arXiv subjects

Thomas H. Parker

Publications and source records attributed to Thomas H. Parker.

14 recordsLinked to original sources

arXiv2019

Relating VFCs on thin compactifications

Many moduli spaces that occur in geometric analysis admit "Fredholm-stratified thin compactifications" in the sense of [IP1] and hence admit a relative fundamental class (RFC), also as defined in [IP1]. We extend these results, emphasizing the naturality of the RFC, eliminating the need for a stratification, and proving three compatibility results: the invariants defined by the RFC agree with those defined by pseudo-cycles, the RFC is compatible with cutdown moduli spaces, and the RFC agrees with the virtual fundamental class (VFC) constructed by J. Pardon via implicit atlases in all cases where both are defined.

math.SG↗

arXiv2018

Thin compactifications and relative fundamental classes

We define a notion of relative fundamental class that applies to moduli spaces in gauge theory and in symplectic Gromov-Witten theory. For universal moduli spaces over a parameter space, the relative fundamental class specifies an element of the Cech homology of the compactification of each fiber; it is defined if the compactification is "thin" in the sense that its boundary has homological codimension at least two.

math.SG↗

arXiv2017

The Gopakumar-Vafa formula for symplectic manifolds

The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa formula holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.

math.SG↗

arXiv2015

A natural Gromov-Witten virtual fundamental class

We describe a program for proving that the Gromov-Witten moduli spaces of compact symplectic manifolds carry a unique virtual fundamental class that satisfies certain naturality conditions. The virtual fundamental class is constructed using only Ruan-Tian perturbations by introducing stabilizing divisors, using Cech homology, and systematically applying naturality conditions. In high dimensions or low genus, no gluing theorems are needed.

math.SG↗

arXiv2015

Corrigendum: The symplectic sum formula for Gromov-Witten invariants

We correct an error and an oversight in [IP]. The sign of the curvature in (8.7) is wrong, requiring a new proof of Proposition 8.1. Also, several lemmas addressed only the basic case of maps with intersection multiplicity s=1; the general case follows by applying the pointwise estimates in [IP] with a modified Sobolev norm. These corrections do not affect the results of the paper. We thank M. Tehrani and A. Zinger for pointing out these issues.

math.SG↗

arXiv2012

Recursion Formulas for Spin Hurwitz Numbers

The classical Hurwitz numbers which count coverings of a complex curve have an analog when the curve is endowed with a theta characteristic. These "spin Hurwitz numbers", recently studied by Eskin, Okounkov and Pandharipande, are interesting in their own right. By the authors' previous work, they are also related to the Gromov-Witten invariants of Kahler surfaces. We prove a recursive formula for spin Hurwitz numbers, which then gives the dimension zero GW invariants of Kahler surfaces with positive geometric genus. The proof uses a degeneration of spin curves, an invariant defined by the spectral flow of certain anti-linear deformations of the d-bar operator, and an interesting localization phenomenon for eigenfunctions that shows that maps with even ramification points cancel in pairs.

math.SG↗

arXiv2009

An obstruction bundle relating Gromov-Witten invariants of curves and Kahler surfaces

In [LP] the authors defined symplectic "Local Gromov-Witten invariants" associated to spin curves and showed that the GW invariants of a Kahler surface X with p_g>0 are a sum of such local GW invariants. This paper describes how the local GW invariants arise from an obstruction bundle (in the sense of Taubes) over the space of stable maps into curves. Together with the results of [LP], this reduces the calculation of the GW invariants of complex surfaces to computations in the GW theory of curves.

math.SG↗

arXiv2006

A Structure Theorem for the Gromov-Witten Invariants of Kahler Surfaces

We prove a structure theorem for the Gromov-Witten invariants of compact Kahler surfaces with geometric genus $p_g>0$. Under the technical assumption that there is a canonical divisor that is a disjoint union of smooth components, the theorem shows that the GW invariants are universal functions determined by the genus of this canonical divisor components and the holomorphic Euler characteristic of the surface. We compute special cases of these universal functions.

math.SG↗

arXiv2005

Symplectic Gluing and Family Gromov-Witten Invariants

This article describes the use of symplectic cut-and-paste methods to compute Gromov-Witten invariants. Our focus is on recent advances extending these methods to Kahler surfaces with geometric genus p_g>0, for which the usual GW invariants vanish for most homology classes. This involves extending the Splitting Formula and the Symplectic Sum Formula to the family GW invariants introduced by the first author. We present applications to the invariants of elliptic surfaces and to the Yau-Zaslow Conjecture. In both cases the results agree with the conjectures of algebraic geometers and yield a proof, to appear in [LL1], of previously unproved cases of the Yau-Zaslow Conjecture.

math.AG↗

arXiv2004

Relative Gromov-Witten Invariants

We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of a compact space of `V-stable' maps. Simple special cases include the Hurwitz numbers for algebraic curves and the enumerative invariants of Caporaso and Harris.

math.SG↗

arXiv2000

The Symplectic Sum Formula for Gromov-Witten Invariants

In the symplectic category there is a `connect sum' operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula for the Gromov-Witten invariants of a symplectic sum Z=X#Y in terms of the relative GW invariants of X and Y. Several applications to enumerative geometry are given.

math.SG↗

arXiv1998

Gromov-Witten Invariants of Symplectic Sums

The natural sum operation for symplectic manifolds is defined by gluing along codimension two submanifolds. Specifically, let X be a symplectic 2n-manifold with a symplectic (2n-2)-submanifold V. Given a similar pair (Y,W) with a symplectic identification V=W and a complex anti-linear isomorphism between the normal bundles of V and W, we can form the symplectic sum Z=X # Y. This note announces a general formula for computing the Gromov-Witten invariants of the sum Z in terms of relative Gromov-Witten invariants of (X,V) and (Y,W). Two applications are presented: a short derivation of the Caporaso-Harris formula [CH], and new proof that the rational enumerative invariants of the rational elliptic surface are given by the ``modular form'' (5.2).

math.SG↗

arXiv1997

Gromov Invariants and Symplectic Maps

Given a symplectomorphism f of a symplectic manifold X, one can form the `symplectic mapping cylinder' $X_f = (X \times R \times S^1)/Z$ where the Z action is generated by $(x,s,t)\mapsto (f(x),s+1,t)$. In this paper we compute the Gromov invariants of the manifolds $X_f$ and of fiber sums of the $X_f$ with other symplectic manifolds. This is done by expressing the Gromov invariants in terms of the Lefschetz zeta function of f and, in special cases, in terms of the Alexander polynomials of knots. The result is a large set of interesting non-Kahler symplectic manifolds with computational ways of distinguishing them. In particular, this gives a simple symplectic construction of the `exotic' elliptic surfaces recently discovered by Fintushel and Stern and of related `exotic' symplectic 6-manifolds.

dg-ga↗

arXiv1997

The Gromov Invariants of Ruan-Tian and Taubes

Taubes has recently defined Gromov invariants for symplectic four-manifolds and related them to the Seiberg-Witten invariants. Independently, Ruan and Tian defined symplectic invariants based on ideas of Witten. In this note, we show that Taubes' Gromov invariants are equal to certain combinations of Ruan-Tian invariants. This link allows us to generalize Taubes' invariants. For each closed symplectic four-manifold, we define a sequence of symplectic invariants $Gr_δ$, $δ=0,1,2,...$. The first of these, $Gr_0$, generates Taubes' invariants, which count embedded J-holomorphic curves. The new invariants $Gr_δ$ count immersed curves with $δ$ double points. In particular, these results give an independent proof that Taubes' invariants are well-defined. They also show that some of the Ruan-Tian symplectic invariants agree with the Seiberg-Witten invariants.

alg-geom↗