复射影簇自同构的刚性

刚性是指一个几何对象可以在一定程度上被它的拓扑性质决定，是复代数几何领域的一个重要概念。本项目致力于研究复射影簇自同构的刚性，即一个自同构的拓扑性质在多大程度上决定了它本身？自同构的刚性问题可以转化为数值平凡自同构的刚性问题。本项目将围绕数值平凡自同构展开，研究内容包括：. (1)各类三维簇及稳定曲面数值平凡自同构群的一致上界，. (2)微分拓扑平凡自同构的判定，. (3)自同构群在周群上的诱导作用。.这些课题的成果将能揭示复射影簇及其自同构的一些基本的拓扑性质，并在复射影簇的 Teichmüller空间、模空间及周群等课题的研究中有进一步应用。

Rigidity means that a geometric object is determined, to a certain extent, by its topological properties. It is an important concept in complex algebraic geometry. This project aims to study the rigidity of automorphisms of complex projective varieties, that is, to what extent the topological properties of an automorphism determine the automorphism itself. The rigidity problem of automorphisms can be reduced to that of numerically trivial automorphisms. This project will center around numerically trivial automorphisms, and the research topics include:. (1) uniform upper bounds for the numerically trivial automorphism groups of various classes of threefolds and stable surfaces,. (2) criteria for an automorphism to be differential-topologically trivial, . (3) the induced action of the automorphism group on the Chow groups..Results on this topics will reveal some basic topological properties of complex projective varieties and their automorphisms, and lead to further applications in the study of the Teichmüller spaces, the moduli spaces and the Chow groups of complex projective varieties.