BIMSA > Seminar on Algebraic, Complex Geometry and Singularities Higher order Hessian matrix theory and its applications in singularity theory and Calabi-Yau manifolds
Higher order Hessian matrix theory and its applications in singularity theory and Calabi-Yau manifolds
Organizer
Shing Toung Yau
Speaker
Shuanghe Fan
Time
Saturday, March 23, 2024 10:30 AM - 11:30 AM
Venue
清华理科楼A-304 & Online
Abstract
One of the fundamental questions in algebraic geometry and singularity theory is to determine whether two given smooth projective manifolds, denoted as $X$ and $Y$, are projectively equivalent. When the order of their defining equations is 2, one can utilize quadratic form theory and classical Hessian matrix theory. However, when considering cases where $n>2$, a novel higher order Hessian matrix theory based on higher order Jacobian matrix theory needs to be developed in order to provide an answer to this question. This new theory serves as a generalization of classical Hessian matrices. In this talk, we present our recent progress on higher order Hessian matrices theory. As an application, we introduce different invariants for smooth projective manifolds (including Calabi-Yau manifolds) under projective transformations. Notably, one invariant sequence bears a striking resemblance to the well-known $j$-invariant. Furthermore, the theory can also be applied to investigate whether two sets defined by convergent power series in $mathbb{C}left{x_1,x_2,cdots,x_l ight}$ are equivalent up to an invertible linear transformation.