Cycle spaces associated to group actions

Toshiki Mabuchi

Osaka U., Japan

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Abstract

In this talk, we discuss the notion of second fundamental form in the geometry of Kähler potentials. Let M be a compact complex manifold with a Kähler class k. For an embedding

i : N N M

of a compact complex submanifold N into M,  let P_M denote the space of all Kähler potentials on M for the class k, and let P_N denote the space of all Kähler potentials on N for the class i^k. Then by pulling back by i, we have a natural projection of P_M onto P_N. This then allows us to define the second fundamental form for this projection.

For the Chow norm studied by S. Zhang in relation to the stability problem, its seond variation can be actually written in terms of the second fundamental form thus defined, where i is chosen as the Kodaira embedding of a polarized algebraic manifold (N, L) in the complex projective space M = P^*(H⁰(N, L^m ) ). This then gives us a clear picture in the study of uniqueness, modulo biholomorphisms, of an extremal Kähler metric in a polarization class on a polarized projective algebraic manifold.