Colloquium
Seminar information archive ~11/05|Next seminar|Future seminars 11/06~
Organizer(s) | ASUKE Taro, TERADA Itaru, HASEGAWA Ryu, MIYAMOTO Yasuhito (chair) |
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URL | https://www.ms.u-tokyo.ac.jp/seminar/colloquium_e/index_e.html |
2014/01/24
16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Bo Berndtsson (Chalmers University of Technology)
Complex Brunn-Minkowski theory (ENGLISH)
Bo Berndtsson (Chalmers University of Technology)
Complex Brunn-Minkowski theory (ENGLISH)
[ Abstract ]
The classical Brunn-Minkowski theory deals with the volume of convex sets.
It can be formulated as a statement about how the volume of slices of a convex set varies when the slice changes. Its complex counterpart deals with slices of pseudo convex sets, or more generally fibers of a complex fibration. It describes how $L^2$-norms of holomorphic functions, or sections of a line bundle, vary when the fibers change, and says essentially that a certain associated vector bundle has positive curvature. In the presence of enough symmetry this implies convexity properties of volumes; the real Brunn-Minkowski theorem corresponding to maximal symmetry. There are also applications and relations in other directions, like variations of Kahler metrics, variations of complex structures and the study of plurisubharmonic functions.
The classical Brunn-Minkowski theory deals with the volume of convex sets.
It can be formulated as a statement about how the volume of slices of a convex set varies when the slice changes. Its complex counterpart deals with slices of pseudo convex sets, or more generally fibers of a complex fibration. It describes how $L^2$-norms of holomorphic functions, or sections of a line bundle, vary when the fibers change, and says essentially that a certain associated vector bundle has positive curvature. In the presence of enough symmetry this implies convexity properties of volumes; the real Brunn-Minkowski theorem corresponding to maximal symmetry. There are also applications and relations in other directions, like variations of Kahler metrics, variations of complex structures and the study of plurisubharmonic functions.