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 space of vector bundles on X of suitable rank. In characteristic two we provide a complete classification of rank two semi-stable vector bundles whose Frobenius pull-back is not semi-stable. We also obtain fairly good information about the strata of the Frobenius stratification, including the irreducibility and the dimension of each non-empty Frobenius stratum. In particular we show that the locus of Frobenius destabilized bundles has dimension 3g-4 in the moduli space of semi-stable bundles of rank two. We also construct stable bundles that are destabilized by Frobenius in the following situations: characteristic p=2 and rank four, (2) characteristic p=rank=3, (3) characteristic p=rank=5 and g at least three.
We also explore (in any characteristic) the connection between Frobenius destabilized bundles and (pre)-opers, this approach allows us to reinterpret some of our results in terms of pre-opers and also allows us to construct Frobenius destablised bundles from certain pre-opers (or opers).
The other result we obtain is (for characteristic two): we show that the Gunning bundle descends under Frobenius when genus g is even. If g is odd, then the Gunning bundle twisted by any odd degree line bundle also descends.
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 destabilized by the Frobenius morphism. Then we prove that such bundles of rank 2 exist over any curve in characteristic 3, and are unique up to twisting by a line bundle. We also give an application of such bundles to the study of ample vector bundles, which is valid in all characteristics.
Let X be a smooth projective curve of genus g \textgreater{}1 defined over an algebraically closed field k of characteristic p \textgreater{}0. For p sufficiently large (explicitly given in terms of r,g) we construct an atlas for the locus of all Frobenius-destabilized bundles (i.e. we construct all Frobenius-destabilized bundles of degree zero up to isomorphism). This is done by exhibiting a surjective morphism from a certain Quot-scheme onto the locus of stable Frobenius-destabilized bundles. Further we show that there is a bijective correspondence between the set of stable vector bundles E …
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 F: X \to X_1 induces by pull-back a rational map V: M_{X_1} \to M_{X}. In this paper we show the following results. 1) For any line bundle L over X, the rank-p vector bundle F_*L is stable. 2) The rational map V has base points, i.e., there exist stable bundles E over X_1 such that F^* E is not semistable. 3) Let B \subset M_{X_1} denote the scheme-theoretica…
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 the Frobenius pull-back may destory stability of a vector bundle. This paper introduces and studies a Harder-Narasimhan type stratification on the moduli scheme and proves that the family of semi-stable rank-2 vector bundles (with a fixed degree) whose Frobenius pull-back are not semi-stable is parameterized by an irreducible subscheme of dimension 3g-4.
Let X be a smooth projective curve over an algebraic closed field of characteristic p and F be the Frobenius morphism of X. Here, I give a negative answer to the guess that the length of the Harder-Narasimhan of F*W is not bigger than p, where W is a semistable bundle on X.
Here we prove that for a smooth projective variety $X$ of arbitrary dimension and for a vector bundle $E$ over $X$, the Harder-Narasimhan filtration of a Frobenius pull back of $E$ is a refinement of the Frobenius pull-back of the Harder-Narasimhan filtration of $E$, provided there is a lower bound on the characteristic $p$ (in terms of rank of $E$ and the slope of the destabilising sheaf of the cotangent bundle of $X$). We also recall some examples, due to Raynaud and Monsky,to show that some lower bound on $p$ is necessary. We further prove an analogue of this result for principal $G$-bundle…
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 exact differential forms and we derive a scheme-theoretic description of the Frobenius inverse image of the semi-stable boundary of the moduli space.
Let $X$ be a smooth projective curve of genus $g(X)\geq 1$ over an algebraically closed field $k$ of characteristic $p>0$, $\M^s_X(r,d)$ the moduli space of stable vector bundles of rank $r$ and degree $d$ on $X$. We study the Frobenius stratification of $\M^s_X(r,d)$ in terms of Harder-Narasimhan polygons of Frobenius pull backs of stable vector bundles and obtain the irreducibility and dimension of each non-empty Frobenius stratum in case $(p,g,r)=(3,2,3)$.
Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$, $F_X:X\rightarrow X$ the absolute Frobenius morphism. Let $\M^s_X(r,d)$ be the moduli space of stable vector bundles of rank $r$ and degree $d$ on $X$. We study the Frobenius stratification of $\M^s_X(3,0)$ in terms of Harder-Narasimhan polygons of Frobenius pull backs of stable vector bundles and obtain the irreducibility and dimension of each non-empty Frobenius stratum in the case $(p,g)=(3,2)$.
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 ρ 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 ρ. Moreover we quantify the number of such frobenius pullbacks required.
We use the theory of p-curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the Frobenius morphism. We take two approaches, first using explicit formulas for p-curvature to analyze low-characteristic cases, and then using degeneration techniques to obtain an answer for a general curve by degenerating to an irreducible rational nodal curve, and applying the results of math.AG/0409145 and math.AG/0407445. We also apply our explicit formulas to give a new description of the strata of curves of genus 2 of different p-ran…
This article is the expanded version of a talk given at the conference: Algebraic geometry in East Asia 2008, Seoul. In this notes, I intend to give a brief survey of results on the behavior of semi-stable bundles under the Frobenius pullback and direct images. Some results are new.
We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…
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 princ…
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 tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.
We study Hilbert-Kunz multiplicity of non-singular curves in positive characteristic. We analyse the relationship between the Frobenius semistability of the kernel sheaf associated with the curve and its ample line bundle, and the HK multiplicity. This leads to a lower bound, achieved iff the kernel sheaf is Frobenius semistable, and otherwise to formulas for the HK multiplicity in terms of parameters measuring the failure of Frobenius semistability. As a byproduct, an explicit example of a vector bundle on a curve is given whose $n$-th iterated Frobenius pullback is not semistable, while its …
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 space of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 with determinant equal to a theta characteristic whose Frobenius pull-back is not stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.
In this note we show that the restriction of the cotangent bundle $Ω_{\mathbb P}^2$ of the projective plane to a Fermat curve $C$ of degree $d$ in characteristic $p \equiv -1 \mod 2d$ is, up to tensoration with a certain line bundle, isomorphic to its Frobenius pull-back. This leads to a Frobenius periodicity $F^*({\mathcal E}) \cong {\mathcal E} $ on the Fermat curve of degree 2d, where ${\mathcal E}= {\rm Syz}(U^2,V^2,W^2)(3)$.
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 map F : X \to X_1 induces by pull-back a rational map V: M_{X_1}(r) \to M_X(r). We determine the equations of V in the following two cases (1) (g,r,p) = (2,2,2) and X non-ordinary with Hasse-Witt invariant equal to 1 (see math.AG/0005044 for the case X ordinary), and (2) (g,r,p) = (2,2,3). We also show, for any triple (g,r,p), the existence of base points of …
We analyze the deformation theory of equivariant vector bundles. In particular, we provide an effective criterion for verifying whether all infinitesimal deformations preserve the equivariant structure. As an application, using rigidity of the Frobenius homomorphism of general linear groups, we prove that projectivizations of Frobenius pullbacks of tautological vector bundles on Grassmanians are arithmetically rigid, that is, do not lift over rings where $p \neq 0$. This gives the same conclusion for Totaro's examples of Fano varieties violating Kodaira vanishing. We also provide an alternativ…
We define the $p^m$-curvature map on the sheaf of differential operators of level $m$ on a scheme of positive characteristic $p$ as dual to some divided power map on infinitesimal neighborhhods. This leads to the notion of $p^m$-curvature on differential modules of level $m$. We use this construction to recover Kaneda's description of a semi-linear Azumaya splitting of the sheaf of differential operators of level $m$. Then, using a lifting modulo $p^2$ of Frobenius, we are able to define a Frobenius map on differential operators of level $m$ as dual to some divided Frobenius on infinitesimal n…
Let $X$ be genus 2 curve defined over an algebraically closed field of characteristic $p$ and let $X\_1$ be its $p$-twist. Let $M\_X$ (resp. $M\_{X\_1}$) be the (coarse) moduli space of semi-stable rank 2 vector bundles with trivial determinant over $X$ (resp. $X\_1$). The moduli space $M\_X$ is isomorphic to the 3-dimensional projective space and is endowed with an action of the group $J[2]$ of order 2 line bundles over $X$. When $3\leq p \leq 7$, we show that the Verschiebung (i.e., the separable part of the action of Frobenius by pull-back) $V : M\_{X\_1} \dashrightarrow M\_X$ is completely…
In this paper we describe the Frobenius pull-backs of the syzygy bundles $Syz_C(X^a, Y^a, Z^a)$, $a \geq 1$, on the projective Fermat curve C of degree n in characteristics coprime to n, either by giving their strong Harder-Narasimhan Filtration if $Syz_C(X^a, Y^a, Z^a)$ is not strongly semistable or in the strongly semistable case by their periodicity behavior. Moreover, we apply these results to Hilbert-Kunz functions, to find Frobenius periodicities of the restricted cotangent bundle $Ω_{P^2}|_C$ of arbitrary length and a problem of Brenner regarding primes with strongly semistable reductio…
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 …
The base space of a semi-universal unfolding of a hypersurface singularity carries a rich geometric structure, which was axiomatized as a CDV-structure by C. Hertling. For any CDV-structure on a Frobenius manifold M, the pull-back of the (1,0)-tangent bundle of M to the product of M by the complex line carries two natural holomorphic structures equipped with flat meromorphic connections. We show that, for any semi-simple CDV-structure, there is a formal isomorphism between these two bundles compatible with connections. Moreover, if we assume that the super-symmetric index Q vanishes, we give a…
The Tango bundle T over P^5 is proved to be the pull-back of the twisted Cayley bundle C(1) via a map f : P^5 --> Q_5 existing only in characteristic 2. The Frobenius morphism F factorizes via such f. Using f the cohomology of T is computed in terms of F^*(C), Sym^2(C), C and the tensor product of S by C, while these are computed by applying Borel-Bott-Weil theorem. By machine-aided computation the mimimal resolutions of C and T are given; incidentally the matrix presenting the spinor bundle S over Q_5 is shown.
This paper is devoted to the study of the Higgs bundle associated with the universal abelian variety over the good reduction of a Shimura curve of PEL type. Due to the endomorphism structure, the Higgs bundle decomposes into the direct sum of Higgs subbundles of rank two. They are basically divided into two type: uniformizing type and unitary type. As the first application we obtain the mass formula counting the number of geometric points of the degeneracy locus in the Newton polygon stratification. We show that each Higgs subbundle is Higgs semistable. Furthermore, for each Higgs subbundle of…
We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where $k$ is an algebraically closed field of characteristic $p>0$. Let $E$ be a uniform vector bundle over $G$ of rank $r\le d$. We show that $E$ is either a direct sum of line bundles or a twist of a pull back of the universal bundle $H_d$ or its dual $H_d^{\vee}$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles…
Suppose B=F[x,y,z]/h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous (x,y,z)-primary ideal and n -> e_n be the Hilbert-Kunz function of B with respect to J. Let q=p^n. When J=(x,y,z), Pardue (see R. Buchweitz, Q. Chen. Hilbert-Kunz functions of cubic curves and surfaces. J. Algebra 197 (1997). 246-267) showed that e_n=(7q^2)/3-q/3-R where R=5/3 if q is congruent to 2 (3), and is 1 otherwise. We generalize this, showing that e_n= (mu q^2) + (alpha q) - R where R only depends on q mod 3. We describe alpha and R in te…
