Abstract not found

We prove a localization formula for the virtual fundamental class in the general context of C∗-equivariant perfect obstruction theories. Let X be an algebraic scheme with a C∗-action and a C∗-equivariant perfect obstruction theory. The virtual fundamental class [X] vir in

Gromov-Witten invariants for arbitrary projective varieties and arbitrary genus are constructed using the techniques from [K. Behrend, B. Fantechi. The Intrinsic Normal Cone.]

Let Mg,n be the nonsingular moduli stack of genus g, n-pointed, Deligne-Mumford stable curves. For each marking i, there is an associated cotangent line bundle Li → Mg,n with fiber T ∗ C,pi over the moduli point [C, p1,...,pn]. Let ψi = c1(Li) ∈ H ∗ (Mg,n, Q). The integrals of products of the ψ classes

We briefly review the formal picture in which a Calabi-Yau n-fold is the complex analogue of an oriented real n-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of [LT], [BF] in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in P 3, and Donaldson – and Gromov-Witten – like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold X, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general K3 fibration X, enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the “Mukai-dual ” 3-fold for others. As an example the invariant is shown to distinguish Gross ’ diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of X. 1

We describe a formalism based on quantization of quadratic hamiltonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about Gromov-Witten invariants of compact symplectic manifolds and, more generally, Frobenius structures at higher genus. We state several results illustrating the formalism and its use. In particular, we establish Virasoro constraints for semisimple Frobenius structures and outline a proof of the Virasoro conjecture for Gromov – Witten invariants of complex projective spaces and other Fano toric manifolds. Details will be published elsewhere.

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We find a new relation among codimension 2 algebraic cycles in the moduli space M1,4, and use this to calculate the elliptic Gromov-Witten invariants of projective spaces CP² and CP³.

The Gromov-Witten theory of a 3-fold X is defined via integrals over the moduli space of stable maps. The Donaldson-Thomas theory of X is defined

Let W/C be a degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first construct the stack of expanded degenerations of W. We then construct the moduli space of stable morphisms to this stack, which provides a degeneration of the moduli spaces of stable morphisms associated to W/C. Using a similar technique, for a pair (Z, D) of smooth variety and a smooth divisor, we construct the stack of expanded relative pairs and then the moduli spaces of relative stable morphisms to (Z, D). This is the algebro-geometric analogue of Donaldson-Floer theory in gauge theory. The construction of relative Gromov-Witten invariants and the degeneration formula of Gromov-Witten invariants will be treated in the subsequent paper.