Some recent transcendental techniques in algebraic and complex geometry

By Y. Siu. 2002

71 citations recorded by the source corpus.

This article discusses the recent transcendental techniques used in the proofs of the following three conjectures. (1)~The plurigenera of a compact projective algebraic manifold are invariant under holomorphic deformation. (2)~There exists no smooth Leviflat hypersurface in the complex projective plane. (3)~A generic hypersurface of sufficiently high degree in the complex projective space is hyperbolic in the sense that there is no nonconstant holomorphic map from the complex Euclidean line to it.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2c64ea040abff810ab2ca550674e94157bcb4312

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In Applied Mathematics

1. J. Demailly
2. Y. Siu
3. Joël Merker
4. Simone Diverio
5. Erwan Rousseau
6. Damian Brotbek
7. Qi’an Guan
8. Junyan Cao

In Geometry and Topology

1. J. Demailly
2. Joël Merker
3. Y. Siu
4. Damian Brotbek
5. Simone Diverio
6. Erwan Rousseau
7. Qi’an Guan
8. Xiangyu Zhou

Citations in the corpus

• Hyperbolicity of generic high-degree hypersurfaces in complex projective space (2012, methodology): “1 in [22], for the last step discussed on p.”
• Effective algebraic degeneracy (2008, background): “As in [ 20 , 21], we thereby confirm, for generic projective hypersurfaces of high degree, the Green-Griffiths-Lang conjecture.”
• Holomorphic Morse Inequalities and the Green-Griffiths-Lang Conjecture (2010, methodology): “This strategy first devised by [Siu02, Siu04] has indeed been successful in some cases for the study of generic algebraic degeneracy (e.g. for hypersurfaces of very large degree in Pn+1C ).”
• Recent results on the Kobayashi and Green-Griffiths-Lang conjectures (2018, methodology): “Around 2000, Siu [Siu02, 04] showed that generic hyperbolicity results in the direction of the Kobayashi conjecture could be investigated by combining the algebraic techniques of Clemens, Ein and Voisin with the existence of certain “vertical” meromorphic vector fields on the jet space of the…”