An important problem in complex geometry is to classify complex manifolds up to biholomorphism. Deformation theory approaches this problem by studying how complex structures vary in holomorphic families and relating these variations to the geometry of their parameter spaces.

In this talk, we use compact Riemann surfaces as our main examples to explain this viewpoint. Kodaira–Spencer theory describes infinitesimal changes of complex structure, and the L² pairing of their harmonic representatives defines the Weil–Petersson metric. The curvature of this metric provides information about the geometry of the moduli space of Riemann surfaces.

These ideas extend to families of higher-dimensional compact Kähler manifolds through higher direct image sheaves, which organize the cohomology of the fibers. Their natural L² metrics provide a broader framework for understanding the relationship between deformation and curvature. We will present general results on the curvature of these metrics and discuss several applications.