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Seminar on Geometric Complex Analysis
Seminar information archive ～04/08｜Next seminar｜Future seminars 04/09～
| Date, time & place | Monday 10:30 - 12:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Kengo Hirachi, Shigeharu Takayama |
2026/04/20
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Yusaku Tiba (Ochanomizu Univ.)
$C^{\ell}$-estimates for the $\overline{\partial}$-equation on high tensor powers of positive line bundles (Japanese)
https://forms.gle/8ERsVDLuKHwbVzm57
Yusaku Tiba (Ochanomizu Univ.)
$C^{\ell}$-estimates for the $\overline{\partial}$-equation on high tensor powers of positive line bundles (Japanese)
[ Abstract ]
Let $M$ be a compact complex manifold, $L$ be a positive holomorphic line bundle over $M$, and $E$ be a holomorphic vector bundle over $M$. It is known that the cohomology groups $H^i(M, L^k \otimes E)$ vanish for $i > 0$ when $k$ is sufficiently large. This vanishing theorem is typically proved by solving the $\overline{\partial}$-equation using H\"ormander’s $L^2$-estimates in the complex geometry. In this talk, we solve the $\overline{\partial}$-equation not by H\"ormander’s method, but by means of weighted integral formulas. In particular, we apply the weighted integral formula of Andersson--Berndtsson (1982) in a semi-classical setting and obtain $C^{\ell}$-norm estimates for solutions of the $\overline{\partial}$-equation.
[ Reference URL ]Let $M$ be a compact complex manifold, $L$ be a positive holomorphic line bundle over $M$, and $E$ be a holomorphic vector bundle over $M$. It is known that the cohomology groups $H^i(M, L^k \otimes E)$ vanish for $i > 0$ when $k$ is sufficiently large. This vanishing theorem is typically proved by solving the $\overline{\partial}$-equation using H\"ormander’s $L^2$-estimates in the complex geometry. In this talk, we solve the $\overline{\partial}$-equation not by H\"ormander’s method, but by means of weighted integral formulas. In particular, we apply the weighted integral formula of Andersson--Berndtsson (1982) in a semi-classical setting and obtain $C^{\ell}$-norm estimates for solutions of the $\overline{\partial}$-equation.
https://forms.gle/8ERsVDLuKHwbVzm57
