Related papers: A quantum splitting principle and an application
We prove a conjecture of Artur Elezi in a generalized form suggested by Givental. Namely, our main result relates genus-0 Gromov--Witten invariants of a bundle space with such invariants of the base, provided that the fiber is a toric…
We prove genus $g$ invariants in quantum $K$-theory are determined by genus zero invariants of a smooth stack in the spirit of K.~Costello's result in Gromov--Witten theory.
We study the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of n points of the plane. Since the equivariant quantum cohomology is semisimple, the higher genus theory is determined by an R-matrix via the Givental-Teleman…
We study stable maps to normal crossings pairs with possibly negative tangency orders. There are two independent models: punctured Gromov-Witten theory of pairs and orbifold Gromov-Witten theory of root stacks with extremal ages. Exploiting…
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…
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 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 prove a decomposition theorem for the quantum cohomology of variations of GIT quotients. More precisely, for any reductive group $G$ and a simple $G$-VGIT wall-crossing $X_- \dashrightarrow X_+$ with a wall $S$, we show that the quantum…
This is the third of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, then the given vector field can be straightened (ie, made parallel to the given R direction) by an…
Let X be a smooth complex projective variety, and let Y in X be a smooth very ample hypersurface such that -K_Y is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) 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 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…
In this article, we study the change of genus zero Gromov-Witten invariants under cubic extremal transitions, following Lee-Lin-Wang [arXiv:1705.04799]. We use the language of quantum $D$-modules.
In this work, we adopt the Gradient Projection Method (GPM) to problems of quantum control. For general $N$-level closed and open quantum systems, we derive the corresponding adjoint systems and gradients of the objective functionals, and…
Givental has defined a Lagrangian cone in a symplectic vector space which encodes all genus-zero Gromov-Witten invariants of a smooth projective variety X. Let Y be the subvariety in X given by the zero locus of a regular section of a…
Let $G$ be a complex reductive group and let $X$ and $E$ be two linear representations of $G$. Let $Y$ be a complete intersection in $X$ equal to the zero locus of a $G$-equivariant section of the trivial bundle $E \times X \to X$. We…
We generalize Horrocks' criterion for the splitting of vector bundles on projective space. We establish an analogous splitting criterion for vector bundles on a class of smooth complex projective varieties of dimension at least four, over…
In this article we propose a definition of super Gromov-Witten invariants by postulating a torus localization property for the odd directions of the moduli spaces of super stable maps and super stable curves of genus zero. That is, we…
This paper reviews recent work on a new geometric object called a bundle gerbe and discusses some new examples arising in quantum field theory. One application is to an Atiyah-Patodi-Singer index theory construction of the bundle of…
We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums…