## Advanced Studies in Pure Mathematics

### ADHM sheaf theory and wallcrossing

Wu-yen Chuang

#### Abstract

In this article we survey the recent developments in ADHM sheaf theory on a smooth projective variety $X$. When $X$ is a curve the theory is an alternative construction of stable pair theory of Pandharipande and Thomas or Gromov–Witten theory on local curve geometries. The construction relies on relative Beilinson spectral sequence and Fourier–Mukai transformation. We will present some applications of the theory, including the derivations of the wallcrossing formulas, higher rank Donaldson–Thomas invariants on local curves, and the coholomogies of the moduli of stable Hitchin pairs.

#### Article information

Dates
Revised: 6 August 2012
First available in Project Euclid: 19 October 2018

https://projecteuclid.org/ euclid.aspm/1539916447

Digital Object Identifier
doi:10.2969/aspm/06510083

Mathematical Reviews number (MathSciNet)
MR3380776

Zentralblatt MATH identifier
1360.14129

#### Citation

Chuang, Wu-yen. ADHM sheaf theory and wallcrossing. Algebraic Geometry in East Asia — Taipei 2011, 83--106, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06510083. https://projecteuclid.org/euclid.aspm/1539916447