Scattering Diagrams, Sheaves, and Curves

@article{Bousseau2021ScatteringDS,
  title={Scattering Diagrams, Sheaves, and Curves},
  author={Pierrick Bousseau},
  journal={Acta Mathematica Sinica, English Series},
  year={2021}
}
  • Pierrick Bousseau
  • Published 20 February 2020
  • Mathematics
  • Acta Mathematica Sinica, English Series
We review the recent proof of the N.Takahashi's conjecture on genus $0$ Gromov-Witten invariants of $(\mathbb{P}^2, E)$, where $E$ is a smooth cubic curve in the complex projective plane $\mathbb{P}^2$. The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov-Witten invariants of $(\mathbb{P}^2, E)$ and the world of moduli spaces of coherent sheaves on $\mathbb{P}^2$. Using this bridge, the N.Takahashi's conjecture can be translated into a… 

References

SHOWING 1-10 OF 48 REFERENCES

A proof of N.Takahashi's conjecture for $(\mathbb{P}^2,E)$ and a refined sheaves/Gromov-Witten correspondence

We prove N.Takahashi's conjecture determining the contribution of each contact point in genus-$0$ maximal contact Gromov-Witten theory of $\mathbb{P}^2$ relative to a smooth cubic $E$. This is a new

Holomorphic anomaly equation for $({\mathbb P}^2,E)$ and the Nekrasov-Shatashvili limit of local ${\mathbb P}^2$

Abstract We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor,

A theory of generalized Donaldson–Thomas invariants

This book studies generalized Donaldson-Thomas invariants $\bar{DT}{}^\alpha(\tau)$. They are rational numbers which 'count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern

Decomposition of degenerate Gromov–Witten invariants

We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields

Contributions of degenerate stable log maps

Let $X$ be a smooth surface and $D$ a (possibly reducible or non-reduced) curve on $X$. An $\mathbb{A}^1$-curve on $(X, D)$ is an irreducible curve $C$ on $X$ such that the normalization of

A PROOF OF N. TAKAHASHI’S CONJECTURE FOR (P2,E) AND A REFINED SHEAVES/GROMOV-WITTEN CORRESPONDENCE

We prove N. Takahashi’s conjecture determining the contribution of each contact point in genus-0 maximal contact Gromov–Witten theory of P relative to a smooth cubic E. This is a new example of a

Curve counting via stable pairs in the derived category

For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C⊂X is an embedded curve and D⊂C is a divisor. A virtual class is constructed on the

Sheaves of maximal intersection and multiplicities of stable log maps

A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J

Curves on K3 surfaces and modular forms

We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equivalences relating the reduced Gromov–Witten invariants of K3 surfaces to

Local BPS Invariants: Enumerative Aspects and Wall-Crossing

We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We