Xiaotao Sun

Direct images of bundles under Frobenius morphism

Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k)=p>0 and F:X→X1 be the relative Frobenius morphism. For any vector bundle W on X, we prove that instability of F*W is bounded by instability of W⊗Tℓ(Ω1X) (0≤ℓ≤n(p-1)) (Corollary 4.9). When X is a smooth projective curve of genus g≥2, it implies F*W being stable whenever W is stable.

On Nori’s Fundamental Group Scheme

The aim of this note is to give a structure theorem on Nori’s fundamental group scheme of a proper connected variety defined over a perfect field and endowed with a rational point.

Remarks on Semistability of G-Bundles in Positive Characteristic

We analyzes a notion of strong semistability of principal G-bundles by including reduction to nonreduced parabolic subgroup schemes. It turns out that strong semistability is equivalent to the Frobenius semistability of Ramanan and Rananathan. We also give a bound for nonstrongly semsitability of a semistable GL(n)-bundle improving a previous result of Shepherd-Barron.

Factorization of generalized theta functions in the reducible case

We proved the factorization of generalized theta functions when the curve has two irreducible components meeting at one node.

Remarks on lines and minimal rational curves

We determine all of lines in the moduli space M of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section.

On canonical fibrations of algebraic surfaces

LetS be an algebraic surface of general type. If the canonical system |KS| ofS is a pencil of genusg, we hope to find the largestc(g) such thatKS2≥c(g)pg+constant. We have known thatc(3)≤6. In this paper, we proved thatc(3)≥5.25.

LOGARITHMIC HEAT PROJECTIVE OPERATORS

Abstract The sheaf of differential operators on log schemes is defined and studied. Then logarithmic differential operators on compactified Jacobians of singular curves are studied. In particular, a logarithmic heat equation for theta functions is produced geometrically.

On the cohomology of regular differential forms and dualizing sheaves

If Y is a local Dedekind scheme and X/Y is a projective Cohen– Macaulay variety of relative dimension 1, then Rf∗ω X/Y is torsionfree if and only if X/Y is arithmetically Cohen–Macaulay for a suitable embedding in Pk . If X is regular then Rf∗ω X/Y is torsionfree whenever the multiplicity of the special fibre is not a multiple of the characteristic of the residue class field. Regular differential forms play an important role in the local and global duality theory of varieties and morphisms, and have been studied extensively over the last few years (cf. [12], [5]). Given a complex projective manifold X and a morphism f : X → Y to a reduced variety Y , J. Kollar has shown that Rf∗ωX is torsionfree for any i ∈ N (cf. [10]). Little is known in the case of mixed characteristics, though in particular in the case of arithmetic varieties the torsionfreeness of the cohomology of canonical sheaves is of great interest. It appears in Bloch’s work on the de Rham discriminant [2], and the second author has shown that it allows one to control the ramification of the base change of arithmetic surfaces, hence giving explicit formulas and estimates for the self-intersection number of canonical sheaves (cf. [14]). In this note we develop some general criteria which reduce the problem of torsionfreeness of the cohomology of canonical sheaves to a question on homogeneous coordinate rings. Using the special situation in the case of arithmetic surfaces, we are able to settle the question in the case that the multiplicity of the special fibre is not a multiple of the characteristic of the residue class field. The work was done while the second author was a visiting fellow at the ICTP. He would like to express his thanks to the mathematics section of the ICTP for its hospitality. Both authors are grateful to Professor M. S. Narasimhan for helpful discussions. §1. Regular differential forms and residual complexes Let Y be an integral noetherian scheme and let f : X → Y be a projective morphism, Cohen–Macaulay, equi–dimensional of relative dimension d and with geometrically connected fibres. We assume that X is reduced and f is generically smooth, i.e. that the extension K(X)/K(Y ) of meromorphic functions is a direct Received by the editors December 18, 1996. 1991 Mathematics Subject Classification. Primary 13N05, 14F10. The first author was partially supported by a Heisenberg–Stipendium of the Deutsche Forschungsgemeinschaft. c ©1998 American Mathematical Society 1931 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1932 REINHOLD HUBL AND XIAOTAO SUN product of separable field extensions. In this situation the sheaf ω X/Y of relative regular differential forms of X/Y is well defined (cf. [11]), and it comes equipped with an intergral ∫ X/Y : Rf∗ω X/Y −−−−→ OY such that (ω X/Y , ∫ X/Y ) is a dualizing pair for f , i.e. ω X/Y is quasi–isomorphic to Grothendieck’s f !OY (cf. [9]) in the derived category. In order to study the torsion of the higher direct images Rf∗ω X/Y we may assume that Y = Spec(D) for some complete discrete valuation ring (D, π) whose residue class field k = D/πD is algebraically closed. Write X = Proj(S) for some graded reduced D–algebra S = ⊕ Sn. By ω d+1 S/D we denote the module of regular differential forms of S/D, and by AS we denote the system of denominators consisting of those sequences F1, . . . , Fl of homogeneous elements whose residue classes modulo π form an S/πS–active sequence. For details on systems of denominators, modules C AS (M) of generalized fractions of M with respect to AS and (unaugmented) Cousin-complexes C• AS (M) we refer the reader to [8] and [16], and also to [4]. Furthermore let E be a Dinjective hull of k and set E := Ẽ, the sheaf on Y associated to E. Furthermore let K• S(m) := (C• AS (ω S/D ⊗D E), d•)[d+ 1] be the Cousin complex of ω S/D ⊗D E with respect to AS , and set K• X(m) := K• S(m)[−1] . 1.1 Proposition. C• AS (S(l)⊗DE) is a flasque resolution of f∗E(l), and K• X(m) is a complex of injective OX–modules and a representative of f !E. Proof. First note that C• AS (M) is exact and a resolution of M if and only if AS consists of poor M–regular sequences. Thus C• AS (S ⊗D E) is in general not exact; however, for any homogeneous elementG ∈ S+ of positive degree, SG/D is a Cohen– Macaulay–algebra, and hence all sequences (F) ∈ AS are E⊗D SG–poor sequences (cf. [4] , §1), implying that C• AS (S(l)⊗DE) is a resolution of (S(l)⊗DE) = f∗E . As C AS (S ⊗D E) ∼= ⊕ C AS (S ⊗ E)Ph , where the sum runs over all homogeneous primes of S/πS of height p, it follows that C• AS (S ⊗D E) also is flasque. Similarly, for any homogeneous G ∈ S+ of positive degree, (ω S/D)G is a Gorenstein module, implying that (C• AS (ω S/D ⊗D E))G is a graded injective resolution of ω SG/D ⊗ E; hence K• X(m) is an injective resolution of (ω S/R ⊗D E) = (ω S/D) ⊗OX f∗E . As ω X/Y = (ω S/D) by [5], (2.6), the proposition follows. 1.2 Corollary. For i 0 we have 1.3 Proposition. For i > 0 the following conditions are equivalent: i) The sheaf Rf∗ω X/Y is torsion free. ii) The morphism μπ : HomOY (R f∗ω X/Y , E) → HomOY (Rf∗ω X/Y , E) is surjective. iii) The morphism μπ : Hom cont D (H i+1 S+ (ω S/D), E)0 → Hom D (H S+ (ω S/D), E)0 is surjective. iv) The map μπ : H −(i+1)(HomS(ωd+1 S/D,K S(m)))0 → H−(i+1)(HomS(ωd+1 S/D,K S(m)))0 is surjective. v) The morphism μπ : H d−i(C• AS (S ⊗D E))0 → Hd−i(C• AS (S ⊗D E))0