# Counting curves on surfaces in Calabi–Yau 3-folds

@article{Gholampour2014CountingCO, title={Counting curves on surfaces in Calabi–Yau 3-folds}, author={Amin Gholampour and Artan Sheshmani and Richard P. Thomas}, journal={Mathematische Annalen}, year={2014}, volume={360}, pages={67-78} }

Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of two-dimensional torsion sheaves, enumerating pairs $$Z\subset H$$Z⊂H in a Calabi–Yau threefold $$X$$X. Here $$H$$H is a member of a sufficiently positive linear system and $$Z$$Z is a one-dimensional subscheme of it. The associated sheaf is the ideal sheaf of $$Z\subset H$$Z⊂H, pushed forward to $$X$$X and considered as a certain Joyce–Song pair in the derived category of $$X…

## 13 Citations

Localized Donaldson-Thomas theory of surfaces

- Mathematics
- 2017

Let $S$ be a projective simply connected complex surface and $\mathcal{L}$ be a line bundle on $S$. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces…

Donaldson-Thomas Invariants of 2-Dimensional sheaves inside threefolds and modular forms

- Mathematics, Physics
- 2013

Motivated by the S-duality conjecture, we study the Donaldson-Thomas invariants of the 2 dimensional Gieseker stable sheaves on a threefold. These sheaves are supported on the fibers of a nonsingular…

Flops and the S-duality conjecture

- Mathematics
- 2013

We prove the transformation formula of Donaldson-Thomas (DT) invariants counting two dimensional torsion sheaves on Calabi-Yau 3-folds under flops. The error term is described by the Dedekind eta…

LOCALIZED DONALDSON-THOMAS THEORY OF SURFACES AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND SHING-TUNG YAU

- 2019

Let S be a projective simply connected complex surface and L be a line bundle on S. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of L. The moduli…

Generalized Donaldson-Thomas Invariants of 2-Dimensional sheaves on local P^2

- Mathematics, Physics
- 2013

Let X be the total space of the canonical bundle of P^2. We study the generalized Donaldson-Thomas invariants, defined in the work of Joyce-Song, of the moduli spaces of the 2-dimensional Gieseker…

Donaldson-Thomas invariants, linear systems and punctual Hilbert schemes

- Mathematics
- 2019

We study certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of a linear system of a fixed line bundle. When the canonical bundle…

Intersection numbers on the relative Hilbert schemes of points on surfaces

- Mathematics
- 2015

We study certain top intersection products on the Hilbert scheme of points on a nonsingular surface relative to an effective smooth divisor. We find a formula relating these numbers to the…

Moduli and Invariants

- 2020

This workshop focused on the newest of results in the theory of moduli spaces and applications to enumerative invariants in algebraic geometry. The workshop showed new advances in computations of…

Hilbert schemes, Donaldson–Thomas theory, Vafa–Witten and Seiberg–Witten theory

- Mathematics, PhysicsNotices of the International Congress of Chinese Mathematicians
- 2019

This article provides a summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of nested Hilbert schemes of points and curves on algebraic surfaces and…

Scaling Black Holes and Modularity

- Physics, Mathematics
- 2021

Scaling black holes are solutions of supergravity with multiple black hole singularities, which can be adiabatically connected to a single center black hole solution. We develop techniques to…

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