Quebec Mathematical Sciences Colloquium

February 26, 2021 from 15:00 to 16:00 (Montreal/EST time)

Analytic solutions to algebraic equations, and a conjecture of Kobayashi

Colloquium presented by Jean-Pierre Demailly (Université Grenoble Alpes, France)

A projective algebraic variety is defined as the zero locus of a finite family of homogeneous polynomials.  Over the field of complex numbers, the geometry of such varieties is governed to a large extent by the sign, in a suitable sense, of the Ricci curvature form.  When this sign is negative, the variety is expected to exhibit certain hyperbolicity properties in the sense of Kobayashi - as well as further very deep number-theoretic properties that are mostly conjectural, in the arithmetic situation.  In particular, all entire holomorphic curves drawn on it should be contained in a proper algebraic subvariety: this is a famous conjecture of Green-Griffiths and Lang.  Following recent ideas of D.  Brotbek, we will try to explain here a rather elementary proof of a related conjecture of Kobayashi, stating that a general algebraic hypersurface of sufficiently high degree is hyperbolic, i.e. does not contain any entire holomorphic curve.