On the Smith–Thom deficiency of Hilbert squares

We give an expression for the Smith–Thom deficiency of the Hilbert square $X^{\left[2\right]}$ 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 $X^{\left[2\right]}$ 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.