Higher Fano Manifolds

Authors

  • Carolina Araujo IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil
  • Roya Beheshti Department of Mathematics & Statistics, Washington University in St. Louis, St. Louis, MO 63130, USA
  • Ana-Maria Castravet Universit´e Paris-Saclay, UVSQ, Laboratoire de Math´ematiques de Versailles, 78000, Versailles, France
  • Kelly Jabbusch Department of Mathematics & Statistics, University of Michigan–Dearborn, 4901 Evergreen Rd, Dearborn, MI 48128, USA
  • Svetlana Makarova Department of Mathematics, University of Pennsylvania, 209 S 33rd St, Philadelphia, PA 19104, USA
  • Enrica Mazzon Max-Planck-Institut f¨ur Mathematik, Vivatsgasse 7, 53111, Bonn, Germany
  • Libby Taylor Stanford University, 380 Serra Mall, Stanford, CA 94305, USA
  • Nivedita Viswanathan School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3FD, UK

DOI:

https://doi.org/10.33044/revuma.2921

Abstract

We address in this paper Fano manifolds with positive higher Chern characters, which are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo the Brauer obstruction). Aiming at finding new examples of higher Fano manifolds, we investigate positivity of higher Chern characters of rational homogeneous spaces. We determine which rational homogeneous spaces of Picard rank 1 have positive second Chern character, and show that the only rational homogeneous spaces of Picard rank 1 having positive second and third Chern characters are projective spaces and quadric hypersurfaces. We also classify Fano manifolds of large index having positive second and third Chern characters. We conclude by discussing conjectural characterizations of projective spaces and complete intersections in terms of these higher Fano conditions.

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Published

2022-05-10