Abstract: |
Consider an open holomorphic map $\Phi: Z \to S$. We want to study the De Rham cohomology groups of the fibers of the map parametrized by $S$. If a fiber over a point has an isolated critical point, then it is well-known and well-studied that the nearby fibers get geometrically middle dimensional (hence the Lagrangean) vanishing cycles. The study of associated De Rham cohomology theory, and it semi-infinite Hodge structure (primitive forms, etc) is nowadays relatively well studied. If the critical set is not isolated but have positive dimension, then associated vanishing cycles are no-longer middle dimensional. I don't know any systematic study of such "mixed semi-infinite Hodge theory" (whatever it may be, or exist), and it could be quite a mess. However, recently, I started to study such cases. As all over the first step, I show that the coherence, i.e. finiteness, of the relative De Rham cohomology groups over $S$. Its study indicates already several new feature of interests. |