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.

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.

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 $X$ 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.

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.

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

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.

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 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, ${\rm d} f \wedge \omega = 0$.

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.

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.

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.

In this paper we prove that the moduli spaces $MI_{\scriptstyle {\bb P}^{2n+1}}(k)$ of mathematical instanton bundles on ${\bb P}^{2n+1}_{{\bb C}}$ with quantum number $k$ are singular for $n \geqslant 2$ and $k \geqslant 3$, giving a positive answer to a conjecture made by Ancona and Ottaviani in 1993.