Related papers: Frobenius morphism and semi-stable bundles

We show the Frobenius pullback of a general semi-stable vector bundle in the moduli space of vector bundles with fixed rank and degree is still semi-stable by deformation trick. We then present several applications of the main theorem.

Let X be a smooth projective curve of genus g>1 over an algebraically closed field of characteristic 2. Pull-back by the (absolute) Frobenius on X only defines a rational morphism on the moduli scheme of rank-2 vector bundles on X, because…

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. For any vector bundle $W$ on $X$, we prove that instability of…

Let X be a smooth projective curve of genus $g\geq 2$ defined over an algebraically closed field k of characteristic $p>0$ and let $F:X\rightarrow X_{1}$ be the relative k-linear Frobenius map. We prove (Theorem 1.1) E is a stable bundle on…

We study parabolic bundles on an algebraic curve in positive characteristic. Our motivation is to properly formulate Frobenius pull-backs of parabolic bundles in a way that extends various previous facts and arguments for the usual…

We study stable rank 2 vector bundles with trivial determinant whose Frobenius pull back is non stable over a general curve of genus g>1. In genus 2, we apply recent results about the theta divisor associated to the bundle B of locally…

Let $X$ be a smooth projective variety over an algebraically field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. When ${\rm dim}(X)=1$, we prove that $F_*W$ is a stable bundle for any stable bundle $W$…

We generalize, explain and simplify Langer's results concerning Frobenius direct images of line bundles on quadrics, describing explicitly the decompositions of higher Frobenius push-forwards of arithmetically Cohen-Macaulay bundles into…

We give a new, shorter computation of Frobenius push-forwards of line bundles on toric varieties.

Let $X$ be a smooth projective curve of genus $g(X)\geq 1$ over an algebraically closed field $k$ of characteristic $p>0$ and $F_{X/k}:X\rightarrow X^{(1)}$ be the relative Frobenius morphism. Let $\mathfrak{M}^{s(ss)}_X(r,d)$ (resp.…

We study the various arithmetic and geometric Frobenius morphisms on the moduli stack of principal bundles over a smooth projective algebraic curve and determine explicitly their actions on the $\ell-$adic cohomology of the moduli stack in…

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to…

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…

Let X be a smooth projective curve of genus g>1 defined over an algebraically closed field k of characteristic p>0. Let M_X(r) be the moduli space of semi-stable rank r vector bundles with fixed trivial determinant. The relative Frobenius…

In this article, we study the behavior of the stability of pullback of a vector bundle under a finite morphism from a (not necessarily smooth) stacky curve to an orbifold curve. We establish a categorical equivalence between proper formal…

Vector bundles in positive characteristics have a tendency to be destabilized after pulling back by the Frobenius morphism. In this paper, we closely examine vector bundles over curves that are, in an appropriate sense, maximally…

In this paper we describe the action of the Frobenius morphism on the indecomposable vector bundles on cycles of projective lines. This gives an answer on a question of Paul Monsky, which appeared in his study of the Hilbert--Kunz theory…

Let X be an irreducible smooth projective curve of genus at least two over an algebraically closed field k of characteristic p>0. In this paper we study the natural stratification, defined using the absolute Frobenius of X, on the moduli…

Let X be a smooth projective curve of genus g \geq 2 over an algebraically closed field k of characteristic p > 0. Let M_X be the moduli space of semistable rank-2 vector bundles over X with trivial determinant. The relative Frobenius map…

This paper review one construction of Frobenius manifolds (and slightly weaker structures). It splits it into several steps and discusses the freedom and the constraints in these steps. The steps pass through holomorphic bundles with…