Empty Real Enriques Surfaces and Enriques-einstein-hitchin 4-manifolds

N. Hitchin [H] proved that the Euler characteristic χ(E) and signature σ(E) of a compact orientable 4-dimensional Einstein manifold E satisfy the inequality |σ(E)| 6 2 3χ(E), the equality holding only if either E is flat or the universal covering X of E is a K3-surface and π1(E) = 1, Z/2, or Z/2× Z/2. In the latter cases, E is a K3-surface if π1 = 1, an Enriques surface if π1 = Z/2, or the quotient of an Enriques surface by a free antiholomorphic involution if π1 = Z/2 × Z/2. It is the Einstein manifolds of the last type that we call Enriques-Einstein-Hitchin varieties. The varieties of the other three extremal types (flat, π1 = 1, and π1 = Z/2) are known to form connected families: two varieties of the same type can be deformed continuously into each other. To our knowledge, the number of connected components of the moduli space of Enriques-Einstein-Hitchin varieties was not known. In this paper we give the answer: we prove that their moduli space is connected. As is known (modulo Calabi-Yau theorem this statement is also contained in [H]), the universal covering X of an Enriques-Einstein-Hitchin manifold E carries a canonical complex structure, so that X is a K3-surface, one nontrivial element of π1(E) = Z/2 × Z/2 acts on X holomorphically, and the two others, antiholomorphically. This correspondence establishes a homotopy equivalence between the moduli space of Enriques-Einstein-Hitchin varieties and that of Enriques surfaces with free anti-holomorphic involution (cf. [I]). An Enriques surface with a free anti-holomorphic involution is, by definition, an empty real Enriques surface, and the connectedness of the moduli space of Enriques-Einstein-Hitchin varieties follows from the main result of the present paper: