@Dmitri: I expect that if we have a smooth complex family (over a connected base) in which two fibers are compact simple Kahler manifolds, then the abelian categories of analytic coherent sheaves on these two manifolds are equivalent. Of course, you can not expect that this holds for all members of the family. For non-algebraic K3s with no curves this is actually known due to a theorem of Verbitsky. I am not sure if it is known for tori. Also, I am not sure if being 'simple' is enough. Perhaps one should require that we have no connected proper analytic subvarieties of positive dimension.

If $X$ is a Kahler manifold which is projective, then any choice of an ample line bundle on $X$ induces a natural polarization on $J^{k}X$. This is a $(1,1)$ integral class on $J^{k}X$ which as a Hermitian pairing is non-degenerate but need not be definite. The corresponding holomorphic line bundle is non-degenerate and its powers have only one non-trivial cohomology group, namely the cohomology of degree the number of negative eigenvalues. The corresponding cohomology classes can be viewed as forms with coefficients in the line bundle. These are the analogues of the theta functions.

It exists in English - published by John Wiley and Sons in 1976. Unfortunately our library doesn't have it but one can find it through inter library loan.

Hmm. I don't think this is a Serre fibration. It doesn't have the path lifting property. It is a fibration in Theo's sense as far as I can tell. If Theo meant locally trivial fibrations, then the existence of a local Lagrangian section is automatic so I don't understand the question then.

Hi Kevin! It means that the homology class of the fiber is divisible. In the example let $\Delta = D/(\pm 1)$ and let $p : X \to \Delta$ be the projection induced from the projection $D\times E \to D$. Then this is a Lagrangian fibration for which the map $p$ is submersive everywhere outside of $0 \in \Delta$. The fiber over $t \neq 0\in \Delta$ is a smooth copy of $E$. The fiber over $0$ consists of critical points for $p$ but is a submanifold isomorphic to $F = E/(\text{tanslation})$. In $H_{2}(X)$ we have $[E] = 2[F]$ so $p$ can not have local sections in a neighborhood of $0 \in \Delta$.