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.

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 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.

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 poseJ(X, f) = D(X)^{1/2} \int_{T_X(F)\setminus M(F)} f(({\rm Ad}\,x^{-1})(X))\,{\rm d}x, $$J(X, f) = D(X)^{1/2} \int_{T_X(F)\setminus M(F)} f(({\rm Ad}\,x^{-1})(X))\,{\rm d}x,$$ oú D(X) = \vert \det ({\rm ad} X \vert m(F)/t_X (F))\vert_F, $$D(X) = \vert \det ({\rm ad} X \vert m(F)/t_X (F))\vert_F,$$ $\vert \cdot\vert_F$ désignant la valeur absolue usuelle de $F$.