| masataka(@math.sci.osaka-u.ac.jp) | |
| Research |
Complex geometry, Algebraic geometry, Several complex variables |
| Keywords | Tangent bundle, Cotangent bundle, Foliation, Singular Hermitian metric, Second Chern class, Structure theorem |
| URL | https://masataka123.github.io/blog3_e/ |
I study projective manifolds (i.e., submanifolds of complex projective space) by using some methods from complex geometry, algebraic geometry, and several complex variables. More specifically, I focus on a classification problem which predicts that projective manifolds can be decomposed into 3 types of manifolds classified by their Ricci curvature: positive, zero(flat), and negative. I approach this problem from the viewpoint of the tangent bundle, the cotangent bundle, and the second Chern class.
In my earlier work, I established a structure theorem stating that projective manifolds whose tangent bundle has non-negative (possibly singular) curvature decompose into a manifold with positive Ricci curvature and a torus (which has Ricci-flat curvature). In this proof, I used singular Hermitian metrics, which arise naturally in several complex variables. This result was later extended to the setting of foliations.
Using foliation structures, I showed that projective manifolds whose cotangent bundle has non-negative curvature decompose into a manifold with negative Ricci curvature and a torus, under the assumption that the second Chern class vanishes. In this work, I also proved the abundance conjecture, which is a major open problem in algebraic geometry, in the case where the second Chern class vanishes. Currently, I am studying the relationship between the second Chern class and the structure theorem of projective manifolds.
