Room 515, Cosmology Building, National Taiwan University + Zoom, Physical+Online Seminar

(實體+線上演講 台灣大學次震宇宙館515研討室+ Zoom)

Hyperbolicity of Projective Hypersurfaces Via Green-Griffiths Jet Differentials

Benoît Cadorel (University of Lorraine)

**Abstract**

A complex projective variety X is said to be (Brody) hyperbolic if it admits no entire curve, i.e. no non-constant holomorphic map starting from the complex plane. The Kobayashi conjecture asserts that a generic hypersurface of high degree should be Brody hyperbolic (we even expect the bound on the degree to be linear in the dimension). The first complete proof of this conjecture was only given by Brotbek in 2017, with a very high bound on the degree. It has become clear in the last few years that an efficient way of proving hyperbolicity results for hypersurfaces is to apply jet differential techniques, via Siu's so called strategy of slanted vector fields : this strategy has been refined by many authors (Diverio-Merker-Rousseau, Darondeau, Brotbek, Deng, Demailly, Riedl-Yang...). Until quite recently, the best known bounds on the degree were at least exponential in the dimension.

A core part of the strategy of slanted vector fields consists in picking an adequate jet space above the hypersurface, before studying the base locus of a natural tautological line bundle on it. Quite recently, Bérczi-Kirwan have managed to construct a new jet space via techniques of non-reductive GIT, which allowed them to spectacularly improve the previously known bounds, to a polynomial in the dimension.

Another candidate for jet spaces has existed for quite a long time, but seems to have been a bit overlooked in this problem: the Green-Griffiths jet space, introduced by these authors around 1980. As we will explain, the latter can also be used in the strategy above, and, surprisingly enough, it again gives a polynomial bound.

**Organizer:** Hsueh-Yung Lin ( NTU)