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Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems

Volume 117 / 2019

Jarosław Buczyński, Giovanni Moreno Banach Center Publications 117 (2019), 145-176 MSC: 14M17, 53D10, 14M20, 53C26, 58A15. DOI: 10.4064/bc117-5

Abstract

Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the first case we also discuss a distinguished contact cone structure, arising as the variety of minimal rational tangents. We discuss the repercussion of the aforementioned classification theorems for the geometry of quaternion-Kähler manifolds with positive scalar curvature and for the geometry of second-order PDEs imposed on hypersurfaces.

Authors

  • Jarosław BuczyńskiInstitute of Mathematics of Polish Academy of Sciences
    ul. Śniadeckich 8
    00-656 Warszawa, Poland
    e-mail
  • Giovanni MorenoFaculty of Mathematics, Computer Science and Mechanics
    University of Warsaw
    ul. Banacha 2
    02-097 Warszawa, Poland
    e-mail

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