Maximal real varieties from moduli constructions

Abstract: For a complex manifold equipped with an anti-holomorphic involution, which is referred to as a real variety, the Smith-Thom inequality states that the total $\mathbb{F}_2$-Betti number of the real locus is not greater than the total $\mathbb{F}_2$-Betti number of the ambient complex manifold. A real variety is called maximal if the equality holds. In this paper, we present a series of new constructions of maximal real varieties by exploring moduli spaces of certain objects on a maximal real variety. Our results establish the maximality of the following real varieties: - Moduli spaces of stable vector bundles of coprime rank and degree over a maximal smooth projective real curve (known as Brugall\'e-Schaffhauser's theorem, with a short new proof presented in this work); the same result holds for moduli spaces of stable parabolic vector bundles. - Moduli spaces of stable Higgs bundles of coprime rank and degree over a maximal smooth projective real curve, providing maximal hyper-K\"ahler examples. - If a real variety has non-empty real locus and maximal Hilbert square, then the variety itself and its Hilbert cube are maximal. This is always the case for maximal real smooth cubic threefolds, but never the case for maximal real smooth cubic fourfolds. - Punctual Hilbert schemes on a maximal real projective surface with vanishing first $\mathbb{F}_2$-Betti number and connected real locus, such as $\mathbb{R}$-rational maximal real surfaces and some generalized Dolgachev surfaces. - Moduli spaces of stable sheaves on the real projective plane, or more generally, on an $\mathbb{R}$-rational maximal Poisson surface. We also observe that maximality is a motivic property when interpreted as equivariant formality. Furthermore, any smooth projective real variety motivated by maximal ones is also maximal.