The $K$-energy on hypersurfaces and stability

Gang Tian
1994 Communications in analysis and geometry  
GANG TIAN GANG TIAN singularity, then it may not be semistable. The simplest examples are those cubic surfaces in CP 3 with one singularity of type other than Ax or ^42-The K-energy is a functional on the space of admissible Kahler metrics in the Kahler class given by the polarization. It is in fact a Donaldson functional on a "virtual" holomorphic bundle and is defined in terms of the Bott-Chern class associated to the invariant polynomial Ch n+1 defining the (n+l)-th Chern Charactor (cf.
more » ... on 3). Here is our first theorem, In fact, the proof here yields a stronger result which can be described as follows: let g F s be the Fubini-Study metric on CP n+1 , for any a in G, a*g F s restricts to an admissible Kahler metric on S/, we denote by Ggps the set of such admissible Kahler metrics, then S/ is stable if the K-energy is proper on GQFS, and £/ is semistable if the K-energy is bounded from below on Ggps-An easy corollary of Theorem 0.1 is the following Theorem 0.2. Let £/ be a normal hypersurface in CP n+1 of degree d. Then (1) if3 n + 2 and £/ has only log-terminal singularities, then £/ is stable. The definition of a log-terminal singularity will be given in section 5. After some preparations, we give the proof for Theorem 0.1 in section 4 and for Theorem 0.2 in section 5. In section 6, we will briefly discuss some generalizations of above theorems without proof. In particular, Theorem 0.1, 0.2 will still be true for complete intersections in projective spaces. The details of these generalizations will appear in a forthcoming paper [Tl], where we will deal with subvarieties in CP^ with canonical or anti-canonical polarization and higher codimensions. It was S.T.Yau who brought the stability to my attention more than six years ago. I would like to thank him for sharing his insight that there must be a
doi:10.4310/cag.1994.v2.n2.a4 fatcat:v6mfayl7gngvhk7xumn26uylmi