In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic $p>0$. Using my characterization, I show that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion-free. I also show that a conjecture of KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic $p>0$ is true for smooth projective surfaces. I give a new proof of a result by Li and prove a refinement of it. Based on my characterization of abelian varieties, I propose modifications of Li’s conjecture, which I expect to be true.

Let $X,\,Y$ be reduced and irreducible compact complex spaces and $S$ the set of all isomorphism classes of reduced and irreducible compact complex spaces $W$ such that $X\,\times \,Y\,\cong \,X\,\times \,W$. Here we prove that $S$ is at most countable. We apply this result to show that for every reduced and irreducible compact complex space $X$ the set $S(X)$ of all complex reduced compact complex spaces $W$ with $X\,\times \,{{X}^{\sigma }}\,\cong \,W\,\times \,{{W}^{\sigma }}$ (where ${{A}^{\sigma }}$ denotes the complex conjugate of any variety $A$) is at most countable.

Let $(X,L)$ be a polarized manifold over the complex number field with dim$X=n$. In this paper, we consider a conjecture of M. C. Beltrametti and A. J. Sommese and we obtain that this conjecture is true if $n=3$ and ${{h}^{0}}\,(L)\,\ge \,2$, or $\dim\,\text{Bs}|L|\le 0$ for any $n\ge 3$. Moreover we can generalize the result of Sommese.

Let X ⊂ Rnn be an irreducible nonsingular algebraic set and let Z be an algebraic subset of X with dim Z ≦ dim X — 2. In this paper it is shown that there exists an irreducible algebraic subset Y of X satisfying the following conditions: dim Y = dim X — 1, Z ⊂ Y and that the ideal of regular functions on X vanishing on Y is principal.