In complex analysis and geometry it is vital to be able to construct many holomorphic sections of a given line bundle over a complex manifold. A line bundle has curvature and it is well known that positive curvature is very favourable. This project will focus on two specific problems that can be summarized as constructing and quot; many and quot; holomorphic sections in two new important situations where previous techniques have not been successful: 1. The curvature of the line bundle is positive, but the curvature has complicated singularities. 2. The curvature is smooth and positive, but the manifold X has a boundary with negative curvature.
The main method for both problems will be to obtain Morse inequalities that estimate the obstructions to construct holomorphic sections (they estimate the dimensions of cohomology groups with values in high powers of the line bundle). In my PhD thesis a new approach to such inequalities was introduced. It used Bergman kernels and will be further developed in this project. While my work up to now has been purely analytical and concerned with smooth curvature Prof. Demailly and his group at the host institute are leading experts on the theory of positive currents, its use in the study of singular curvature and the combination analysis/algebra in complex geometry.
The main training objectives are to enable me to:
(a) acquire new expertise concerning currents
(b) become acquainted with their numerous recent applications in complex and algebraic geometry
(c) apply microlocal analysis in complex geometry.
This training will ideally complement my expertise and guide my future research into new promising areas. This project is closely related to various current research areas including embedding and deformation problems for complex and CR-manifolds, almost complex methods in symplectic geometry, constant scalar curvature metrics, the study of electrons in magnetic fields and recent developments in string/M-theory.
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