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Seminar on Geometric Complex Analysis
Seminar information archive ～06/26｜Next seminar｜Future seminars 06/27～
| Date, time & place | Monday 10:30 - 12:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Kengo Hirachi, Shigeharu Takayama |
Future seminars
2026/06/29
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Chin-Yu Hsiao (National Taiwan University)
Heat kernel asymptotics for the $\bar{\partial}$-Neumann Laplacian on manifolds with boundary (English)
https://forms.gle/8ERsVDLuKHwbVzm57
Chin-Yu Hsiao (National Taiwan University)
Heat kernel asymptotics for the $\bar{\partial}$-Neumann Laplacian on manifolds with boundary (English)
[ Abstract ]
We study the heat kernel asymptotics of the $\bar{\partial}$-Neumann Laplacian associated with high tensor powers of a holomorphic line bundle. Specifically, for a relatively compact complex submanifold with smooth boundary in a complex manifold, we consider the $\bar{\partial}$-Neumann Laplacian acting on holomorphic sections of a holomorphic bundle over the submanifold. As the tensor power of the line bundle approaches infinity, we obtain explicit asymptotic expansions of the heat kernel in the submanifold's interior, on its boundary, and near the boundary. This is achieved by explicitly solving the heat equation for the weighted $\bar{\partial}$-Neumann Laplacian in domains that are not necessarily strongly pseudoconvex and showing uniform convergence of the associated scaled Laplacian's heat kernel to this solution. As an application, we establish analogue holomorphic Morse inequalities of Demailly on complex manifolds with boundary.
[ Reference URL ]We study the heat kernel asymptotics of the $\bar{\partial}$-Neumann Laplacian associated with high tensor powers of a holomorphic line bundle. Specifically, for a relatively compact complex submanifold with smooth boundary in a complex manifold, we consider the $\bar{\partial}$-Neumann Laplacian acting on holomorphic sections of a holomorphic bundle over the submanifold. As the tensor power of the line bundle approaches infinity, we obtain explicit asymptotic expansions of the heat kernel in the submanifold's interior, on its boundary, and near the boundary. This is achieved by explicitly solving the heat equation for the weighted $\bar{\partial}$-Neumann Laplacian in domains that are not necessarily strongly pseudoconvex and showing uniform convergence of the associated scaled Laplacian's heat kernel to this solution. As an application, we establish analogue holomorphic Morse inequalities of Demailly on complex manifolds with boundary.
https://forms.gle/8ERsVDLuKHwbVzm57
2026/07/06
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Xiaojun Wu (Tsukuba Univ.)
Generalised Ueda Obstruction Classes and Non-Semipositive Line Bundles (English)
https://forms.gle/8ERsVDLuKHwbVzm57
Xiaojun Wu (Tsukuba Univ.)
Generalised Ueda Obstruction Classes and Non-Semipositive Line Bundles (English)
[ Abstract ]
Serre’s classical example provides a fundamental instance of a nef but non-semipositive line bundle and motivated the analytic definition of nefness introduced by Demailly–Peternell–Schneider. Building on subsequent developments by Koike, the classical Ueda obstruction classes provide a natural criterion for non-semipositivity. In this talk, we introduce a natural generalisation of the Ueda obstruction classes that is always well defined and for which the Chern curvature naturally determines representatives. As an application, we obtain an elementary and systematic method for constructing nef but non-semipositive line bundles.
[ Reference URL ]Serre’s classical example provides a fundamental instance of a nef but non-semipositive line bundle and motivated the analytic definition of nefness introduced by Demailly–Peternell–Schneider. Building on subsequent developments by Koike, the classical Ueda obstruction classes provide a natural criterion for non-semipositivity. In this talk, we introduce a natural generalisation of the Ueda obstruction classes that is always well defined and for which the Chern curvature naturally determines representatives. As an application, we obtain an elementary and systematic method for constructing nef but non-semipositive line bundles.
https://forms.gle/8ERsVDLuKHwbVzm57
2026/07/13
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Shuho Kanda (The Univ. of Tokyo)
Holomorphic polynomial crystallographic actions of nilpotent groups (Japanese)
https://forms.gle/8ERsVDLuKHwbVzm57
Shuho Kanda (The Univ. of Tokyo)
Holomorphic polynomial crystallographic actions of nilpotent groups (Japanese)
[ Abstract ]
It is a natural and still open question whether every simply connected nilpotent Lie group endowed with a left-invariant complex structure is biholomorphic to $\mathbb{C}^n$. In this talk, we give an affirmative answer under the additional assumption that the complex structure is nilpotent. Moreover, we construct such a biholomorphism explicitly by polynomial maps in exponential coordinates. As a consequence, every lattice in such a Lie group admits a free, properly discontinuous and cocompact action on $\mathbb{C}^n$ by holomorphic polynomial automorphisms. We interpret this as a holomorphic analogue of polynomial crystallographic actions, namely actions on $\mathbb{R}^n$ by polynomial diffeomorphisms that are free, properly discontinuous and cocompact, as introduced by Dekimpe, Igodt, and Lee.
[ Reference URL ]It is a natural and still open question whether every simply connected nilpotent Lie group endowed with a left-invariant complex structure is biholomorphic to $\mathbb{C}^n$. In this talk, we give an affirmative answer under the additional assumption that the complex structure is nilpotent. Moreover, we construct such a biholomorphism explicitly by polynomial maps in exponential coordinates. As a consequence, every lattice in such a Lie group admits a free, properly discontinuous and cocompact action on $\mathbb{C}^n$ by holomorphic polynomial automorphisms. We interpret this as a holomorphic analogue of polynomial crystallographic actions, namely actions on $\mathbb{R}^n$ by polynomial diffeomorphisms that are free, properly discontinuous and cocompact, as introduced by Dekimpe, Igodt, and Lee.
https://forms.gle/8ERsVDLuKHwbVzm57
