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Asphericity of moduli spaces via curvature. (English) Zbl 1067.53028

A simply way to build a cover \(\widehat Y/\) of a space \(\widehat X\) branched over \(\Delta\subseteq\widehat X\) is to take any covering space \(Y\) of \(\widehat X\setminus\Delta\) and define \(\widehat Y= Y\cup\Delta\). The cover \(\widehat Y\) is called a simple branched cover of \(\widehat X\) over \(\Delta\). The author proves that if \(\widehat X\) satisfies Alexandrov’s curvature condition \(\text{CAT}(k)\) and \(\Delta\) is complete and satisfies a convexity condition, then the natural metric on \(\widehat Y\) also satisfies \(\text{CAT}(k)\). In other words, the branched cover satisfies the same upper curvature bounds as its base space. Then the author studies branched covers of a Riemannian manifold over suitable mutually orthogonal submanifolds. The results are applied to hyperplane complements in several Hermitian symmetric spaces of nonpositive sectional curvature. This permits to prove that the module spaces of the smooth cubic surfaces in \(\mathbb{C} P^3\) and of the smooth complex Enriques surfaces are aspherical (i.e., has contractible universal cover).

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
14J28 \(K3\) surfaces and Enriques surfaces
57N65 Algebraic topology of manifolds
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