Relative Gromov-Witten Research Articles

The Doran-Harder-Thompson conjecture for toric complete intersections

Given a Tyurin degeneration of a Calabi-Yau complete intersection in a toric variety, we prove gluing formulas relating the generalized functional invariants, periods, and I-functions of the mirror Calabi-Yau family and those of the two mirror Landau-Ginzburg models. Our proof makes explicit the “gluing/splitting” of fibrations in the Doran-Harder-Thompson mirror conjecture. Our gluing formula implies an identity, obtained by composition with their respective mirror maps, that relates the absolute Gromov-Witten invariants for the Calabi-Yaus and relative Gromov-Witten invariants for the quasi-Fanos.

Logarithmic Gromov–Witten theory with expansions

We construct a version of relative Gromov-Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we construct virtually smooth and proper moduli spaces of curves in $X$ with prescribed boundary conditions along $D$. Each point in such a moduli space parameterizes maps from nodal curves to expanded degenerations of $X$ that are dimensionally transverse to the strata. We use the expanded formalism to reconstruct the virtual class attached to a tropical map in terms of spaces of maps to expansions attached to the vertices.

Open/closed correspondence via relative/local correspondence

We establish a correspondence between the disk invariants of a smooth toric Calabi-Yau 3-fold X with boundary condition specified by a framed Aganagic-Vafa outer brane (L,f) and the genus-zero closed Gromov-Witten invariants of a smooth toric Calabi-Yau 4-fold X˜, proving the open/closed correspondence proposed by Mayr and developed by Lerche-Mayr. Our correspondence is the composition of two intermediate steps:•First, a correspondence between the disk invariants of (X,L,f) and the genus-zero maximally-tangent relative Gromov-Witten invariants of a relative Calabi-Yau 3-fold (Y,D), where Y is a toric partial compactification of X by adding a smooth toric divisor D. This correspondence can be obtained as a consequence of the topological vertex (Li-Liu-Liu-Zhou) and Fang-Liu where the all-genus open Gromov-Witten invariants of (X,L,f) are identified with the formal relative Gromov-Witten invariants of the formal completion of (Y,D) along the toric 1-skeleton. Here, we present a proof without resorting to formal geometry.•Second, a correspondence in genus zero between the maximally-tangent relative Gromov-Witten invariants of (Y,D) and the closed Gromov-Witten invariants of the toric Calabi-Yau 4-fold X˜=OY(−D). This can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat in the non-compact setting.

Convergence of the mirror to a rational elliptic surface

The construction introduced by Gross, Hacking and Keel allows one to construct a formal mirror family to a pair $(S,D)$ where $S$ is a smooth rational projective surface and $D$ a certain type of Weil divisor supporting an ample or anti-ample class. In that paper they proved two convergence results. Firstly that if the intersection matrix of $D$ is not negative semi-definite then the family they construct lifts to an algebraic family. Secondly they prove that if the intersection matrix is negative definite then their construction lifts along certain analytic strata on the base, and then over a formal neighbourhood of this. In the original version of that paper they claimed that if the intersection matrix were negative semi-definite then family in fact extends over an analytic neighbourhood of the origin but gave an incorrect proof. In this paper we correct this error. We explain how the general Gross-Siebert program can be used to reduce construction of the mirror to such a surface to calculating certain relative Gromov-Witten invariants. We then relate these invariants to the invariants of a new space where we can find explicit formulae for the invariants. From this we deduce analytic convergence of the mirror family, at least when the original surface has an $I_4$ fibre.

On gerbe duality and relative Gromov–Witten theory

We formulate and study an extension of gerbe duality to relative Gromov-Witten theory.

Higher genus relative and orbifold Gromov–Witten invariants

Given a smooth projective variety $X$ and a smooth divisor $D\subset X$. We study relative Gromov-Witten invariants of $(X,D)$ and the corresponding orbifold Gromov-Witten invariants of the $r$-th root stack $X_{D,r}$. For sufficiently large $r$, we prove that orbifold Gromov-Witten invariants of $X_{D,r}$ are polynomials in $r$. Moreover, higher genus relative Gromov-Witten invariants of $(X,D)$ are exactly the constant terms of the corresponding higher genus orbifold Gromov-Witten invariants of $X_{D,r}$. We also provide a new proof for the equality between genus zero relative and orbifold Gromov-Witten invariants, originally proved by Abramovich-Cadman-Wise \cite{ACW}. When $r$ is sufficiently large and $X=C$ is a curve, we prove that stationary relative invariants of $C$ are equal to the stationary orbifold invariants in all genera.

Gromov–Witten theory of elliptic fibrations : Jacobi forms and holomorphic anomaly equations

We conjecture that the relative Gromov-Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all genera and curve classes numerically. The generating series are quasi-Jacobi forms for the lattice $E_8$. We also show the compatibility of the conjecture with the degeneration formula. As Corollary we deduce that the Gromov-Witten potentials of the Schoen Calabi-Yau threefold (relative to $\mathbb{P}^1$) are $E_8 \times E_8$ quasi-bi-Jacobi forms and satisfy a holomorphic anomaly equation. This yields a partial verification of the BCOV holomorphic anomaly equation for Calabi-Yau threefolds. For abelian surfaces the holomorphic anomaly equation is proven numerically in primitive classes. The theory of lattice quasi-Jacobi forms is reviewed. In the Appendix the conjectural holomorphic anomaly equation is expressed as a matrix action on the space of (generalized) cohomological field theories. The compatibility of the matrix action with the Jacobi Lie algebra is proven. Holomorphic anomaly equations for K3 fibrations are discussed in an example.

Relative Gromov-Witten invariants of projective completions of vector bundles

It was proved by Fan--Lee and Fan that the absolute Gromov--Witten invariants of two projective bundles $\mathbb P(V_i)\rightarrow X$ are identified canonically when the total Chern classes $c(V_1)=c(V_2)$ for two bundles $V_1$ and $V_2$ over a smooth projective variety $X$. In this note we show that for the two projective completions $\mathbb P(V_i\oplus\mathcal O)$ of $V_i$ and their infinity divisors $\mathbb P(V_i)$, the relative Gromov--Witten invariants of $(\mathbb P(V_i\oplus\mathcal O),\mathbb P(V_i))$ are identified canonically when $c(V_1)=c(V_2)$.

Three Dimensional Tropical Correspondence Formula

A tropical curve in $\mathbb R^{3}$ contributes to Gromov-Witten invariants in all genus. Nevertheless, we present a simple formula for how a given tropical curve contributes to Gromov-Witten invariants when we encode these invariants in a generating function with exponents of $\lambda$ recording Euler characteristic. Our main modification from the known tropical correspondence formula for rational curves is as follows: a trivalent vertex, which before contributed a factor of $n$ to the count of zero-genus holomorphic curves, contributes a factor of $2\sin(n\lambda/2)$. We explain how to calculate relative Gromov-Witten invariants using this tropical correspondence formula, and how to obtain the absolute Gromov-Witten and Donaldson-Thomas invariants of some $3$-dimensional toric manifolds including $\mathbb CP^{3}$. The tropical correspondence formula counting Donaldson-Thomas invariants replaces $n$ by $i^{-(1+n)}q^{n/2}+i^{1+n}q^{-n/2}$.

Absolute vs. relative Gromov–Witten invariants

We compare the absolute and relative Gromov-Witten invariants of compact symplectic manifolds when the symplectic hypersurface contains no relevant holomorphic curves. We show that these invariants are then the same, except in a narrow range of dimensions of the target and genera of the domains, and provide examples when they fail to be the same.

Equivalence of ELSV and Bouchard–Mariño conjectures for $$r$$ r -spin Hurwitz numbers

We propose two conjectures on Huwritz numbers with completed $(r+1)$-cycles, or, equivalently, on certain relative Gromov-Witten invariants of the projective line. The conjectures are analogs of the ELSV formula and of the Bouchard-Mari\~no conjecture for ordinary Hurwitz numbers. Our $r$-ELSV formula is an equality between a Hurwitz number and an integral over the space of $r$-spin structures, that is, the space of stable curves with an $r$th root of the canonical bundle. Our $r$-BM conjecture is the statement that $n$-point functions for Hurwitz numbers satisfy the topological recursion associated with the spectral curve $x = -y^r + \log y$ in the sense of Chekhov, Eynard, and Orantin. We show that the $r$-ELSV formula and the $r$-BM conjecture are equivalent to each other and provide some evidence for both.

Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations

We consider four approaches to relative Gromov-Witten theory and Gromov-Witten theory of degenerations: Jun Li's original approach, Bumsig Kim's logarithmic expansions, Abramovich-Fantechi's orbifold expansions, and a logarithmic theory without expansions due to Gross-Siebert and Abramovich-Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov-Witten invariants associated to all four of these theories are identical.

Positive divisors in symplectic geometry

In this paper, we first prove a vanishing theorem of relative Gromov-Witten invariant of ℙ1-bundle. Based on this vanishing theorem and degeneration formula, we obtain a comparison theorem between absolute and relative Gromov-Witten invariant under some positive condition of the symplectic divisor.

Logarithmic Gromov-Witten invariants

The goal of this paper is to give a general theory of logarithmic Gromov-Witten invariants. This gives a vast generalization of the theory of relative Gromov-Witten invariants introduced by Li-Ruan, Ionel-Parker, and Jun Li and completes a program first proposed by the second named author in 2002. One considers target spaces X X carrying a log structure. Domains of stable log curves are log smooth curves. Algebraicity of the stack of such stable log maps is proven, subject only to the hypothesis that the log structure on X X is fine, saturated, and Zariski. A notion of basic stable log map is introduced; all stable log maps are pull-backs of basic stable log maps via base-change. With certain additional hypotheses, the stack of basic stable log maps is proven to be proper. Under these hypotheses and the additional hypothesis that X X is log smooth, one obtains a theory of log Gromov-Witten invariants.

Finite group action and Gromov-Witten invariants in symplectic four-manifolds

Let a finite group act semi-freely on a closed symplectic four-manifold with a 2-dimensional fixed point set. Then we show that the relative Gromov-Witten invariants are the same as the invariants on the quotient set-up with respect to the fixed point set.

The tropical vertex

Elements of the tropical vertex group, introduced by Kontsevich and Soibelman, are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory.

Gromov–Witten theory of 𝒜n–resolutions

We give a complete solution for the reduced Gromov-Witten theory of resolved surface singularities of type A_n, for any genus, with arbitrary descendent insertions. We also present a partial evaluation of the T-equivariant relative Gromov-Witten theory of the threefold A_n x P^1 which, under a nondegeneracy hypothesis, yields a complete solution for the theory. The results given here allow comparison of this theory with the quantum cohomology of the Hilbert scheme of points on the A_n surfaces. We discuss generalizations to linear Hodge insertions and to surface resolutions of type D,E. As a corollary, we present a new derivation of the stationary Gromov-Witten theory of P^1.

The Caporaso–Harris formula and plane relative Gromov-Witten invariants in tropical geometry

Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d, g) of complex plane curves of degree d and genus g through 3d + g − 1 general points with the help of relative Gromov-Witten invariants. Recently, Mikhalkin has found a way to reinterpret the numbers N(d, g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes. We relate these two results by defining an analogue of the relative Gromov-Witten invariants and rederiving the Caporaso–Harris formula in terms of both tropical geometry and lattice paths.

Minimality and symplectic sums

Let X1, X2 be symplectic 4-manifolds containing symplectic sur- faces F1, F2 of identical positive genus and opposite squares. Let Z denote the symplectic sum of X1 and X2 along the Fi. Using relative Gromov-Witten theory, we determine precisely when the symplectic 4-manifold Z is minimal (i.e., cannot be blown down); in particular, we prove that Z is minimal unless either: one of the Xi contains a ( 1)-sphere disjoint from Fi; or one of the Xi admits a ruling with Fi as a section. As special cases, this proves a conjecture of Stipsicz asserting the minimality of fiber sums of Lefschetz fibrations, and implies that the non-spin examples constructed by Gompf in his study of the geography problem are minimal. Let (X1,!1), (X2,!2) be symplectic 4-manifolds, and let F1 ⊂ X1, F2 ⊂ X2 be two-dimensional symplectic submanifolds with the same genus whose homology classes satisfy (F1) 2 + (F2) 2 = 0, with the !i normalized to give equal area to the surfaces Fi. For i = 1,2, a neighborhood of Fi is symplectically identified by Weinstein's symplectic neighborhood theorem (19) with the disc normal bundlei of Fi in Xi. Choose a smooth isomorphismof the normal bundle to F1 in X1 (which is a complex line bundle) with the dual of the normal bundle to F2 in X2. According to (2) (and independently (11)), the symplectic sum

Virasoro constraints for target curves

We prove generalized Virasoro constraints for the relative Gromov-Witten theories of all nonsingular target curves. Descendents of the even cohomology classes are studied first by localization, degeneration, and completed cycle methods. Descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations. As an outcome of our results, the relative theories of target curves are completely and explicitly determined.