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Geometry Colloquium
Seminar information archive ～03/17｜Next seminar｜Future seminars 03/18～
| Date, time & place | Friday 10:00 - 11:30 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|
2013/01/30
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Ryoichi Kobayashi (Nagoya University)
Hamiltonian Volume Minimizing Property of Maximal Torus Orbits in the Complex Projective Space (JAPANESE)
Ryoichi Kobayashi (Nagoya University)
Hamiltonian Volume Minimizing Property of Maximal Torus Orbits in the Complex Projective Space (JAPANESE)
[ Abstract ]
We prove that any $U(1)^n$-orbit in $\\Bbb P^n$ is volume minimizing under Hamiltonian deformation.
The idea of the proof is :
- (1) We extend one $U(1)^n$-orbit to the momentum torus fibration $\\{T_t\\}_{t\\in\\Delta^n}$ and consider its Hamiltonian deformation $\\{\\phi(T_t)\\}_{t\\in\\Delta^n}$ where $\\phi$ is a Hamiltobian diffeomorphism of $\\Bbb P^n$,
and then :
- (2) We compare each $U(1)^n$-orbit and its Hamiltonian deformation by compaing the large $k$ asymptotic behavior of the sequence of projective embeddings defined, for each $k$, by the basis of $H^0(\\Bbb P^n,\\Cal O(k))$ obtained by semi-classical approximation of the $\\Cal O(k)$ Bohr-Sommerfeld tori of the Lagrangian torus fibration $\\{T_t\\}_{t\\in\\Delta^n}$ and its Hamiltonian deformation $\\{\\phi(T_t)\\}_{t\\in\\Delta^n}$.
We prove that any $U(1)^n$-orbit in $\\Bbb P^n$ is volume minimizing under Hamiltonian deformation.
The idea of the proof is :
- (1) We extend one $U(1)^n$-orbit to the momentum torus fibration $\\{T_t\\}_{t\\in\\Delta^n}$ and consider its Hamiltonian deformation $\\{\\phi(T_t)\\}_{t\\in\\Delta^n}$ where $\\phi$ is a Hamiltobian diffeomorphism of $\\Bbb P^n$,
and then :
- (2) We compare each $U(1)^n$-orbit and its Hamiltonian deformation by compaing the large $k$ asymptotic behavior of the sequence of projective embeddings defined, for each $k$, by the basis of $H^0(\\Bbb P^n,\\Cal O(k))$ obtained by semi-classical approximation of the $\\Cal O(k)$ Bohr-Sommerfeld tori of the Lagrangian torus fibration $\\{T_t\\}_{t\\in\\Delta^n}$ and its Hamiltonian deformation $\\{\\phi(T_t)\\}_{t\\in\\Delta^n}$.
