Geometric Analysis on complex manifolds

Overview: The PI proposes to use techniques from geometric analysis and nonlinear partial differential equations to investigate problems about the geometry of complex (and almost-complex) manifolds. The first project is about applications to analysis to the construction of currents on complex manifolds, which are used to study the geometry of (1,1) cohomology classes on compact Kahler manifolds. In the second project the PI will develop new analytic techniques to construct special metrics on non-Kahler complex manifolds. The third project is about understanding collapsed limits of Ricci-flat Calabi-Yau manifolds. This is closely related to the theory of mirror symmetry, which was inspired by physical considerations. The fourth project is centered on Donaldson’s program to extend Yau’s solution of the Calabi Conjecture in Kahler geometry to symplectic four-manifolds. This would provide a new analytic tool to construct symplectic forms four-manifolds as solutions of a highly nonlinear PDE, and would have striking applications in symplectic topology. Intellectual Merit : The project on (1,1) classes on complex manifolds is directly related to the study of base loci of line bundles on projective varieties, a classical topic in algebraic geometry, which will be approached using transcendental tools. The problem of defining and constructing special Hermitian metrics on non-Kahler complex manifolds is a fundamental and hard problem in geometry, with ties to mathematical physics. The study of Calabi-Yau manifolds is a central topic in geometry, having ramifications in fields as diverse as algebraic geometry, number theory and theoretical physics. The proposed work on symplectic four-manifolds will deepen our understanding of the geometry of these spaces and will introduce new analytic tools in the field. Broader Impacts : The PI will continue to organize a learning seminar at his institution, to introduce undergraduate and graduate students to these topics, and training them to give their own presentations. Thanks to the rich interplay between geometry and physics, geometric aspects closely related to the proposed research have found applications in various areas of theoretical physics.