We prove a splitting formula that reconstructs the logarithmic Gromov- Witten invariants of simple normal crossing varieties from the punctured Gromov- Witten invariants of their irreducible components, under the assumption of the gluing strata being toric varieties. The formula is based on the punctured Gromov-Witten theory developed in arXiv:2009.07720.
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 construction of virtual fundamental chains via specialization. Comments on the equivalence of the degeneration formula of closed Gromov-Witten invariants by Li-Ruan/Li versus Ionel-Parker are given in the Appendix.
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 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.
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 define a formalism for computing open orbifold GW invariants of [C^3/G] where G is any finite abelian group. We prove that this formalism and a suitable gluing algorithm can be used to compute GW invariants in all genera of any toric CY orbifold of dimension 3. We conjecture a correspondence with the DT orbifold vertex of Bryan-Cadman-Young.
We give a summation over graphs type formula for the permutation-equivariant K-theoretic Gromov-Witten total potential of a projective manifold X in terms of cohomological Gromov-Witten (GW) invariants of X. We achieve this by describing combinatorially the Kawasaki strata of the moduli spaces of stable maps to X.
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 pro…
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 quas…
We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete formula for the change of GW-invariants of any genus transform under flop and a general type I extremal transition. Other extremal transition will be handled in a subsequent paper.
We develop a gluing algorithm for Gromov-Witten invariants of toric Calabi-Yau threefolds based on localization and gluing graphs. The main building block of this algorithm is a generating function of cubic Hodge integrals of special form. We conjecture a precise relation between this generating function and the topological vertex at fractional framing.
Log Gromov-Witten invariants have recently been defined separately by Gross and Siebert and Abramovich and Chen. This paper provides a dictionary between log geometry and holomorphic exploded manifolds in order to compare Gromov-Witten invariants defined using exploded manifolds or log schemes. The gluing formula for Gromov-Witten invariants of exploded manifolds suggests an approach to proving analogous gluing formulas for log Gromov-Witten invariants.
In this paper, using the gluing formula of Gromov-Witten invariants under symplectic cutting, due to Li and Ruan, we studied the Gromov-Witten invariants of blow-ups at a smooth point or along a smooth curve. We established some relations between Gromov-Witten invariants of M and its blow-ups at a smooth point or along a smooth curve.
We construct 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 show that there exist proper moduli spaces of curves in $X$ with prescribed boundary conditions along $D$, equipped with virtual classes. Each point in such a moduli space parameterizes a map from a nodal curve to an expanded degeneration of $X$ that is dimensionally transverse to the strata. In the context of maps to a simple normal crossings degeneration, the virtual fundamen…
We compute section class relative equivariant Gromov-Witten invariants of the total space of P^2-bundles of the form P(O+L1+L2)-->C where C is a genus g curve, O is the trivial bundle, and L1 (resp. L2) is an arbitrary line bundle of degree k1 (resp. k2) over C. We prove a gluing formula for the partition functions of these invariants. Using this gluing formula together with localization techniques, we construct three explicit 3x3 matrices G, U1 and U2 with entries in Q((u))(t0,t1,t2), where u is the genus parameter, and t0,t1,t2 are the equivariant parameters. Then we prove that the partition…
We prove three fundamental properties of counting holomorphic cylinders in log Calabi-Yau surfaces: positivity, integrality and the gluing formula. Positivity and integrality assert that the numbers of cylinders, defined via virtual techniques, are in fact nonnegative integers. The gluing formula roughly says that cylinders can be glued together to form longer cylinders, and the number of longer cylinders equals the product of the numbers of shorter cylinders. Our approach uses Berkovich geometry, tropical geometry, deformation theory and the ideas in the proof of associativity relations of Gr…
Let $V$ be a smooth, projective, convex variety. We define tautological $ψ$ and $κ$ classes on the moduli space of stable maps $\M_{0,n}(V)$, give a (graphical) presentation for these classes in terms of boundary strata, derive differential equations for the generating functions of the Gromov-Witten invariants of $V$ twisted by these tautological classes, and prove that these intersection numbers are completely determined by the Gromov-Witten invariants of $V$. This results in families of Frobenius manifold structures on the cohomology ring of $V$ which includes the quantum cohomology as a spe…
The study of open/closed string duality and large $N$ duality suggests a Gromov-Witten theory for conifolds that sits on the border of both a closed Gromov-Witten theory and an open Gromov-Witten theory. In this work we employ the result of Jun Li on Gromov-Witten theory for a projective singular variety of the gluing form $Y_1\cup_D Y_2$ to extract Gromov-Witten invariants of a conifold.
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 Capor…
In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form $Y_1\cup_D Y_2$ that arises from a degeneration $W/{\Bbb A}^1$ and a theory of relative Gromov-Witten invariants for a codimension-1 relative pair $(Y,D)$. As a summit, he derived a degeneration formula that relates a finite summation of the usual Gromov-Witten invariants of a general smooth fiber $W_t$ of $W/{\Bbb A}^1$ to the Gromov-Witten invariants of the singular fiber $W_0=Y_1\cup_D Y_2$ via gluing the relative pairs $(Y_1,D)$ and $(Y_2,…
We prove that the fermionic form of the generating function of the Gromov-Witten invariants of the resolved conifold is a Bogoliubov transform of the fermionic vacuum; in particular, it is a tau function of the KP hierarchy. Our proof is based on the gluing rule of the topological vertex and the formulas of the fermionic representations of the framed one-legged and two-legged topological vertex which were conjectured by Aganagic et al and proved in our recent work.
This manuscript describes in detail the symplectic sum formulas in Gromov-Witten theory and related topological and analytic issues. In particular, we analyze and compare two analytic approaches to these formulas. The Ionel-Parker formula contains two unique features, rim tori refinements of relative invariants and the so-called S-matrix, which have been a mystery in GW-theory over the past decade. We explain why the latter, which appears due to imprecise reasoning, should not be present and how the former should be interpreted. While the gluing argument in the IP work attempts to address all …
In this paper, we propose a definition of genus one real Gromov-Witten invariants for certain symplectic manifolds with real a structure, including Calabi-Yau threefolds, and use equivariant localization to calculate certain genus one real invariants of the projective space. For this definition, we combine three moduli spaces corresponding to three possible types of involutions on a symplectic torus, by gluing them along common boundaries, to get a moduli space without codimension-one boundary and then study orientation of the total space. Modulo a technical conjectural lemma, we can prove tha…
