Moduli spaces of principal bundles on singular varieties

Let k be an algebraically closed field of characteristic zero. Let f:X-->S be a flat, projective morphism of k-schemes of finite type with integral geometric fibers. We prove existence of a projective relative moduli space for semistable singular principal bundles on the fibres of f. This generalizes the result of A. Schmitt who studied the case when X is a nodal curve.Comment: 25 pages; dedicated to the memory of Professor Masaki Maruyam

Semistable principal G-bundles in positive characteristic

Let $X$ be a normal projective variety defined over an algebraically closed field $k$ of positive characteristic. Let $G$ be a connected reductive group defined over $k$. We prove that some Frobenius pull back of a principal $G$-bundle admits the canonical reduction $E_P$ such that its extension by $P\to P/R_u(P)$ is strongly semistable. Then we show that there is only a small difference between semistability of a principal $G$-bundle and semistability of its Frobenius pull back. This and the boundedness of the family of semistable torsion free sheaves imply the boundedness of semistable principal $G$-bundles.Comment: 23 pages; The final version of this article will be published in the Duke Mathematical Journal, published by Duke University Pres

A note on restriction theorems for semistable sheaves

We prove a new restriction theorem for semistable sheaves on varieties in all characteristics strengthening previous results. We also prove restriction theorem for strong semistability for varieties with some non-negativity constrains on the cotangent bundle (e.g., most of Fano and Calabi-Yau varieties).Comment: 12 pages, to appear in Mathematical Research Letters, appendix contained serious error pointed out by Trivedi and it is removed in this versio