Formulating…

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@ -199,14 +199,14 @@ The Green-Griffiths conjecture (concerning the Zariski closure of holomorphic
entire curves or more generally, of families of holomorphic disks on varieties
of general type) continues to keep busy many complex geometers. Probably the
most complete result in this field is due to A. Bloch (more than 100 years ago),
who -in modern language- showed that the Zariski closure of a map
$\varphi:\mathbb C\to A$ to a complex tori $A$ is the translate of a sub-tori. A
decade ago, K. Yamanoi established the Green-Griffiths conjecture for projective
manifolds general type, which admit a generically finite map into an abelian
variety. This represents a very nice generalisation of Bloch's theorem.
who -in modern language- showed that the Zariski closure of a map $\varphi:
\mathbb C \to A$ to a complex tori $A$ is the translate of a sub-tori. A decade
ago, K.~Yamanoi established the Green-Griffiths conjecture for projective
manifolds general type, which admit a generically finite map into an Abelian
variety. This represents a very nice generalization of Bloch's theorem.
In the last couple of years the field is taking a very interesting direction, by
combining techniques from Hodge theory with the familiar Nevalinna theory and
combining techniques from Hodge theory with the familiar Nevanlinna theory and
jet differentials, cf. the articles by D. Brotbek, Y. Deng, Y. Brunebarbe, B.
Cadorel and A. Javanpeykar.
@ -216,13 +216,58 @@ Green-Griffiths conjecture for manifolds whose fundamental group admits a
representation having certain natural properties (echoing the case of curves of
genus at least two).
Techniques from birational geometry, in connection with the work of F. Campana
Techniques from birational geometry, in connection with the work of F.~Campana
are also present in the field via the -long awaited- work of E. Rousseau and its
collaborators.
\subsubsection{Complex hyperbolicity. Mark II}
The 1979 Green-Griffiths-Lang conjecture asserts that every complex-projective
variety $X$ of general type contains a proper subvariety $Y \subsetneq X$, such
that every non-constant entire holomorphic curve $\mathbb C \to X$ takes its
values in $Y$. Its beginnings date back to 1926, when André Bloch showed that
the Zariski closure of entire holomorphic curve $\varphi: \mathbb C \to A$ to a
complex torus $A$ is the translate of a sub-torus. Today, the conjecture still
drives much of the research in complex geometry. We highlight several advances
that will be relevant for our workshop.
\paragraph{Hypersurfaces in projective space}
A remarkable paper of Bérczi and Kirwan, \cite{MR4688701} published in September
last year, establishes hyperbolicity and proves the Green-Griffiths-Lang for
generic hypersurfaces of the projective space, $X \subsetneq \mathbb P^n$,
provided that the degree of $X$ is larger than an explicit polynomial in $n$.
These are significant improvements of earlier degree bounds, which involve
non-polynomial bounds of order $(\sqrt{n} \log n)^n$ or worse. The proof builds
on a strategy of Diverio-Merker-Rousseau and combines non-reductive geometric
invariant theory with the ``Grassmannian techniques'' of Riedl-Yang. A very
recent preprint of Cadorel simplifies the proof Bérczi-Kirwan substantially, but
still needs to undergo peer review, \cite{arXiv:2406.19003}.
\paragraph{Hyperbolicity and representations of fundamental groups}
Using recent advances in the theory of harmonic maps (due to
Daskalopoulos-Mese), Y. Deng and K. Yamanoi were able to confirm the
Green-Griffiths conjecture for manifolds whose fundamental group admits a
representation having certain natural properties (echoing the case of curves of
genus at least two).
\paragraph{Material collections}
In the last couple of years the field is taking a very interesting direction, by
combining techniques from Hodge theory with the familiar Nevanlinna theory and
jet differentials, cf. the articles by D. Brotbek, Y. Deng, Y. Brunebarbe, B.
Cadorel and A. Javanpeykar.
\bibstyle{alpha}
\bibliographystyle{alpha}
\bibliography{general}
\end{document}

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