Let $K$ be a number field, and let $X \subseteq P^{s-1}_K$ be a smooth complete intersection defined over $K$. In this paper, weak approximation is shown to hold for $X$ provided $s$ exceeds some function of the degree and codimension of $X$. This is a corollary of a more general result about the number of integral points on certain affine varieties in homogeneously expanding regions. This general result is established via a suitable adaptation of the Hardy-Littlewood Circle Method.

For a smooth projective variety $X$ of dimension $n$ in a projective space ${\bb P}^N$ defined over an algebraically closed field $k$, the Gauss map is a morphism from $X$ to the Grassmannian of $n$-plans in ${\bb P}^N$sending $x \in X$ to the embedded tangent space $T_xX \subset {\bb P}^N$. The purpose of this paper is to prove the generic injectivity of Gauss maps in positive characteristic for two cases; (1) weighted complete intersections of dimension $n \geqslant 3$ of general type; (2) surfaces or 3-folds with $\mu$-semistable tangent bundles; based on a criterion of Kaji by looking at the stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman-Piene is true in case $X$ is of general type of dimension $n \geqslant 3$. The second result is a generalization of the injectivity for curves.

Let $f:X\rightarrow Y$ be a morphism of noetherian schemes, generically smooth and equidimensional of dimension $d, \iota : X^{\prime} \rightarrow X$ a closed embedding such that $f \circ \iota : X^{\prime} \rightarrow Y$ is generically smooth and equidimensional of dimension $d^{\prime}$, and $X^{\prime}, X$ and $Y$ are excellent schemes without embedded components. We exhibit a concrete morphism\[{\rm Res}_{X^{\prime}/X}: {\rm det}\,{\cal N}_{X^{\prime}/X} \otimes_{{\cal O}_{X^{\prime}}} \iota^* \omega^d_{X/Y} \rightarrow \omega^{d^{\prime}}_{X^{\prime}/Y},\] which transforms the integral of $X/Y$ into the integral of$X^{\prime}/Y$. Here ${\cal N}_{X^{\prime}/X}$ denotes the normal sheaf of $X^{\prime}/X$ and$\omega^d_{X/Y}$ resp. $\omega^{d^{\prime}}_{X^{\prime}/Y}$ denotes the sheaf of regular differential forms of $X/Y$ resp. $X^{\prime}/Y$. Using generalized fractions we provide a canonical description of residual complexes and residue pairs of Cohen-Macaulay varieties, and obtain a very explicit description of fundamental classes and their traces.

Explicit formulas for $K$-types of a $({\frak g}; K)$ module, such as Blattner's formula, are well known. However, the formulas are often too complex to make the $K$-type structure transparent. In this note we make use of some ideas of Vogan linking the ‘edges’ of the set of $K$-types to a certain restriction map from ${\frak u}$ to ${\frak u} \cap {\frak t}$ cohomology. It is hoped that such ideas will lead to tighter control of the set of $K$-types.

One-to-one correspondences are established between the set of all nondegenerate graded Jacobi operators of degree $-1$ defined on the graded algebra $\Omega(M)$ of differential forms on a smooth, oriented, Riemannian manifold $M$, the space of bundle isomorphisms $L{\A}TM{\to} TM$, and the space of nondegenerate derivations of degree $1$ having null square. Derivations with this property, and Jacobi structures of odd $\Bbb Z_2$-degree are also studied through the action of the automorphism group of $\Omega(M)$.

We study the torsion in the Mordell-Weil group of the Jacobian of the Fermat curve of exponent $p$ over the cyclotomic field obtained by adjoining a primitive $p$-th root of 1 to $Q$. We show that for all (except possibly one) proper subfields of this cyclotomic field, the torsion parts of the corresponding Mordell-Weil groups are elementary abelian $p$-groups.

In the thesis [Pe1] it was introduced, studied and applied a general theory of Weierstrass loci for vector bundles on a smooth curve. The results, proofs, background, examples and motivations of this thesis are contained in [Pe2]. We believe that this theory, at least in characteristic 0, is the ‘right’ one. The aim of this paper is to introduce and study an extension of [Pe1] to the case of higher dimensional varieties. At least two possible theories seem to be useful and natural; see the discussion just after Remark 1.1 and Section 4. We strongly prefer the ‘symmetric’ one (see Definition 1.5). In the first section we introduce the general theory and give the main general results. In the second section we study in details the case of $P^2$ for three reasons: it is nice; it shows how to use the general theory and what could be expected in more general situations and (last but not least) to convince the reader that it is technically easier and often more interesting to work in the ‘symmetric’ set up. Then in the third section we apply the method of Section 2 to a much more general situation (essentially, any variety $X$ as base of the vector bundle). In the fourth section we give the set up and start the analysis of specific examples of what happens near a specific point $P$ of the base variety $X$ (even when $X$ is singular at $P$). Here, except at the first step we are able to work only with the ‘symmetric’ definition.

In this paper we compute the cohomology with compact supports of a Siegel threefold as a virtual module over the product of the Galois group of$\bar{\bb Q}$ over ${\bb Q}$ and the Hecke algebra. We use a method which has been developed by Ihara, Langlands and Kottwitz: comparison of the Grothendieck—Lefschetz formula and the Arthur—Selberg trace formula.

We show that the birational classification in positive characteristic of smooth Fano threefolds $X$ with Picard number 1 is the same as in characteristic zero. In particular, there are no exotic such Fanos; as a consequence of the classification, $X$ is shown to be liftable without ramification to characteristic zero and to contain a line. The main techniques employed are those of Ekedahl and of Mori and Takeuchi.

Let ${E}$ be a nonconstant elliptic curve, over a global field ${K}$ of positive, odd characterisitc. Assuming the finiteness of the Shafarevich-Tate group of ${E}$, we show that the order of the Shafarevich-Tate group of ${E}$, is given by${O}({N}^{1/2+6\,\log(2)/\log({q})})$, where ${N}$ is the conductor of ${E}, {q}$ is the cardinality of the finite field of constants of ${K}$, and where the constant in the bound depends only on ${K}$. The method of proof is to work with the geometric analog of the Birch-Swinnerton Dyer conjecture for the corresponding elliptic surface over the finite field, as formulated by Artin-Tate, and to examine the geometry of this elliptic surface.

Let $X$ be a complex analytic manifold, $M \subset X$ a $C^2$ submanifold, $\Omega\subset M$ an open set with $C^2$ boundary $S=\partial\Omega$. Denote by $\mu_M({\cal O}_X)$ (resp. $\mu_\Omega({\cal O}_X)$) the microlocalization along $M$ (resp. $\Omega$) of the sheaf ${\cal O}_X$ of holomorphic functions.

In the literature (cf. [A-G], [K-S 1,2]) one encounters two classical results concerning the vanishing of the cohomology groups$H^j\mu_M({\cal O}_X)_p$ for $p\in \dot{T}^*_MX$. The most general gives the vanishing outside a range of indices $j$ whose length is equal to $s^0(M,p)$ (with $s^{+,-,0}(M,p)$ being the number of respectively positive, negative and null eigenvalues for the ‘microlocal’ Levi form $L_M(p)$). The sharpest result gives the concentration in a single degree, provided that the difference $s^-(M,p^{\prime})-\gamma(M,p^{\prime})$ is locally constant for $p^{\prime}\in T^*_MX$ near $p$ (with $\gamma(M,p)=\dim^{\rm C}(T^*_MX\cap iT^*_MX)_z$ for $z$ the base point of $p$).

The first result was restated for the complex $\mu_\Omega({\cal O}_X)$ in [D'A-Z 2], in the case ${\rm codim}_MS=1$. We extend it here to any codimension and moreover we also restate for $\mu_\Omega({\cal O}_X)$ the second vanishing theorem.

We also point out that the principle of our proof, related to a criterion for constancy of sheaves due to [K-S 1], is a quite new one.

We extend a result of Vojta on height inequalities for algebraic points on curves over function fields to include the case of positive characteristic. The main tool used is the Kodaira—Spencer map and destabilizing flags for vector bundles on curves.

We show that the cohomology of the moduli space of flat SU(2) connections on a two-manifold may be computed using a perfect Morse function.

A strategy is proposed for applying Chabauty's Theorem to hyperelliptic curves of genus ${>} 1$. In the genus 2 case, it shown how recent developments on the formal group of the Jacobian can be used to give a flexible and computationally viable method for applying this strategy. The details are described for a general curve of genus 2, and are then applied to find ${\bm C}({\bb Q})$ for a selection of curves. A fringe benefit is a more explicit proof of a result of Coleman.

Soit $F$ un corps local non archimédien de caractéristique nulle, à corps résiduel fini. Si $M$ est un groupe réductif connexe défini sur $F$, on note par la lettre minuscule $m$ son algébre de Lie. On note $m_{\rm reg}$ l'ensemble des $X \in m$ dont le centralisateur$T_X$ dans $M$ est un tore. Si $X, X^{\prime} \in m_{\rm reg}(F)$, on dit que $X$ et $X^{\prime}$ sont conjugués, resp. stablement conjugués, s'il existe $x \in M(F)$, resp. $x \in M(\bar{F})$ où $\bar{F}$ est la clôture algébrique de $F$, tel que $({\rm Ad}\, x)(X) = X^{\prime}$. On note $C^{\infty}_c (m(F))$ l'espace des fonctions sur $m(F)$. à valeurs complexes, localement constantes et à support compact. Soient $X \in m_{\rm reg}(F)$ et $f \in C^{\infty}_c (m(F))$. Supposons $T_X(F)$ et $M(F)$ munis de mesures de Haar. On pose\[J(X, f) = D(X)^{1/2} \int_{T_X(F)\setminus M(F)} f(({\rm Ad}\,x^{-1})(X))\,{\rm d}x,\] oú \[D(X) = \vert \det ({\rm ad} X \vert m(F)/t_X (F))\vert_F,\]$\vert \cdot\vert_F$ désignant la valeur absolue usuelle de $F$.

A zero set of a holomorphic vector field is totally degenerate, if the endomorphism of the conormal sheaf induced by the vector field is identically zero. By studying a class of foliations generalizing foliations of ${\rm C}^*$-actions, we show that if a projective manifold admits a holomorphic vector field with a smooth totally degenerate zero component, then the manifold is stably birational to that component of the zero set. When the vector field has an isolated totally degenerate zero, we prove that the manifold is rational. This is a special case of Carrell's conjecture.

We introduce some blow-analytic invariants of real analytic function-germs and discuss their properties. As a consequence, we obtain, for instance, the multiplicity of function-germs is a blow-analytic invariant.

The first main result of this paper is a bijective correspondence between the strictly full triangulated subcategories dense in a given triangulated category and the subgroups of its Grothendieck group (Thm. 2.1). Since every strictly full triangulated subcategory is dense in a uniquely determined thick triangulated subcategory, this result refines any known classification of thick subcategories to a classification of all strictly full triangulated ones. For example, one can thus refine the famous classification of the thick subcategories of the finite stable homotopy category given by the work of Devinatz–Hopkins–Smith ([Ho], [DHS], [HS] Thm. 7, [Ra] 3.4.3), which is responsible for most of the recent advances in stable homotopy theory. One can likewise refine the analogous classification given by Hopkins and Neeman ([Ho] Sect. 4, [Ne] 1.5) of the thick subcategories of $D(R)_{\rm parf}$, the chain homotopy category of bounded complexes of finitely generated projective $R$-modules, where $R$ is a commutative noetherian ring.

Let $K$ be a function field and $C$ a non-isotrivial curve of genus $g\geqslant 2$ over $K$. In this paper, we will show that if $C$ has a global stable model with only geometrically irreducible fibers, then Bogomolov conjecture over function fields holds.

A Pfaffian equation on $X$ is an invertible subsheaf $L \hookrightarrow \Omega^1_X$ of the sheaf of differential 1-forms on $X$. It can thus be thought of as a global section of $\Omega^1_X \otimes L^{-1}$ or, by choosing an isomorphism $L^{-1} \simeq {\cal O}_X(E) \subset K$, as a meromorphic differential form $\omega$ on $X$. We say that the Pfaffian equation has a first integral if there is a non-empty open subset $U\subset X$ and a smooth map $f: U\rightarrow {\rm P}^1$ such that $L_U \simeq f^* \Omega^1_{{\rm P}^1}$ as subsheaves of $\Omega^1_X$; that is, if $L$ ‘comes from’ a rational map to a curve. In this case, $f$, viewed as a rational function, is called a first integral. Note that as rational differential forms, ${\rm d} f \wedge \omega = 0$.