In this paper, we present an extension theorem which is equivalent to an injectivity theorem on the cohomology groups of vector bundles equipped with singular Hermitian metrics over holomorphically convex Kähler manifolds.

we study some automorphic cohomology classes of degree one on the griffiths–schmid varieties attached to some unitary groups in three variables. using partial compactifications of those varieties, constructed by kato and usui, we define for such a cohomology class some analogues of fourier–shimura coefficients, which are cohomology classes on certain elliptic curves. we show that a large space of such automorphic classes can be generated by those with rational ‘coefficients’. more precisely, we consider those cohomology classes that come from picard modular forms, via some penrose-like transform studied in a previous article: we prove that the coefficients of the classes thus obtained can be computed from the coefficients of the picard form by a similar transform defined at the level of the elliptic curve.

For an integer $n\,\ge \,3$, let ${{M}_{n}}$ be the moduli space of spatial polygons with $n$ edges. We consider the case of odd $n$. Then ${{M}_{n}}$ is a Fano manifold of complex dimension $n\,-\,3$. Let ${{\Theta }_{{{M}_{n}}}}$ be the sheaf of germs of holomorphic sections of the tangent bundle $T{{M}_{n}}$. In this paper, we prove ${{H}^{q}}\left( {{M}_{n}},\,{{\Theta }_{{{M}_{n}}}} \right)\,=\,0$ for all $q\,\ge \,0$ and all odd $n$. In particular, we see that the moduli space of deformations of the complex structure on ${{M}_{n}}$ consists of a point. Thus the complex structure on ${{M}_{n}}$ is locally rigid.