Relative Gromov Research Articles

Bavard duality for the relative Gromov seminorm

The relative Gromov seminorm is a finer invariant than stable commutator length where a relative homology class is fixed. We show a duality result between bounded cohomology and the relative Gromov seminorm, analogously to Bavard duality for scl. We give an application to computations of scl in graphs of groups. We also explain how our duality result can be given a purely algebraic interpretation via a relative version of the Hopf formula. Moreover, we show that this leads to a natural generalisation of a result of Calegari on a connection between scl and the rotation quasimorphism.

The Proper Landau–Ginzburg Potential, Intrinsic Mirror Symmetry and the Relative Mirror Map

Given a smooth log Calabi–Yau pair (X, D), we use the intrinsic mirror symmetry construction to define the mirror proper Landau–Ginzburg potential and show that it is a generating function of two-point relative Gromov–Witten invariants of (X, D). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of Fan et al. (Sel Math (NS) 25(4): Art. 54, 25, 2019. https://doi.org/10.1007/s00029-019-0501-z) to compute two-point relative invariants. When D is nef, we compute the proper Landau–Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety X, this implies the conjecture of m Gräfnitz et al. (2022) that the proper Landau–Ginzburg potential is the open mirror map. When X is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.

Gromov–Witten theory of complete intersections via nodal invariants

AbstractWe 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 invariants, we introduce the new notion of nodal relative Gromov–Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov–Witten theory are stated in complete generality and are of independent interest.

Curve counting in genus one: Elliptic singularities and relative geometry

We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of Zinger's reduced Gromov--Witten invariants. We relate the relative and absolute theories by degeneration of the tangency conditions, and the resulting formulas generalise a well-known recursive calculation scheme put forward by Gathmann in genus zero. The geometric input is a desingularisation of the principal component of the moduli space of genus one logarithmic stable maps to a very ample pair, using the geometry of elliptic singularities. Our study passes through general techniques for calculating integrals on logarithmic blowups of moduli spaces of stable maps, which may be of independent interest.

Refined floor diagrams from higher genera and lambda classes

We show that, after the change of variables q=e^{iu}, refined floor diagrams for {mathbb {P}}^2 and Hirzebruch surfaces compute generating series of higher genus relative Gromov–Witten invariants with insertion of a lambda class. The proof uses an inductive application of the degeneration formula in relative Gromov–Witten theory and an explicit result in relative Gromov–Witten theory of {mathbb {P}}^1. Combining this result with the similar looking refined tropical correspondence theorem for log Gromov–Witten invariants, we obtain a non-trivial relation between relative and log Gromov–Witten invariants for {mathbb {P}}^2 and Hirzebruch surfaces. We also prove that the Block–Göttsche invariants of {mathbb {F}}_0 and {mathbb {F}}_2 are related by the Abramovich–Bertram formula.

Gromov–Witten invariants of root stacks with mid-ages and the loop axiom

We study orbifold Gromov–Witten invariants of the r-th root stack XD,r with a pair of mid-ages when r is sufficiently large. We prove that genus g invariants with a pair of mid-ages ka/r and 1−ka/r are polynomials in ka and the kai-coefficients are polynomials in r with degree bounded by 2g. Moreover, genus zero invariants with a pair of mid-ages are independent of the choice of mid-ages. As an application, we obtain an identity for relative Gromov–Witten theory which can be viewed as a modified version of the usual loop axiom.

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 Gromov–Witten theory and double ramification cycles

We extend the definition of relative Gromov--Witten invariants with negative contact orders to all genera. Then we show that relative Gromov--Witten theory forms a partial CohFT. Some cycle relations on the moduli space of stable maps are also proved.

On the rim tori refinement of relative Gromov–Witten invariants

We construct Ionel–Parker’s proposed refinement of the standard relative Gromov–Witten invariants in terms of abelian covers of the symplectic divisor and discuss in what sense it gives rise to invariants. We use it to obtain some vanishing results for the standard relative Gromov–Witten invariants. In a separate paper, we describe to what extent this refinement sharpens the usual symplectic sum formula and give further qualitative applications.

Relative Gromov–Witten invariants and the enumerative meaning of mirror maps for toric Calabi–Yau orbifolds

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.

The refined symplectic sum formula for Gromov–Witten invariants

We describe the extent to which Ionel–Parker’s proposed refinement of the standard relative Gromov–Witten invariants (GW-invariants) sharpens the usual symplectic sum formula. The key product operation on the target spaces for the refined invariants is specified in terms of abelian covers of symplectic divisors, making it suitable for studying from a topological perspective. We give several qualitative applications of this refinement, which include vanishing results for GW-invariants.

Witten–Dijkgraaf–Verlinde–Verlinde Equation and its Application to Relative Gromov–Witten Theory

Abstract We derive a recursive formula for certain relative Gromov–Witten invariants with a maximal tangency condition via the Witten–Dijkgraaf–Verlinde–Verlinde equation. For certain relative pairs, we get explicit formulae of invariants using the recursive formula.

The relative Gromov width of Lagrangian cobordisms between Legendrians

We obtain upper and lower bounds for the relative Gromov width of Lagrangian cobordisms between Legendrian submanifolds. Upper bounds arise from the existence of $J$-holomorphic disks with boundary on the Lagrangian cobordism that pass through the center of any given symplectically embedded ball. The areas of these disks --- and hence the sizes of these balls --- are controlled by a real-valued fundamental capacity, a quantity derived from the algebraic structure of filtered linearized Legendrian Contact Homology of the Legendrian at the top of the cobordism. Lower bounds come from explicit constructions that use neighborhoods of Reeb chords in the Legendrian ends. We also study relationships between the relative Gromov width and another quantitative measurement, the length of a Lagrangian cobordism.

Structures in genus‐zero relative Gromov–Witten theory

In this paper, we define genus-zero relative Gromov--Witten invariants with negative contact orders. Using this, we construct relative quantum cohomology rings and Givental formalism. A version of Virasoro constraints also follows from it.

Mirror theorems for root stacks and relative pairs

Given a smooth projective variety $X$ with a smooth nef divisor $D$ and a positive integer $r$, we construct an $I$-function, an explicit slice of Givental's Lagrangian cone, for Gromov--Witten theory of the root stack $X_{D,r}$. As an application, we also obtain an $I$-function for relative Gromov--Witten theory following the relation between relative and orbifold Gromov--Witten invariants.

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 $c(V_1)=c(V_2)$.

Chamber Structure for Some Equivariant Relative Gromov–Witten Invariants of ℙ1 in Genus 0

In this paper, we study genus 0 equivariant relative Gromov–Witten invariants of ℙ1 whose corresponding relative stable maps are totally ramified over one point. For fixed number of marked points, we show that such invariants are piecewise polynomials in some parameter space. The parameter space can then be divided into polynomial domains, called chambers. We determine the difference of polynomials between two neighbouring chambers. In some special chamber, which we called the totally negative chamber, we show that such a polynomial can be expressed in a simple way. The chamber structure here shares some similarities to that of double Hurwitz numbers.

Gromov–Witten Theory of $\text{K3} \times {\mathbb{P}}^1$ and Quasi-Jacobi Forms

Abstract Let $S$ be a K3 surface with primitive curve class $\beta$. We solve the relative Gromov–Witten theory of $S \times {\mathbb{P}}^1$ in classes $(\beta,1)$ and $(\beta,2)$. The generating series are quasi-Jacobi forms and equal to a corresponding series of genus $0$ Gromov–Witten invariants on the Hilbert scheme of points of $S$. This proves a special case of a conjecture of Pandharipande and the author. The new geometric input of the paper is a genus bound for hyperelliptic curves on K3 surfaces proven by Ciliberto and Knutsen. By exploiting various formal properties we find that a key generating series is determined by the very first few coefficients. Let $E$ be an elliptic curve. As collorary of our computations, we prove that Gromov–Witten invariants of $S \times E$ in classes $(\beta,1)$ and $(\beta,2)$ are coefficients of the reciprocal of the Igusa cusp form. We also calculate several linear Hodge integrals on the moduli space of stable maps to a K3 surface and the Gromov–Witten invariants of an abelian threefold in classes of type $(1,1,d)$.

Exact Lagrangian caps and non-uniruled Lagrangian submanifolds

We make the elementary observation that the Lagrangian submanifolds of $\mathbb{C}^n$, for each $n \ge 3$, constructed by Ekholm, Eliashberg, Murphy and Smith are non-uniruled and moreover have infinite relative Gromov width. The construction of these submanifolds use exact Lagrangian caps, which obviously are non-uniruled in themselves. This property is also used to show that if a Legendrian submanifold inside a 1-jet space admits an exact Lagrangian cap then its Legendrian contact homology DGA is acyclic.

RELATIVE RUAN AND GROMOV–TAUBES INVARIANTS OF SYMPLECTIC 4-MANIFOLDS

We define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed symplectic hypersurface V in a symplectic 4-manifold (X, ω) at prescribed points with prescribed contact orders (in addition to insertions on X\V). We obtain invariants of the deformation class of (X, V, ω). Two large issues must be tackled to define such invariants: (1) curves lying in the hypersurface V and (2) genericity results for almost complex structures constrained to make V pseudo-holomorphic (or almost complex). Moreover, these invariants are refined to take into account rim-tori decompositions. In the latter part of the paper, we extend the definition to disconnected submanifolds and construct relative Gromov–Taubes invariants.