On a compact Kähler manifold $X$ with a holomorphic 2-form $\a$, there is an almost complex structure associated with $\a$. We show how this implies vanishing theorems for the Gromov-Witten invariants of $X$. This extends the approach, used in \cite{lp} for Kähler surfaces, to higher dimensions.
We provide an enumerative meaning of the mirror maps for toric Calabi-Yau orbifolds in terms of relative Gromov-Witten invariants of the toric compactifications. As a consequence, we obtain an equality between relative Gromov-Witten invariants and open Gromov-Witten invariants. Therefore, the instanton corrected mirrors for toric Calabi-Yau orbifolds can be constructed using relative Gromov-Witten invariants.
In the paper, we study the wall-crossing phenomenon of reduced open Gromov-Witten invariants on K3 surfaces with rigid special Lagrangian boundary condition. As a corollary, we derived the multiple cover formula for the reduced open Gromov-Witten invariants.
We study genus 1 Gromov-Witten invariants of Fano complete intersections in the projective spaces. Among other things, we show a reconstruction theorem for genus 1 invariants with only ambient insertions, and compute the genus 1 invariants with 1 marked point. For cubic hypersurfaces of dimension $\neq 4$ and odd dimensional intersections of two quadrics, we obtain a complete reconstruction theorem for genus 1 Gromov-Witten invariants.
We describe generating functions for arbitrary-genus Gromov-Witten invariants of the projective space with any number of marked points explicitly. The structural portion of this description gives rise to uniform energy bounds and vanishing results for these invariants. They suggest deep conjectures relating Gromov-Witten invariants of symplectic manifolds to the energy of pseudo-holomorphic maps and the expected dimension of their moduli space.
We provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov-Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten in…
Degeneration and gluing of Kuranishi structures in Gromov-Witten theory and the degeneration/gluing axioms for open Gromov-Witten invariants under a symplectic cut
We show that it is possible to define the contribution of degree one covers of a disk to open Gromov-Witten invariants. We build explicit sections of obstruction bundles in order to extend the algebro-geometric techniques of Pandharipande to the case of domains with boundary.
In our previous work, we provided an algebraic proof of the Zinger's comparison formula between genus one Gromov-Witten invariants and reduced invariants when the target space is a complete intersection of dimension two or three in a projective space. In this paper, we extend the result in any dimensions and for descendant invariants.
In this paper, we study holomorphic discs in K3 surfaces and defined the open Gromov-Witten invariants. Using this new invariant, we can establish a version of correspondence between tropical discs and holomorphic discs with non-trivial invariants. We give an example of wall-crossing phenomenon of the invariant and expect it satisfies Kontsevich-Soibelman wall-crossing formula.
We prove two tropical gluing formulae for Gromov-Witten invariants of exploded manifolds, useful for calculating Gromov-Witten invariants of a symplectic manifold using a normal-crossing degeneration. The first formula generalizes the symplectic-sum formula for Gromov-Witten invariants. The second formula is stronger, and also generalizes Kontsevich and Manin's splitting and genus-reduction axioms. Both tropical gluing formulae have versions incorporating gravitational descendants.
We derive a closed formula for the generating function of genus two Gromov-Witten invariants of quintic 3-folds and verify the corresponding mirror symmetry conjecture of Bershadsky, Cecotti, Ooguri and Vafa.
Polyfold theory, as developed by Hofer, Wysocki, and Zehnder, is a relatively new approach to resolving transversality issues that arise in the study of $J$-holomorphic curves in symplectic geometry. This approach has recently led to a well-defined Gromov-Witten invariant for $J$-holomorphic curves of arbitrary genus, and for all closed symplectic manifolds. The Gromov-Witten axioms, as originally described by Kontsevich and Manin, give algebraic relationships between the Gromov-Witten invariants. In this paper, we prove the Gromov-Witten axioms for the polyfold Gromov-Witten invariants.
Notes for a short lecture series, covering exploded manifolds, the moduli stack of curves in exploded manifolds, and a tropical gluing formula for Gromov-Witten invariants: a gluing formula providing a degeneration formula for Gromov-Witten invariants in normal-crossing degenerations. I gave the original lecture series in April 2016 at the Simons Center for Geometry and Physics at Stonybrook. Video of the lectures is available on the SCGP website, http://scgp.stonybrook.edu/video_portal/video.php?id=2595
We compute the $g=1, n=1$ B-model Gromov-Witten invariant of an elliptic curve E directly from the derived category D(E). More precisely, we carry out the computation of the categorical Gromov-Witten invariant defined by Costello using as target a cyclic $A_\infty$ model of D(E) described by Polishchuk. This is the first non-trivial computation of a positive genus categorical Gromov-Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov-Witten invariants of a symplectic 2-torus computed by Dijkgraaf.
The first part of this work constructs positive-genus real Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the present part focuses on their properties that are essential for actually working with these invariants. We determine the compatibility of the orientations on the moduli spaces of real maps constructed in the first part with the standard node-identifying immersion of Gromov-Witten theory. We also compare these orientations with alternative ways of orienting the moduli spaces of real maps that are available in special cases. In a sequel, we …
We present a gluing formula for Gromov-Witten invariants in the case of a triple product. This gluing formula is a simple case of a much more general gluing formula proved and stated using exploded manifolds. We present this simple case because it is relatively easy to explain without any knowledge of exploded manifolds or log schemes.
