Monday April 29th, 2024 | |
| Time: | 10:00 AM - 11:00 AM |
| Title: | A framework for the curvature of $L^2$ metrics |
| Speaker: | Pranav Upadrashta, Stony Brook University |
| Abstract: | |
| A result of Berndtsson states that the Chern connection associated with the $L^2$ metric on certain infinite rank vector bundles has positive curvature in the sense of Nakano. Following the ideas of Lempert and Szőke, together with Varolin, we developed a framework to define the curvature of families of Hilbert spaces that might not fit together to form a holomorphic vector bundle. This enables us to derive curvature formulas for $L^2$ metrics on families of Hilbert spaces associated with a general holomorphic submersion $pi: X rightarrow B$ and a holomorphic hermitian vector bundle $(E, h) rightarrow X$. In the first part of the thesis, we obtain a curvature formula for the $L^2$ metric on a family of Bergman spaces when the fibers of $pi$ are domains and $E$ is a line bundle. Additionally, we establish a lower bound on the curvature, from which we recover Berndtsson's aforementioned result when $X$ is a product and $pi$ is projection onto a factor. In the second part, we get another curvature formula when the fibers of $pi$ are compact Kähler manifolds. We show that from this formula, well-known curvature formulas of $L^2$ metrics can be recovered, including the curvature of: (i) the $L^2$ metric on the direct image of a family of holomorphic vector bundles due to To and Weng, (ii) the $L^2$ metric on the higher direct images of a family of line bundles due to Berndtsson, Păun, and Wang (iii) the Weil-Petersson on the moduli space of Hermite-Einstein vector bundles due to Schumacher and Toma. | |