Formulating…

MFO26.tex

@@ -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
+who -in modern language- showed that the Zariski closure of a map $\varphi:
-$\varphi:\mathbb C\to A$ to a complex tori $A$ is the translate of a sub-tori. A
+\mathbb C \to A$ to a complex tori $A$ is the translate of a sub-tori. A decade
-decade ago, K. Yamanoi established the Green-Griffiths conjecture for projective
+ago, K.~Yamanoi established the Green-Griffiths conjecture for projective
-manifolds general type, which admit a generically finite map into an abelian
+manifolds general type, which admit a generically finite map into an Abelian
-variety. This represents a very nice generalisation of Bloch's theorem.
+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}
 
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