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We study the properness of the functor of $F$-trivial bundles by relating it to the base change question for the fundamental group scheme of Nori., Vikram Mehta passed away on 4th June 2014. This is a revised version

Let $k$ be an algebraically closed field of characteristic $p>0$, $W$ the ring of Witt vectors over $k$ and ${R}$ the integral closure of $W$ in the algebraic closure ${\bar{K}}$ of $K:=Frac(W)$; let moreover $X$ be a smooth, connected and projective scheme over $W$ and $H$ a relatively very ample line bundle over $X$. We prove that when $dim(X/{W})\geq 2$ there exists an integer $d_0$, depending only on $X$, such that for any $d\geq d_0$, any $Y\in |H^{\otimes d}|$ connected and smooth over ${W}$ and any $y\in Y({W})$ the natural ${R}$-morphism of fundamental group schemes $\pi_1(Y_R,y_R)\to \pi_1(X_R,y_R)$ is faithfully flat, $X_R$, $Y_R$, $y_R$ being respectively the pull back of $X$, $Y$, $y$ over $Spec(R)$. If moreover $dim(X/{W})\geq 3$ then there exists an integer $d_1$, depending only on $X$, such that for any $d\geq d_1$, any $Y\in |H^{\otimes d}|$ connected and smooth over ${W}$ and any section $y\in Y({W})$ the morphism $\pi_1(Y_R,y_R)\to \pi_1(X_R,y_R)$ is an isomorphism., Comment: 11 pages

We prove that the Local Fundamental Group Scheme satisfies the Lefschetz - Bott theorems in characteristic p. The proofs are standard applications of the Enriques-Severi -Zariski-Serre vanishing theorems and known facts about the p-curvature

We show that for a Frobenius split variety, there are only finitely many closed subvarieties which are compatibly-split., 3 pages

On $X$ projective smooth over an algebraically closed field, we show that if Nori's fundamental group scheme is trivial, then there are no nontrivial Nori semistable bundles of degree 0, that is the group scheme $��^S(X)$ studied in particular by Langer is trivial., 8 pages

We show that simply connected projective manifolds in characteristic $p>0$ have no nontrivial stratified bundles. This gives a positive answer to a conjecture by D. Gieseker. The proof uses Hrushovski's theorem on periodic points., 16 pages. Revised version contains a more general theorem on torsion points on moduli, together with an illustration in rank 2 due to M. Raynaud. Reference added. Last version has some typos corrected. Appears in Invent.math.

We prove that Local Fundamental Group Scheme satisfies the conditions of Lefschetz-Bott-Grothendieck., Comment: I withdraw this paper as Biswas-Holla already had a more complete version of essentially the same theorems in an earlier article on the arXiv

We define the local fundamental group scheme, defined by the F-trivial vector bundles, and give necessary and sufficient conditions for it to base change., Comment: submitted to the Proceedings of the Indian Academy of Sciences, Mathematical Sciences on the 3rd of April, 2006

In this talk we discuss the relations between representations of algebraic groups and principal bundles on algebraic varieties, especially in characteristic $p$. We quickly review the notions of stable and semistable vector bundles and principal $G$-bundles, where $G$ is any semisimple group. We define the notion of a low height representation in characteristic $p$ and outline a proof of the theorem that a bundle induced from a semistable bundle by a low height representation is again semistable. We include applications of this result to the following questions in characteristic $p$: 1) Existence of the moduli spaces of semistable $G$-bundles on curves. 2) Rationality of the canonical parabolic for nonsemistable principal bundles on curves. 3) Luna's etale slice theorem. We outline an application of a recent result of Hashimoto to study the singularities of the moduli spaces in (1) above, as well as when these spaces specialize correctly from characteristic 0 to characteristic $p$. We also discuss the results of Laszlo-Beauville-Sorger and Kumar-Narasimhan on the Picard group of these spaces. This is combined with the work of Hara and Srinivas-Mehta to show that these moduli spaces are $F$-split for $p$ very large. We conclude by listing some open problems, in particular the problem of refining the bounds on the primes involved.

In this paper we construct the moduli of semistable principal bundles with structure group a semisimple group, over nonsingular curves in positive characteristic, with good bounds on the prime., Comment: LaTeX2e, 13 pages