4 N ov 2 00 8 A note on Kähler - Ricci soliton

In this note we provide a direct proof of the following: Any compact KRS with positive bisectional curvature is CP. As a corollary, we obtain an alternative proof of the Frankel conjecture by using the Kähler-Ricci flow. The purpose of this note is to give a proof of the following theorem, which does not rely on the solution of the Frankel conjecture: Theorem 1. Every compact complex manifold admitting a Kähler-Ricci soliton with positive bisectional curvature is biholomorphic to the complex projective space CP . Remark 2. Since the Futaki invariant of CP n vanishes, the Kähler-Ricci soliton must be Kähler-Einstein. In addition, by a theorem of Berger (cf. [2], [11]), it is actually a constant multiple of the standard Fubini-Study metric. It also follows from the uniqueness theorem in [27]. Remark 3. Using a method which originated in [21], the above theorem was proved in [12] without using the uniformization theorem. The proof given here is different and we use some Morse theory. As a corollary, one can use the method of Kähler-Ricci flow to prove the following Frankel conjecture. Corollary 4. Every compact Kähler manifold with positive bisectional curvature is biholomorphic to the complex projective space. Remark 5. The Frankel conjecture was proved by Siu-Yau ([26]) using harmonic maps and by Mori ([21]) via algebraic methods. In Siu-Yau’s proof, by using the theorem of Kobayashi-Ochiai ([18]), the key thing is to show the existence of a Both first and third authors are partially supported by NSF grants.