Hyperbolicity Of Projective Hypersurfaces (impa Monographs)
by Simone Diverio /
2016 / English / PDF
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This book presents recent advances on Kobayashi hyperbolicity in
complex geometry, especially in connection with projective
hypersurfaces. This is a very active field, not least because of
the fascinating relations with complex algebraic and arithmetic
geometry. Foundational works of Serge Lang and Paul A. Vojta,
among others, resulted in precise conjectures regarding the
interplay of these research fields (e.g. existence of Zariski
dense entire curves should correspond to the (potential) density
of rational points).
This book presents recent advances on Kobayashi hyperbolicity in
complex geometry, especially in connection with projective
hypersurfaces. This is a very active field, not least because of
the fascinating relations with complex algebraic and arithmetic
geometry. Foundational works of Serge Lang and Paul A. Vojta,
among others, resulted in precise conjectures regarding the
interplay of these research fields (e.g. existence of Zariski
dense entire curves should correspond to the (potential) density
of rational points).
Perhaps one of the conjectures which generated most activity in
Kobayashi hyperbolicity theory is the one formed by Kobayashi
himself in 1970 which predicts that a very general projective
hypersurface of degree large enough does not contain any
(non-constant) entire curves. Since the seminal work of Green and
Griffiths in 1979, later refined by J.-P. Demailly, J. Noguchi,
Y.-T. Siu and others, it became clear that a possible general
strategy to attack this problem was to look at particular
algebraic differential equations (jet differentials) that every
entire curve must satisfy. This has led to some several
spectacular results. Describing the state of the art around this
conjecture is the main goal of this work.
Perhaps one of the conjectures which generated most activity in
Kobayashi hyperbolicity theory is the one formed by Kobayashi
himself in 1970 which predicts that a very general projective
hypersurface of degree large enough does not contain any
(non-constant) entire curves. Since the seminal work of Green and
Griffiths in 1979, later refined by J.-P. Demailly, J. Noguchi,
Y.-T. Siu and others, it became clear that a possible general
strategy to attack this problem was to look at particular
algebraic differential equations (jet differentials) that every
entire curve must satisfy. This has led to some several
spectacular results. Describing the state of the art around this
conjecture is the main goal of this work.