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A famous conjecture of Kobayashi from the 1970s asserts that a generic algebraic hypersurface of sufficiently large degree $d\geq d_n$ in the complex projective space of dimension $n+1$ is hyperbolic. Yum-Tong Siu introduced several fundamental ideas that led recently to a proof of the conjecture. In 2016, Damian Brotbek gave a new geometric argument based on the use of Wronskian operators and on an analysis of the geometry of Semple jet bundles. Shortly afterwards, Ya Deng obtained effective degree bounds by means of a refined technique. Our goal here will be to explain a drastically simpler proof that yields an improved (though still non optimal) degree bound, e.g. $d_n=[(en)^{2n+2}/5]$. We will also present a more general approach that could possibly lead to optimal bounds.[-]
A famous conjecture of Kobayashi from the 1970s asserts that a generic algebraic hypersurface of sufficiently large degree $d\geq d_n$ in the complex projective space of dimension $n+1$ is hyperbolic. Yum-Tong Siu introduced several fundamental ideas that led recently to a proof of the conjecture. In 2016, Damian Brotbek gave a new geometric argument based on the use of Wronskian operators and on an analysis of the geometry of Semple jet ...[+]

32Q45 ; 32L10 ; 53C55 ; 14J40

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We will discuss several new ideas that can show the existence of jet differential operators on arbitrary projective varieties, and also on general hypersurfaces of $\mathbb{P}^n$ of sufficiently high degree. These results can be applied to improve degree bounds in several hyperbolicity problems and especially in the proof of the Kobayashi conjecture.

32Q45 ; 32L10 ; 53C55 ; 14J40

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