ANALYSIS ON SUPERMANIFOLDS

[摘要] In this thesis a theory of differential analysis for complex supermanifolds is developed analogous to that for complex manifolds. A natural association is set up between complex supermanifolds and smooth supermanifolds which establishes the framework for obtaining a Dolbeault resolution and theorem for complex supermanifolds. These results are proved by appealling to their classical counterparts on the reduced manifold in combination with a purely algebraic result similar in spirit to the Koszul complex. A Dolbeault theorem is developed for cohomology with coefficients in the canonical sheaf, Ber, which for complex supermanifolds is not the sheaf of top degree forms. It is found that the hypercohomology groups of a sequence of sheaves which is quasi-isomorphic to Ber are representable as the cohomology of the global section sequence of these sheaves in a manner analogous to the abstract de Rham theorem. A theory of currents is developed with regularity theorems that only depend on regularity theorems for the reduced manifold along with the characteristic algebra brought about by the super structure, then the analogue of the classical Grothendieck lemma is realized. Applying methods of functional analysis to the above results in Serre duality for complex supermanifolds. The duality is exemplified in a dimension count of the cohomology of superprojective space with coefficients in the structure sheaf and Ber.

[发布机构] Rice University