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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., Comment: Vikram Mehta passed away on 4th June 2014. This is a revised version

Let $X$ be a smooth projective variety over a perfect field $k$ of characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius pullback $(F^t)^*V$ is a direct sum of strongly semistable bundles. A natural question to ask is whether this still holds in higher dimension. Indranil Biswas, Yogish I. Holla, A.J. Parameswaran, and S. Subramanian showed that there is always a counterexample to this over any algebraically closed field of positive characteristic which is uncountable. However, we will produce a smooth projective variety over $\mathbb Z$ and a rank 2 vector bundle on it, which, restricted to each prime $p$ in a nonempty open subset of $\spec\mathbb Z$, constitutes a counterexample over $p$. Indeed, given any split semisimple simply connected algebraic group $G$ of semisimple rank $>1$ over $\mathbb Z$, we will show that there exists a smooth projective homogeneous space $X_Z$ over $\mathbb Z$ and a vector bundle $V$ on $X_Z$ of rank 2 such that for each prime $p$ in a nonempty open subset of $\spec\mathbb Z$, the restriction $V\otimes\mathbb F_p$ as a vector bundle over $X_Z\otimes\mathbb F_p$ is a counterexample. We only use the Borel-Weil-Bott theorem in characteristic 0 and Frobenius Splitting of $G/B$ in characteristic $p$., Comment: 3 pages

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

In this paper we study the extension of structure group of principal bundles with a reductive algebraic group as structure group on smooth projective varieties defined over algebraically closed field of positive characteristic. Our main result is to show that given a representation {\rho} of a reductive algebraic group G, there exists an integer t such that any semistable G-bundle whose first t frobenius pullbacks are semistable induces a semistable vector bundle on extension of structure group via {\rho}. Moreover we quantify the number of such frobenius pullbacks required.

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

Let $Y$ be a normal and projective variety over an algebraically closed field $k$ and $V$ a vector bundle over $Y$. We prove that if there exist a $k$-scheme $X$ and a finite surjective morphism $g:X\to Y$ that trivializes $V$ then $V$ is essentially finite., Comment: 5 pages

Let $X$ and $Y$ be nonsingular projective varieties over an algebraically closed field $k$ of positive characteristic. If $X$ and $Y$ are birational, we show their $S$-fundamental group schemes are isomorphic., Comment: 7 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 $\pi^S(X)$ studied in particular by Langer is trivial., Comment: 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., Comment: 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 show that for a Frobenius split variety, there are only finitely many closed subvarieties which are compatibly-split., Comment: 3 pages

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

Let $X$ be a smooth projective curve of genus $g \geq 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Given a semistable vector bundle $E$ over $X$, we show that its direct image $F\_*E$ under the Frobenius map $F$ of $X$ is again semistable. We deduce a numerical characterization of the stable rank-$p$ vector bundles $F\_*L$, where $L$ is a line bundle over $X$., Comment: 8 pages