is explored in Appendix B by A. Pixton.Comment: An incorrect example in Appendix A, pointed out to us by Dominic Joyce, has been replaced by a reference to a new paper arXiv:1204.3958 containing a corrected exampl

Witt groups of sheaves on topological spaces

This paper investigates the Witt groups of triangulated categories of sheaves (of modules over a ring R in which 2 is invertible) equipped with Poincare-Verdier duality. We consider two main cases, that of perfect complexes of sheaves on locally compact Hausdorff spaces and that of cohomologically constructible complexes of sheaves on polyhedra. We show that the Witt groups of the latter form a generalised homology theory for polyhedra and continuous maps. Under certain restrictions on the ring R, we identify the constructible Witt groups of a finite simplicial complex with Ranicki's free symmetric L-groups. Witt spaces are the natural class of spaces for which the rational intersection homology groups have Poincare duality. When the ring R is the rationals we show that every Witt space has a natural L-theory, or Witt, orientation and we identify the constructible Witt groups with the 4-periodic colimit of the bordism groups of Witt spaces. This allows us to interpret Goresky and Macpherson's L-classes of singular spaces as stable homology operations from the constructible Witt groups to rational homology.Comment: 38 pages, reformatted, minor corrections and changes as suggested by referee. To appear in Commentarii Mathematici Helvetici no. 8

Degeneracy loci, virtual cycles and nested Hilbert schemes I

Given a map of vector bundles on a smooth variety, consider the deepest degeneracy locus where its rank is smallest. We show it carries a natural perfect obstruction theory whose virtual cycle can be calculated by the Thom-Porteous formula. We show nested Hilbert schemes of points on surfaces can be expressed as degeneracy loci. We show how to modify the resulting obstruction theories to recover the virtual cycles of Vafa-Witten and reduced local DT theories. The result computes some Vafa-Witten invariants in terms of Carlsson-Okounkov operators. This proves and extends a conjecture of Gholampour-Sheshmani-Yau and generalises a vanishing result of Carlsson-Okounkov.Comment: Published version. 29 page

Virtual push-forwards

Let $p:F\to G$ be a morphism of stacks of positive \emph{virtual} relative dimension $k$ and let $\gamma\in H^k(F)$. We give sufficient conditions for $p_*\gamma\cdot[F]^{virt}$ to be a multiple of $[G]^{virt}$. We apply this result to show an analogue of the conservation of number for virtually smooth families. We show implications to Gromov-Witten invariants and give a new proof of a theorem of Marian, Oprea and Pandharipande which compares the virtual classes of moduli spaces of stable maps and moduli spaces of stable quotients