| In recent years, it has become clear that the solvability of the complex Monge Ampere equation on a Fano variety is deeply related to an appropriate stability condition that the manifold must satisfy. In my doctoral dissertation, under the supervision of Gang Tian, I was able to relate the Chow-Mumford stability of a projective manifold X to the properness or lower-boundedness of a certain energy functional on a subset of the Kahler cone of X. More precisely, let X be a smooth n-dimensional complex submanifold of CPN, w the Fubini-Study Kahler form, and P( X, w ) the Kahler cone of X. The Chow form of X, Chow(X), is a point in the projective space of a finite dimensional complex representation, E, of SL(N + 1,C). Endow E with any norm , then the Chow-Mumford stability of X is equivalent to the properness of logsChow X as s varies over SL(N + 1,C). In what follows, let F0w4 be Tian's modified energy functional, 4∈PX,w . One of the main results in my thesis is the following:;Theorem 1. Let X be a smooth submanifold of CPN, then -1n+1 -12p nVF0 w4s =2logsChow X+O 1;Here, O(1) denotes a bounded quantity on SL (N + 1,C), V = Xwn , and s*w=w+66 &d1;4s . As an application of this general statement, I have given a new proof of a result originally due to Mumford.;Theorem 2. Let X be a smooth curve of genus g ≥ 1 embedded into P(H 0(X,Lm) ∨) by an ample line bundle L. For m sufficiently large the Chow form of X is stable. |
