Higher Direct Images Research Articles

-modules on rigid analytic spaces III: weak holonomicity and operations

AbstractWe develop a dimension theory for coadmissible$\widehat {\mathcal {D}}$-modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic$\mathcal {D}$-modules in the algebraic setting. We discuss a number of pathologies contained in this subcategory (modules of infinite length, infinite-dimensional fibres). We prove stability results for closed immersions and the duality functor, and show that all higher direct images of integrable connections restricted to a Zariski open subspace are coadmissible of minimal dimension. It follows that the local cohomology sheaves$\underline {H}^{i}_Z(\mathcal {M})$with support in a closed analytic subset$Z$of$X$are also coadmissible of minimal dimension for any integrable connection$\mathcal {M}$on$X$.

On direct images of twisted pluricanonical sheaves on normal varieties

We study the depth properties of certain direct image sheaves on normal varieties. Let $f: Y\rightarrow X$ be a proper morphism of relative dimension $d$ from a smooth variety onto a normal variety such that the preimage $E$ of the singular locus of $X$ is a divisor. We show that for any integer $m>0$, the higher direct image $R^df_*\omega^{\otimes m}_Y(aE)$ modulo the torsion subsheaf is $S_2$, provided that $a$ is sufficiently large. In case $f$ is birational, we give criteria on $a$ for the direct image $f_*\omega_Y(aE)$ to coincide with $\omega_X$. We also introduce an index measuring the singularities of normal varieties.

On higher direct images of convergent isocrystals

Let $k$ be a perfect field of characteristic $p>0$ and let $\operatorname{W}$ be the ring of Witt vectors of $k$. In this article, we give a new proof of the Frobenius descent for convergent isocrystals on a variety over $k$ relative to $\operatorname{W}$. This proof allows us to deduce an analogue of the de Rham complexes comparison theorem of Berthelot [$\mathscr{D}$-modules arithmétiques. II. Descente par Frobenius, Mém. Soc. Math. Fr. (N.S.) 81 (2000)] without assuming a lifting of the Frobenius morphism. As an application, we prove a version of Berthelot’s conjecture on the preservation of convergent isocrystals under the higher direct image by a smooth proper morphism of $k$-varieties.

Semipositivity of generic ordinary fibration over a curve

Let $f:~X\to~B$ be a generic ordinary proper fibration over a complete curve in positive characteristics, we prove that the dual of higher direct image sheaf $R^1f_*\mathcal{O}_X$ is nef. As a corollary, we show that $f_*\omega_{S/B}$ is nef, if $f:S\rightarrow~B$ is a fibration from a surface to a curve with generic ordinary fibres.

WITHDRAWN: Addendum to: Incarnations of Berthelot's conjecture

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On the semicontinuity problem of fibers and global F-regularity

ABSTRACTIn this article, we discuss the semicontinuity problem of certain properties on fibers for a morphism of schemes. One aspect of this problem is local. Namely, we consider properties of schemes at the level of local rings, in which the main results are established by solving the lifting and localization problems for local rings. In particular, we obtain the localization theorems in the case of seminormal and F-rational rings, respectively. Another aspect of this problem is global, which is often related to the vanishing problem of certain higher direct image sheaves. As a test example, we consider the deformation of the global F-regularity.

Relative cohomology of cuspidal forms on PEL-type Shimura varieties

We present a short proof that, for PEL-type Shimura varieties, subcanonical extensions of automorphic bundles, whose global sections over toroidal compactications of Shimura varieties are represented by cuspidal au- tomorphic forms, have no higher direct images under the canonical morphism to the minimal compactication, in characteristic zero or in positive charac- teristics greater than an explicitly computable bound.

Depth of multiplier ideals, vanishing theorem and multiplicity of minimal centers

The purpose of this note is to give a sufficient condition for the smoothness of the minimal centers at given points in terms of the vanishing of the higher direct image sheaves associated with the ideal quotients of the multiplier ideal sheaves by the maximal ideal sheaves.

Commutative Formal Groups Arising from Schemes

We prove the following criterion for the pro-representability of the deformation cohomology of a commutative formal Lie group. Let f be a flat and separated morphism between noetherian schemes. Assume that the target of f is flat over the integers. For a commutative formal Lie group E, we have the deformation cohomology of f with coefficients in E at our disposal. If the higher direct images of the tangent space of E are locally free and of finite rank then the deformation cohomology is pro-representable by a commutative formal Lie group.

Calabi–Yau varieties with semi-stable fibre structures

Motivated by the Strominger–Yau–Zaslow conjecture, we study Calabi–Yau varieties with semi-stable fibre structures. We use Hodge theory to study the higher direct images of wedge products of relative cotangent sheaves of certain semi-stable families over higher dimensional quasi-projective bases, and obtain some results on positivity. We then apply these results to study non-isotrivial Calabi–Yau varieties fibred by semi-stable Abelian varieties (or hyperkahler varieties).

Hodge modules on Shimura varieties and their higher direct images in the Baily–Borel compactification

We study the degeneration in the Baily–Borel compactification of variations of Hodge structure on Shimura varieties. Our main result, Theorem 2.6, expresses the degeneration of variations given by algebraic representations in terms of Hochschild, and abstract group cohomology. It is the Hodge theoretic analogue of Pink's theorem on degeneration of étale and ℓ-adic sheaves [Math. Ann. 292 (1992) 197], and completes results by Harder and Looijenga–Rapoport [Eisenstein-Kohomologie arithmetischer Gruppen: Allgemeine Aspekte, Preprint, 1986; Proc. of Symp. in Pure Math., vol. 53, 1991, pp. 223–260]. The induced formula on the level of singular cohomology is equivalent to the theorem of Harris–Zucker on the Hodge structure of deleted neighbourhood cohomology of strata in toroidal compactifications [Inv. Math. 116 (1994) 243].

Fourier-Mukai transform and index theory

Given a submersive morphism of complex manifoldsf: X→Y, and a complex vector bundleE onX, there is a relationship between the higher direct images of e (the sheaf of holomorphic sections ofE) and the index of the relative Dolbeault complex twisted byE. This relationship allows one to yield a global and simple proof of the equivalence between the Mukai transform of stable vector bundles on a torusT of complex dimension 2 and the Nahm transform of instantons. We also offer a proof of Mukai’s inversion theorem which circumvents the use of derived categories by introducing spectral sequences of sheaves onT (this is related to Donaldson and Kronheimer’s proof, but is automatically global and somehow simpler). The general framework developed in the first part of this paper may be applied to the study of the Mukai transform for more general varieties.

Calculating Discriminants by Higher Direct Images

The author uses the homological algebra to construct for any line bundle $\mathcal {L}$ on a nonsingular projective variety X the complex $\mathbb {F}(\mathcal {L})$ whose determinant is equal to the equation of the dual variety ${X^{\text {V}}}$. This generalizes the Cayley-Koszul complexes defined by Gelfand, Kapranov and Zelevinski. The formulas for the codimension and degree of ${X^{\text {V}}}$ in terms of complexes $\mathbb {F}(\mathcal {L})$ are given. In the second part of the article the general technique is applied to classical discriminants and hyperdeterminants.

Calculating discriminants by higher direct images

The author uses the homological algebra to construct for any line bundle L \mathcal {L} on a nonsingular projective variety X the complex F ( L ) \mathbb {F}(\mathcal {L}) whose determinant is equal to the equation of the dual variety X V {X^{\text {V}}} . This generalizes the Cayley-Koszul complexes defined by Gelfand, Kapranov and Zelevinski. The formulas for the codimension and degree of X V {X^{\text {V}}} in terms of complexes F ( L ) \mathbb {F}(\mathcal {L}) are given. In the second part of the article the general technique is applied to classical discriminants and hyperdeterminants.

On ?-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification

On ?-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification

Hodge filtrations and the higher direct images of canonical sheaves

Hodge filtrations and the higher direct images of canonical sheaves