Hyperbolicity of Projective Hypersurfaces (IMPA Monographs)

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9783319323145: Hyperbolicity of Projective Hypersurfaces (IMPA Monographs)
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<P>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).</P><P> </P><P>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.</P><P></P>

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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).

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.

Biografía del autor:

Simone Diverio is a 1st class CNRS researcher at the Institute of Mathematics of Jusseau - Paris Rive Gauche, France. He received his PhD (2008) jointly from the University of Grenoble I, France, and Sapienza University of Rome, Italy. In 2010 he was awarded the Prime d'Excellence Scientifique by the CNRS. Erwan Rousseau is a professor at Aix-Marseille University, France. He did his PhD at Brest University, France (2004), with post-doc studies at the University of Quebéc, Canada and research at the University of Strasbourg (2010). In 2007, he was awarded the Cours Peccot du Collége de France.

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Buchbeschreibung Springer International Publishing AG, Switzerland, 2016. Hardback. Zustand: New. 1st ed. 2016. Language: English. Brand new Book. Thisbook presents recent advances on Kobayashi hyperbolicity in complex geometry,especially in connection with projective hypersurfaces. This is a very activefield, not least because of the fascinating relations with complex algebraicand arithmetic geometry. Foundational works of Serge Lang and Paul A. Vojta,among others, resulted in precise conjectures regarding the interplay of theseresearch fields (e.g. existence of Zariski dense entire curves shouldcorrespond to the (potential) density of rational points).Perhapsone of the conjectures which generated most activity in Kobayashi hyperbolicitytheory is the one formed by Kobayashi himself in 1970 which predicts that avery general projective hypersurface of degree large enough does not containany (non-constant) entire curves. Since the seminal work of Green and Griffithsin 1979, later refined by J.-P. Demailly, J. Noguchi, Y.-T. Siu and others, itbecame clear that a possible general strategy to attack this problem was tolook at particular algebraic differential equations (jet differentials) thatevery entire curve must satisfy. This has led to some several spectacularresults. Describing the state of the art around this conjecture is the maingoal of this work. Bestandsnummer des Verkäufers SPR9783319323145

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