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Hyperbolic varieties and algebraic differential equations. (Variétés hyperboliques et équations différentielles algébriques.) (French) Zbl 0901.32019

In this survey article, the author presents the relationship between the existence of entire curves (i.e. holomorphic curves \(f:\mathbb{C} \to X)\) on an algebraic variety \(X\) and global algebraic differential operators on the variety \(X\). We mention that the nonexistence of non constant entire curves is equivalent to the Kobayashi’s hyperbolicity.
The author gives a complete proof of the following vanishing result of M. Green and Ph. Griffiths, presented with an incomplete proof in Proc. Int. Chern Symp., Berkely 1979, 41-74 (1980; Zbl 0508.32010)]: “Let \(X\) be a projective algebraic variety and let \(f:\mathbb{C} \to X\) be a non constant entire curve. Then \(P(f', \dots, f^{(k)}) \equiv 0\) for any algebraic differential operator \(P\) with values in the dual \(L^*\) of a holomorphic line bundle \(L\) on \(X\), with positive curvature”. As an application one obtaines explicit examples of hyperbolic algebraic surfaces of small degree by applying the above vanishing result to wronskian operators.

MSC:

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32H30 Value distribution theory in higher dimensions
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables

Citations:

Zbl 0508.32010
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