Related papers: Minimal Gromov--Witten ring

We describe the quantum cohomology rings of a class of toric varieties. The description includes, in addition to the (already known) ring presentations, the (new) analogues for toric varieties of the sorts of quantum Giambelli formulas…

The quantum cohomology algebra of the (full) flag manifold is a fundamental example in quantum cohomology theory, with connections to combinatorics, algebraic geometry, and integrable systems. Using a differential geometric approach, we…

We use a topological framework to study descendent Gromov-Witten theory in higher genus, non-toric settings. Two geometries are considered: surfaces of general type and the Enriques Calabi-Yau threefold. We conjecture closed formulas for…

We study the degeneration and the gluing of Kuranishi structures in Gromov-Witten theory under a symplectic cut. This leads us to a degeneration axiom and a gluing axiom for open Gromov-Witten invariants. They provide then a route to the…

We study the enumerative geometry of stable maps to Calabi-Yau 5-folds $Z$ with a group action preserving the Calabi-Yau form. In the central case $Z=X \times \mathbb{C}^2$, where $X$ is a Calabi-Yau 3-fold with a group action scaling the…

For any smooth complex projective variety X and smooth very ample hypersurface Y in X, we develop the technique of genus zero relative Gromov-Witten invariants of Y in X in algebro-geometric terms. We prove an equality of cycles in the Chow…

We give a graph-sum algorithm that expresses any genus-$g$ Gromov-Witten invariant of the symmetric product orbifold $\mathrm{Sym}^d\mathbb{P}^r:=[(\mathbb{P}^r)^d/S_d]$ in terms of "Hurwitz-Hodge integrals" -- integrals over (compactified)…

We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of…

We introduce Gromov-Witten invariants with naive tangency conditions at the marked points of the source curve. We then establish an explicit formula which expresses Gromov-Witten invariants with naive tangency conditions in terms of…

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…

We use Noether-Lefschetz theory to study the reduced Gromov--Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$-type. This yields strong evidence for a new conjectural formula that expresses Gromov-Witten invariants of this…

This note compares the usual (absolute) Gromov-Witten invariants of a symplectic manifold with the invariants that count the curves relative to a (symplectic) divisor D. We give explicit examples where these invariants differ even though it…

We prove that genus zero Gromov--Witten invariants of a smooth scheme relative to a smooth divisor coincide with genus zero orbifold Gromov--Witten invariants of an appropriate root stack construction along the divisor.

Let $X$ be a smooth complex projective algebraic variety. Let $\mathcal{G}$ be a $G$-banded gerbe with $G$ a finite abelian group. We prove an exact formula expressing genus $g$ orbifold Gromov-Witten invariants of $\mathcal{G}$ in terms of…

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…

This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology $QH_G(X)$ of a smooth complex projective variety X with the action of a connected complex reductive…

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…

In this paper, we compute the open Gromov-Witten invariants for every compact toric surface X which is semi-Fano (i.e. the anticanonical line bundle is nef). Unlike the Fano case, this involves non-trivial obstructions in the corresponding…

We compute the relative orbifold Gromov-Witten invariants of $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$, with respect to vertical fibers. Via a vanishing property of the Hurwitz-Hodge bundle, 2-point rubber invariants are…

We make precise conjectures relating the genus zero Gromov-Witten theory of a nonabelian GIT quotient X//G to that of the associated abelian quotient X//T by a maximal torus T in G.These conjectures imply in particular closed formulas for…