Symmetric differentials and the fundamental group

By Y. Brunebarbe, Bruno Klingler, B. Totaro. 2012

63 citations recorded by the source corpus.

Esnault asked whether every smooth complex projective variety with infinite fundamental group has a nonzero symmetric differential (a section of a symmetric power of the cotangent bundle). In a sense, this would mean that every variety with infinite fundamental group has some nonpositive curvature. We show that the answer to Esnault's question is positive when the fundamental group has a finite-dimensional representation over some field with infinite image. This applies to all known varieties with infinite fundamental group. Along the way, we produce many symmetric differentials on the base of a variation of Hodge structures. One interest of these results is that symmetric differentials give information in the direction of Kobayashi hyperbolicity. For example, they limit how many rational curves the variety can contain.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8691ee162adef4f2a2b37eb1a8c7894cb5d5d41e

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Citations in the corpus

Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups (2017, background): “cotangent bundles to the size of fundamental groups. To pass from X to Xreg, we consider a maximally quasi-e´tale cover. As for Item (1.3.13), the key point is a result of Brunebarbe-Klingler-Totaro, [BKT13], which links the existence of representations with inﬁnite image to the existence of symmetr”
Topological methods in moduli theory (2014, background): “Brunebarbe, Klingler and Totaro showed indeed in [64] that some ‘linearity’ property of the fundamental group entails the existence of nontrivial symmetric differentials.”