# Birational invariance in logarithmic Gromov–Witten theory

@article{Abramovich2018BirationalII, title={Birational invariance in logarithmic Gromov–Witten theory}, author={Dan Abramovich and Jonathan Wise}, journal={Compositio Mathematica}, year={2018}, volume={154}, pages={595 - 620} }

Gromov–Witten invariants have been constructed to be deformation invariant, but their behavior under other transformations is subtle. We show that logarithmic Gromov–Witten invariants are also invariant under appropriately defined logarithmic modifications.

## 53 Citations

### Quantum mirrors of log Calabi–Yau surfaces and higher-genus curve counting

- MathematicsCompositio Mathematica
- 2020

Gross, Hacking and Keel have constructed mirrors of log Calabi–Yau surfaces in terms of counts of rational curves. Using $q$-deformed scattering diagrams defined in terms of higher-genus log…

### Relative quantum cohomology under birational transformations

- Mathematics
- 2022

. We study how relative quantum cohomology, deﬁned in [TY20b] and [FWY20], varies under birational transformations. For toric complete intersections with simple normal crossings divisors that contain…

### GROMOV – WITTEN INVARIANTS OF BLOWUPS

- Mathematics
- 2021

We prove the Abelian/non-Abelian Correspondence with bundles for target spaces that are partial flag bundles, combining and generalising results by Ciocan-Fontanine–Kim– Sabbah, Brown, and Oh. From…

### Theta functions, broken lines and 2-marked log Gromov-Witten invariants

- Mathematics
- 2022

. Theta functions were deﬁned for varieties with eﬀective anticanonical divisor [GHS] and are related to certain punctured Gromov-Witten invariants [ACGS2]. In this paper we show that in the case of…

### Gromov–Witten theory with maximal contacts

- MathematicsForum of Mathematics, Sigma
- 2022

Abstract We propose an intersection-theoretic method to reduce questions in genus 0 logarithmic Gromov–Witten theory to questions in the Gromov–Witten theory of smooth pairs, in the presence of…

### The Abelian/Nonabelian Correspondence and Gromov–Witten Invariants of Blow-Ups

- MathematicsForum of Mathematics, Sigma
- 2022

Abstract We prove the abelian/nonabelian correspondence with bundles for target spaces that are partial flag bundles, combining and generalising results by Ciocan-Fontanine–Kim–Sabbah, Brown, and Oh.…

### Birational Invariance in Punctured Log Gromov-Witten Theory

- Mathematics
- 2022

Given a log smooth scheme ( X, D ), and a log ´etale modiﬁcation ( ˜ X, ˜ D ) → ( X, D ), we relate the punctured Gromov-Witten theory of ( ˜ X, ˜ D ) to the punctured Gromov-Witten theory of ( X, D…

### Logarithmic Gromov–Witten theory with expansions

- MathematicsAlgebraic Geometry
- 2022

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…

### Theta bases and log Gromov-Witten invariants of cluster varieties

- MathematicsTransactions of the American Mathematical Society
- 2021

Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate…

### The Log Product Formula.

- Mathematics
- 2019

We prove a formula expressing the Log Gromov-Witten Invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of $V$ and $W$. This extends results of F. Qu and Y.P. Lee,…

## References

SHOWING 1-10 OF 36 REFERENCES

### Higher genus Gromov–Witten invariants as genus zero invariants of symmetric products

- Mathematics
- 2003

I prove a formula expressing the descendent genus g Gromov-Witten invariants of a projective variety X in terms of genus 0 invariants of its symmetric product stack Sg+1(X). When X is a point, the…

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

- Mathematics
- 2012

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…

### Gromov-Witten invariants of blow-ups along points and curves

- Mathematics
- 1998

Abstract. 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…

### Stable logarithmic maps to Deligne-Faltings pairs II

- Mathematics
- 2011

We make an observation which enables one to deduce the existence of an algebraic stack of log maps for all generalized Deligne--Faltings log structures (in particular simple normal crossings divisor)…

### Moduli of morphisms of logarithmic schemes

- Mathematics
- 2014

We show that there is a logarithmic algebraic space parameterizing logarithmic morphisms between fixed logarithmic schemes when those logarithmic schemes satisfy natural hypotheses. As a corollary,…

### Boundedness of the space of stable logarithmic maps

- Mathematics
- 2014

We prove that the moduli space of stable logarithmic maps from logarithmic curves to a fixed target logarithmic scheme is a proper algebraic stack when the target scheme is projective with fine and…

### Stable logarithmic maps to Deligne-Faltings pairs I

- Mathematics
- 2010

We introduce a new compactification of the space of relative stable maps. This new method uses logarithmic geoemtry in the sense of Kato-Fontaine-Illusie rather than the expanded degeneration. The…

### Gromov–Witten invariants of blow-ups along submanifolds with convex normal bundles

- Mathematics
- 2009

Given a submanifold Z inside X, let Y be the blow-up of X along Z. When the normal bundle of Z in X is convex with a minor assumption, we prove that genus-zero GW-invariants of Y with cohomology…