On the Smith-Thom deficiency of Hilbert squares
Viatcheslav Kharlamov, Rareş Răsdeaconu
TL;DR
This work establishes a concrete framework to compute the Smith-Thom deficiency of Hilbert squares X^{[2]} of real varieties X, tying maximality to the ranks of Mayer-Vietoris restriction maps. For real nonsingular projective complete intersections, it gives an explicit deficiency formula D(X^{[2]}) in terms of the real Betti numbers β_i(X(R)) and identifies sharp conditions under which X^{[2]} is maximal, including a complete characterization in odd/even dimensions and notable new maximal examples. The authors develop a cut-and-paste model for Hilbert squares over the reals, derive compatibility relations for characteristic classes, and use these tools to connect the topology of X(R) to that of X^{[2]}(R). A key consequence is that maximality of X^{[2]} often forces X to be maximal and constrains the topology of F(X) for cubic hypersurfaces, illustrating deep interactions between real and complex loci via Smith theory and Mayer-Vietoris. The results extend to compact complex manifolds with anti-holomorphic involution and provide practical criteria for determining maximality, with exact formulas for several classical families including projective spaces and quadrics.
Abstract
We give an expression for the Smith-Thom deficiency of the Hilbert square $X^{[2]}$ of a smooth real algebraic variety $X$ in terms of the rank of a suitable Mayer-Vietoris mapping in several situations. As a consequence, we establish a necessary and sufficient condition for the maximality of $X^{[2]}$ in the case of projective complete intersections, and show that with a few exceptions no real nonsingular projective complete intersection of even dimension has maximal Hilbert square. We also provide new examples of smooth real algebraic varieties with maximal Hilbert square.