Miyaoka–Yau inequalities and the topological characterization of certain klt varieties

Greb, Daniel; Kebekus, Stefan; Peternell, Thomas

Comptes Rendus. Mathématique, Complex algebraic geometry, in memory of Jean-Pierre Demailly, Tome 362 (2024), pp. 141-157

Ball quotients, hyperelliptic varieties, and projective spaces are characterized by their Chern classes, as the varieties where the Miyaoka–Yau inequality becomes an equality. Ball quotients, Abelian varieties, and projective spaces are also characterized topologically: if a complex, projective manifold $X$ is homeomorphic to a variety of this type, then $X$ is itself of this type. In this paper, similar results are established for projective varieties with klt singularities that are homeomorphic to singular ball quotients, quotients of Abelian varieties, or projective spaces.

Les quotients de boules, les variétés hyperelliptiques et les espaces projectifs sont caractérisés par leurs classes de Chern, comme les variétés pour lesquelles l’inégalité de Miyaoka–Yau devient une égalité. Les quotients de boules, les variétés abéliennes et les espaces projectifs sont aussi caractérisés topologiquement  : si une variété projective complexe $X$ est homéomorphe à une variété de ce type, alors $X$ est elle-même de ce type. Dans cet article, des résultats similaires sont établis pour les variétés projectives avec des singularités klt qui sont homéomorphes à des quotients de boules singulières, à des quotients de variétés abéliennes, ou à des espaces projectifs.

Classification : 32Q30, 32Q26, 14E20, 14E30
Keywords: Miyaoka–Yau inequality, klt singularities, uniformisation, homeomorphisms
Mots-clés : Inégalité de Miyaoka–Yau, singularités klt, uniformisation, homéomorphismes

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