The second fundamental form of Kodaira embeddings, and quantization

Joel Fine

I will discuss the second fundamental form of a complex submanifold of projective space and the role it plays in the quantization picture of Donaldson, relating constant scalar curvature Kahler metrics and balanced embeddings. In particular, the asymptotic behaviour of the second fundamental form is key to proving that the Hessian of Mabuchi energy (a 4th order operator similar to the square of the Laplacian) can be quantized by the Hessian of balancing energy (a finite dimensional operator acting on the spaces of Hermitian matrices and defined purely in terms of projective geometry). (Note to experts: this is relatively old work, so don’t expect to hear about any cutting edge results!)

http://scgp.stonybrook.edu/video_portal/video.php?id=697

Simons- Program: Large N limit problems in Kahler Geometry