Uniformization, hyperbolicity and symmetric differentials of projective manifolds

University Of Miami, Coral Gables FL

Investigators

Abstract

This project tackles two important global geometric/analytic properties of projective complex manifolds and their universal covers. The first problem is known as the uniformization problem which is concerned with the global geometric/analytic properties of universal covers of complex manifolds. One of the main goals is to obtain information on the space of global holomorphic functions on the universal covers, which is predicted to be very rich by the Shafarevich conjecture. The PI will also address the natural question which was posed by S.T.Yau concerning the embeddability of universal covers of projective manifolds as open subsets of a projective manifold. The second problem that this proposal will tackle is the existence of symmetric differentials on projective manifolds and its impact on hyperbolicity properties. The principal investigator showed previously that there are symmetric differentials on deformations of smooth hypersurfaces in projective 3-space. This is an important and a far reaching surprise (recall that the smooth hypersurfaces do not have symmetric differentials). The PI intends to explore this occurrence to obtain several new hyperbolicity results for hypersurfaces. Complex manifolds are natural objects in mathematics and in particular in the field of complex geometry. They appear in the mathematical models of engineering and physics. The research program proposed will contribute to a better understanding of the properties of complex manifolds and the theory of their functions. Advances in complex geometry will have a significant impact on the development of the models of physics, such as string theory, describing the forces that shape our universe. This project will also allow the principal investigator to organize seminars that will bring experts of the field from other parts of the country to his institution. This will help foster the research in complex geometry in the area and it will be essential to the education of the graduate students at his institution. The principal investigator hopes to foster an environment of scientific curiosity and investigation that will have a wide range of applications.