Fake hermitian symmetric manifolds and analytic approach to some problems in algebraic geometry

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The project focuses on two directions in algebraic and complex geometry. The first one is on the classification and construction of some concrete projective algebraic manifolds which have small topological invariants,play important roles in geography of algebraic varieties, and are equipped with rich geometric and analytic properties. This includes classification of fake projective planes and more generally fake Hermitian symmetric spaces which have the same Betti numbers as the complex projective plane or a compact Hermitian symmetric space. The scheme of classification will lead to understanding of finite group actions on potential candidates of examples,and give rise tonew algebraic manifolds with topological invariants which are not known before. Another project proposed is to clarify the problem about existence ornon-existence of exotic complex structures on the complex quadric of dimension two, a problem which is well-known and has been open for a long time. Themethod proposed is a combination of both transcendental and algebraic techniques. This is also the second direction of the proposal, namely, to develop and apply analytic techniques to study problems which are algebraicin nature. A project proposed is to search for a proof of the existenceof rational curves on a Fano manifold using completely analytic method. Another is to study the problem of deformation rigidity of compact Hermitian symmetric spaces.In short, there are two main themes in the proposal. The first one is to construct and classify geometric models which can provide concrete examples for us to understand various mathematical theories and conjectures from algebraic, geometric or number theoretical point of view. The second is to further develop analytic techniques which have been very successful in recent years in understanding algebraic structures of varieties. Apart from thepossibility of enriching different areas of mathematics involved, including computational aspects of the various theories, it will also help to equip students with a reasonable broad base of mathematical knowledge.