Generalized Donaldson-Thomas theory over fields K $\neq$ C

Abstract

Generalized Donaldson-Thomas invariants defined by Joyce and Song arXiv:0810.5645 are rational numbers which `count' both $τ$-stable and $τ$-semistable coherent sheaves with Chern character $α$ on a Calabi-Yau 3-fold $X$, where $τ$ denotes Gieseker stability for some ample line bundle on $X$. These invariants are defined for all classes $α$, and are equal to the classical Donaldson-Thomas invariant defined by Thomas arXiv:math/9806111 when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $τ$. Joyce and Song use gauge theory and transcendental complex analytic methods, so that their theory of generalized Donaldson-Thomas invariants is valid only in the complex case. This paper will propose a new algebraic method extending the theory to algebraically closed fields K of characteristic zero, and partly to triangulated categories and for non necessarily compact Calabi-Yau 3-folds under some hypothesis.