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**Nonpositive curvature of blow-ups.**
*(English)*
Zbl 0924.53033

Let \(M_{\mathbb{C}}\) be a complex manifold of dimension \(n\) and let \(D_{\mathbb{C}}\) be a union of smooth complex codimension-one submanifolds of \(M_{\mathbb{C}}\). A “real version” of \((M_{\mathbb{C}},D_{\mathbb{C}})\) is a pair \((M,D)\) such that \(M\) is the fixed point set of a smooth involution on \(M_{\mathbb{C}}\) which is locally isomorphic to complex conjugation and such that \(D=D_{\mathbb{C}}\cap M\). In many interesting cases, such a real version exists.

In this paper, the authors are interested in the geometry and topology of \((M,D)\). More precisely, we have the following. The set \(D\) is locally isomorphic to an arrangement of hyperplanes, and it cuts \(M\) into regions which are combinatorially equivalent to convex polytopes. If this cellulation is simple, that is, if \(n\) edges meet at every vertex, then so is its dual cellulation. Moreover, the two cellulations have a common subdivision by cubes.

The authors study this cubical complex, equipped with its natural piecewise Euclidean structure in which each cell is identified with a regular Euclidean cube whose edges are of length one. Gromov has shown that for a cubical complex, there is a simple combinatorial condition that is necessary and sufficient for the complex to be nonpositively curved in the sense of Alexandrov. The condition is that the link of each vertex is a “flag complex”. A consequence of this nonpositive curvature (also due to Gromov) is that the universal cover of the space is contractible. One of the main points in this paper is to show that for many interesting examples, Gromov’s condition holds.

In this paper, the authors are interested in the geometry and topology of \((M,D)\). More precisely, we have the following. The set \(D\) is locally isomorphic to an arrangement of hyperplanes, and it cuts \(M\) into regions which are combinatorially equivalent to convex polytopes. If this cellulation is simple, that is, if \(n\) edges meet at every vertex, then so is its dual cellulation. Moreover, the two cellulations have a common subdivision by cubes.

The authors study this cubical complex, equipped with its natural piecewise Euclidean structure in which each cell is identified with a regular Euclidean cube whose edges are of length one. Gromov has shown that for a cubical complex, there is a simple combinatorial condition that is necessary and sufficient for the complex to be nonpositively curved in the sense of Alexandrov. The condition is that the link of each vertex is a “flag complex”. A consequence of this nonpositive curvature (also due to Gromov) is that the universal cover of the space is contractible. One of the main points in this paper is to show that for many interesting examples, Gromov’s condition holds.

Reviewer: A.Papadopoulos (Strasbourg)