Hyperbolic algebraic varieties and holomorphic differential equations
The long and exhaustive article reviews and explains recent methods and results in the theory of hyperbolic algebraic varieties, including several new results of the author which are focussed on some of the most challenging conjectures about hyperbolicity of algebraic manifolds of general type, the conjectures of Green-Griffiths and Lang. The conjectures of Green-Griffiths state that a projective algebraic variety \(X\) is of general type if and only if \(X\) is measure hyperbolic, and that \(X\) then contains a proper subvariety \(Y\) such that \(f(\mathbb C)\subset Y\) for every non-constant holomorphic map \(f:\mathbb C\rightarrow X\). An affirmative solution for these conjectures would imply that every very generic hypersurface in \(\mathbb P^{n+1}\), \(n\geq 3\), of degree \(d\geq 2n+1\) is hyperbolic, see [\textit{L. Ein}, Invent. Math. 94, No. 1, 163--169 (1988; Zbl 0701.14002); Math. Ann. 289, No. 3, 465--471 (1991; Zbl 0746.14019)] and [\textit{C. Voisin}, J. Differ. Geom. 44, No. 1, 200--213 (1996; Zbl 0883.14022)]. The author studies hyperbolicity problems in a more general setting. He works in the class of directed manifolds and transfers classical hyperbolicity concepts for complex varieties to this category. Special emphasis lies on the application of the Ahlfors-Schwarz lemma, jets of curves, Semple jet bundles, k-jet bundles, holomorphic Morse inequalities, jet differentials and k-jet metrics with negative curvature. A (compact resp. projective) directed manifold is understood as a pair \((X,V)\) of a (compact resp. projective) connected complex manifold \(X\) and an irreducible closed analytic subspace \(V\) of the holomorphic tangent bundle \(T_X\) such that \(V\cap T_{X,x}\) is a linear subspace of \(T_{X,x}\) for every \(x\in X\). Based on the Brody criterion it turns out that a compact directed manifold \((X,V)\) is hyperbolic iff every holomorphic map \(f:\mathbb C\rightarrow X\) with \(f'(\mathbb C)\subset V\) is constant. A projective directed manifold \((X,V)\) is by definition algebraic hyperbolic if \(X\) admits an Hermitian metric with fundamental form \(\omega\) such that for some \(\epsilon> 0\) the inequality \(-\chi(\overline{C})\geq\epsilon \int_C\omega\) is satisfied for every irreducible closed algebraic curve \(C\subset X\) tangent to \(V\) and normalized by \(\overline{C}\). Hyperbolic projective directed manifolds are algebraic hyperbolic, but the converse is not known. The main results of the paper are partial answers to a generalized version of the conjectures mentioned above. Assume that \((X,V)\) is a projective directed manifold and that the canonical bundle of \(V\) is big. The generalized Green-Griffiths-Lang conjecture claims the existence of a proper subvariety \(Y\) of \(X\) with \(f(\mathbb C)\subset Y\) for every non-constant holomorphic \(f:\mathbb C\rightarrow X\) with \(f'(\mathbb C)\subset V\). Under the assumption that \(X\) is of general type the author proves the existence of global algebraic differential operators \(P\) on \(X\) with \(P(f, f',\dots,f^{(k)})=0\) for every such \(f\). Another interesting result of the author is related to the hyperbolicity of generic algebraic hypersurfaces in \(\mathbb P^{n+1}\) of sufficiently high degree. \textit{Y.-T. Siu} [``Hyperbolicity of generic high-degree hypersurfaces in complex projective spaces'', Preprint, \url{arXiv:1209.2723}] has shown that there exists a sequence \((d_n)\) in \(\mathbb N\) with the property that a generic algebraic hypersurface in \(\mathbb P^{n+1}\) of degree \(d\geq d_n\), \(n\geq 2\), is hyperbolic. \textit{S. Diverio, J. Merker} and \textit{E. Rousseau} [Invent. Math. 180, No. 1, 161--223 (2010; Zbl 1192.32014)] proved that \(d_n:= 2^{n^5}\) does the job. The paper under review yields a considerable improvement of these estimates. The theorem of Siu is fulfilled for \(d_2:=286\), \(d_3:=7316\) and \(d_n:=\llcorner\frac{n^4}{3}(n\log(n\log(24n)))^n\lrcorner\) for \(n\geq 4\).
scientific article; zbMATH DE number 6133189