On the Geometry of Kahler-Einstein Manifolds

点击翻译按钮获取中文摘要

AbstractAward: DMS-9971506Principal Investigator: Zhiqin LuThis project will analyze Kaehler-Einstein metrics on a compactKaehler manifold in three different ways. The first one is tostudy the potential function of the Bergman metric of a polarizedcompact Kaehler manifold. The first several coefficients of theasymptotic expansion of Zelditch have been computed by theauthor. The method I used is complex geometric. Comparing to themethod using the paramatrix of the Szego kernel of the unitcircle bundle, our method is more straightforward. We wish tohave some uniform estimates. Even in the case of complexdimension one, this problem is interesting. In dimension two,the problem is already complicated enough. Tian proved that thereis a uniform lower bound for the multi-anticanonical bundle overa Kaehler-Einstein surfaces. In higher dimensions, the problemis very important, pointing towards the stability ofmanifolds. The second one is the computation of the Futakiinvariants. This problem is related to the previous one in thatthe potential function is used in the definition of the Futakiinvariants. Futaki invariants can be used not only as theobstruction towards the existence of the Kaehler-Einstein metric,but also can be used to check the stability of manifolds. Sincewe are only interested in the Futaki invariants for thosevarieties which are embedded into some complex projective space,it is reasonable to think that we can compute the invariantsusing some "extrinsic" information(Such as the polynomials indefining the varieties). In fact, in the case that the variety isa complete intersection, we found a formula to calculate theFutaki invariants. The last one is on the geometry of modulispace of polarized Calabi-Yau manifold. A Kaehler metric on themoduli space has been defined and was found that the Riccicurvature of such a metric is negative away from zero by theauthor. Furthermore, if one can prove that the scalar curvatureof the metric is lower bounded, one would be able to proveViehweg's compactification theorem using differential geometricmethods.Einstein's general theory of relativity interprets the concept ofgravity as a geometric property of space-time. The mainmathematical tool to study the geometry of these spaces isdifferential geometry. Since the discovery of general relativity,differential geometry has been crucial to both mathematicians andphysicists. Our project is in one of the main fields indifferential geometry, and should add to our understanding ofgravity and ultimately the space we are living in.