On the Smith-Thom deficiency of Hilbert squares (Q6564521)

Recall that, for any real algebraic variety~\(X\), one has the so-called Smith-Thom inequality \(\beta_*(X_\mathbb{R})\le\beta_*(X_\mathbb{C})\), where \(\beta_*\) stands for the total Betti number with coefficients \(\mathbb{F}_2\). The even difference \(2d:=\beta_*(X_\mathbb{C})\le\beta_*(X_\mathbb{R})\) is called the Smith-Thom deficiency of~\(X\) in the paper, and \(X\) is called maximal if \(2d=0\). Maximal smooth real algebraic varieties are known to have quite a few very special properties.\N\NThe authors study the deficiency of the (naturally real) Hilbert square of a smooth projective real variety~\(X\), and their principal observation is the fact that \(X^{[2]}\) is almost never maximal. More precisely, if \(X^{[2]}\) is maximal, then so is \(X\) and, in the latter case, there is an explicit formula for the deficiency of \(X^{[2]}\) in terms of the Betti numbers \(\beta_i(X_\mathbb{R})\). In particular, if \(X\) is a maximal complete intersection, then \(X^{[2]}\) is maximal if and only if \(\beta_i(X_\mathbb{R})=1\) for all \(0\le i<\lfloor\dim X/2\rfloor\). The authors derive a few consequences of their findings, both known and new, and analyze a few other special cases (beyond complete intersections) where the (non-)maximality of \(X^{[2]}\) can be asserted.