Ricerc@Sapienza

The present project consists in a series of closely interrelated research problems in the theory of differential and complex varieties, linked together by the common theme of curvature. A wide class of results are expected, ranging from the extension in a suitable form of fundamental results from the smooth Riemannian geometry to the singular context, to the infinitesimal and global description of several moduli spaces of fixed curvature (sub-)manifolds, and from the investigation of the consequences of Kobayashi hyperbolicity in complex geometry, to the possibility of providing explicit locally complete families of projective hyper-Kaehler manifolds of Kummer type and so a new approach to Torelli's theorem for these. Advanced and innovative techniques will be exploited and developed, including the use of entropy bounds, of Ehresmann's geometric structures, of microanalytic techniques for singular elliptic operators, of complex Monge-Ampere equations and of differential graded and L-infinity algebras. Both the expected results and the techniques developed in order to prove them will predictably have a major impact on the subject, advancing the state of the art in the understanding of the local and global geometry of differential and complex varieties.