Effective algebraic degeneracy of entire curves in complements of smooth hypersurfaces in \(P^{n}\)

In this work, it is established that for a generic projective hypersurface $H\subset\mathbb{P}^n(\mathbb{C})$ of degree $d\geq(5n)^2\,n^{n}$, any holomorphic entire curve $f\colon\mathbb{C}\to\mathbb{P}^n(\mathbb{C})\setminus H$ has its image contained in a proper algebraic subvariety $Z\subsetneq\mathbb{P}^{n}(\mathbb{C})$, that does not depend on the curve $f$. Here generic means that the coefficients of the defining equation of $H$ have to lie outside of a proper algebraic subvariety of the projective space of coefficients of homogeneous polynomials of degree $d$ (that parametrizes the algebraic hypersurface of degree $d$ in $\mathbb{P}^n(\mathbb{C})$). The proof closely follows the work of Diverio, Merker and Rousseau (Diverio-Merker-Rousseau 2009), thus it is based on the strategy of Siu (Siu 2002, 2004) and techniques of Demailly (Demailly 1995, Diverio 2009) (already adapted to the logarithmic setting by Dethloff and Lu in 2001). It also include an improved adaptation of the contribution of Berczi (Berczi 2010).