The loss of Smith-Thom maximality by Hilbert square

The famous Smith inequality bounds the total Betti number of the fixed point set of an involution by the total Betti number of the ambient space. This bound plays a fundamental role in topology of real algebraic varieties. The varieties for which this bound turns into equality are called maximal, and they manifest interesting specific topological properties. Constructing maximal varieties is a difficult, largely open, task. On the other hand, typically, every "geometricaly meaningful" construction (like taking a symmetric product) turns a maximal space into a maximal space. In an ongoing joint work with R. Rasdeaconu, we investigate how the maximality behaves under taking the Hilbert square. We found that starting from dimension two many of deformation classes of algebraic varieties do not contain any real variety whose Hilbert square is maximal. For example, this is the case of the so-called K3-surfaces.