Chern-Simons functions on toric Calabi-Yau threefolds and Donaldson-Thomas theory

In this paper, we give a construction of the global Chern-Simons functions for toric Calabi-Yau stacks of dimension three using strong exceptional collections. The moduli spaces of sheaves on such stacks can be identified with critical loci of these functions. We give two applications of these functions. First, we prove Joyce's integrality conjecture of generalized DT invariants on local surfaces. Second, we prove a dimension reduction formula for virtual motives, which leads to two recursion formulas for motivic Donaldson-Thomas invariants.Comment: A rewritten versoin. Some references adde

Classification of free actions on complete intersections of four quadrics

In this paper we classify all free actions of finite groups on Calabi-Yau complete intersection of 4 quadrics in \PP^7, up to projective equivalence. We get some examples of smooth Calabi-Yau threefolds with large nonabelian fundamental groups. We also observe the relation between some of these examples and moduli of polarized abelian surfaces.Comment: 17 pages, 1 tabl

Uniformly bounded components of normality

Suppose that $f(z)$ is a transcendental entire function and that the Fatou set $F(f)\neq\emptyset$. Set $$B_1(f):=\sup_{U}\frac{\sup_{z\in U}\log(|z|+3)}{\inf_{w\in U}\log(|w|+3)}$$ and $$B_2(f):=\sup_{U}\frac{\sup_{z\in U}\log\log(|z|+30)}{\inf_{w\in U}\log(|w|+3)},$$ where the supremum $\sup_{U}$ is taken over all components of $F(f)$. If $B_1(f)<\infty$ or $B_2(f)<\infty$, then we say $F(f)$ is strongly uniformly bounded or uniformly bounded respectively. In this article, we will show that, under some conditions, $F(f)$ is (strongly) uniformly bounded.Comment: 17 pages, a revised version, to appear in Mathematical Proceedings Cambridge Philosophical Societ