The structure of integral invariants, in Riemannian and complex geometries
I will present older and newer work, on the following question:
Assume one has a natural geometric functional, which acts on metrics by forming scalar intrinsic invariants, and then integrating these invariants over a manifold. Assume that the resulting integral presents an invariance under certain deformations of the underlying metric. Then what information can one deduce on the integrand?
I will present answers to this question in two cases: When the class of metrics are Riemannian, or Kahler, and the invariance of the integral is under conformal, or Kahler deformations of the underlying metric. I will discuss some applications of this to renormalized volume and generalized Chern-Gauss Bonnet integrands in the Riemannian setting, and to the structure of the Bergman kernel of ample line bundles in the complex setting.