Oliver Club
The moduli space is the parameter space for manifolds with fixed invariants. Often, the moduli space itself has a geometric structure, such as the moduli space ${\mathcal M}_g$ of genus $g$ Riemann surfaces itself being a complex $(3g-3)$-dimensional space. Many moduli spaces, including ${\mathcal M}_g$, are non-compact, and it is of essential interest to describe their compact complex subvarieties. Geometrically, this corresponds to families of objects over a compact base, which remain smooth and do not degenerate. It is already unknown whether ${\mathcal M}_4$ contains a compact complex surface, i.e. whether smooth genus $4$ curves can vary in a compact family with two complex parameters.
In this talk we describe the joint work with Mondello, Salvati Manni, Tsimerman determining the maximal dimension of compact subvarieties of the moduli space of abelian varieties, and applications to compact subvarieties of moduli of curves.