Related papers: Hypertoric geometry and Gromov-Witten theory

For a gerbe $\Y$ over a smooth proper Deligne-Mumford stack $\B$ banded by a finite group $G$, we prove a structure result on the Gromov-Witten theory of $\Y$, expressing Gromov-Witten invariants of $\Y$ in terms of Gromov-Witten invariants…

Given a smooth projective variety $X$ with a smooth nef divisor $D$ and a positive integer $r$, we construct an $I$-function, an explicit slice of Givental's Lagrangian cone, for Gromov--Witten theory of the root stack $X_{D,r}$. As an…

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov-Witten classes, and a…

We develop differential and symplectic geometry of differentiable Deligne-Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks as…

This is the companion article to the Bourbaki talk of the same name given in March 2009. The main theme of the talk and the article is to explain the interplay between homotopy theory and algebraic geometry through the Hopkins-Miller-Lurie…

We propose an intersection-theoretic method to reduce questions in genus zero logarithmic Gromov-Witten theory to questions in the Gromov-Witten theory of smooth pairs, in the presence of positivity. The method is applied to the enumerative…

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Witten theory of P^1. A TQFT formalism is defined via…

For a complex projective manifold Gromov-Witten invariants can be constructed either algebraically or symplectically. Using the versions of Gromov-Witten theory by Behrend and Fantechi on the algebraic side and by the author on the…

We define a new Gromov-Witten theory relative to simple normal crossing divisors as a limit of Gromov-Witten theory of multi-root stacks. Several structural properties are proved including relative quantum cohomology, Givental formalism,…

K-theoretic Gromov-Witten invariants of a compact Kahler manifold $X$ are defined as super-dimensions of sheaf cohomology of interesting bundles over moduli spaces of n-pointed holomorphic curves in X. With this article, we begin a series…

This expository article is an introduction to logarithmic Gromov--Witten (GW) theory. We discuss how to study the GW theory of a smooth projective variety via simple normal crossings degenerations. We survey several approaches to…

Given an action of a reductive group on a normal variety, we construct all invariant open subsets admitting a good quotient with a quasiprojective or a divisorial quotient space. Our approach extends known constructions like Mumford's…

For G a complex reductive group and X a smooth projective or convex quasi-projective polarized G-variety we construct a formal map in quantum K-theory from the equivariant quantum K-theory $QK^G(X)$ to the quantum K-theory of the git…

For any finite abelian group G, the equivariant Gromov-Witten invariants of C^r/G can be viewed as a certain kind of abelian Hurwitz-Hodge integrals. In this note, we use Tseng's orbifold quantum Riemann-Roch theorem to express this kind of…

We generalize the First Reconstruction Theorem of Kontsevich and Manin in two respects. First, we allow the target space to be a Deligne-Mumford stack. Second, under some convergence assumptions, we show it suffices to check the hypothesis…

We use Gromov-Witten theory to study rational curves in holomorphic symplectic varieties. We present a numerical criterion for the existence of uniruled divisors swept out by rational curves in the primitive curve class of a very general…

In this expository manuscript, we review the construction of Gromov-Witten virtual fundamental class via FOOO's theory of Kuranishi structures for moduli spaces of pseudo-holomorphic maps defined on closed Riemann surfaces. We consider…

We establish a genus zero correspondence between the equivariant Gromov-Witten theory of the Deligne-Mumford stack $[\mathbb{C}^N/G]$ and its blowup at the origin. The relationship generalizes the crepant transformation conjecture of…

Based on the methods used by the author to prove the Riemann-Roch formula for algebraic stacks, this paper contains a description of the rationnal G-theory of Deligne-Mumford stacks over general bases. We will use these results to study…

In \cite{TY18}, higher genus Gromov--Witten invariants of the stack of $r$-th roots of a smooth projective variety $X$ along a smooth divisor $D$ are shown to be polynomials in $r$. In this paper we study the degrees and coefficients of…