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

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.