Hyperbolicity of nodal hypersurfaces.

*(English)*Zbl 1108.14013A compact complex variety is hyperbolic if it does not contain any entire curve and Kobayashi’s famous conjecture claims that a general hypersurface of degree at least five in projective threespace is hyperbolic. In the paper under review the authors consider hyperbolicity properties of nodal hypersurfaces of \(\mathbb P^3\). Singular hypersurfaces are not general hypersurfaces in the sense of Kobayashi’s conjecture of course, so it is reasonable to consider only the weaker property of algebraic quasi-hyperbolicity, that is the hypersurface contains only finitely many rational and elliptic curves. The main theorem states that a nodal hypersurface \(X \subset \mathbb P^3\) of degree \(d\) that has at least \(\frac{8}{3} (d^2 - \frac{5}{2} d)\) nodes is algebraically quasi-hyperbolic. Although this result is quite interesting, the reviewer considers the technique of the proof as even more exciting.

The main ingredient of this proof is an asymptotic estimate of the number of symmetric differentials of the hypersurface, i.e. global sections of a symmetric power of the cotangent sheaf \(\Omega_X\). It turns out that \(h^0(X, \text{Sym}^m \Omega_X)\) grows with order \(m^3\). This is rather surprising, since a smooth hypersurface of projective space does not have any symmetric differentials (e.g. F. Sakai [Lect. Notes Math. 732, 545–563 (1979; Zbl 0415.14020)]). The existence of the symmetric differentials is achieved via a Riemann-Roch formula for surfaces with nodal singularities [J. Wahl, Math. Ann. 295, 81–110 (1993; Zbl 0789.14004) and R.Blache, Math. Z. 222, 7–57 (1996; Zbl 0949.14006)] and a vanishing theorem for the second cohomology of the symmetric powers of \(\Omega_X\). Using logarithmic differentials the authors show how to lift (at least asymptotically) the symmetric differentials to a minimal resolution of singularities \(Y \rightarrow X\). The theory of algebraic foliations can then be used to show that \(Y\), and hence \(X\), is algebraically hyperbolic. Note furthermore that \(Y\) can be realised as the central fibre of a family of smooth surfaces such that the general fibre is a smooth hypersurface of \(\mathbb P^3\). In contrast to the plurigenera of the surfaces which are invariant under deformation, the central fibre of this family has ‘many’ symmetric differentials, while the general fibre has none.

The main ingredient of this proof is an asymptotic estimate of the number of symmetric differentials of the hypersurface, i.e. global sections of a symmetric power of the cotangent sheaf \(\Omega_X\). It turns out that \(h^0(X, \text{Sym}^m \Omega_X)\) grows with order \(m^3\). This is rather surprising, since a smooth hypersurface of projective space does not have any symmetric differentials (e.g. F. Sakai [Lect. Notes Math. 732, 545–563 (1979; Zbl 0415.14020)]). The existence of the symmetric differentials is achieved via a Riemann-Roch formula for surfaces with nodal singularities [J. Wahl, Math. Ann. 295, 81–110 (1993; Zbl 0789.14004) and R.Blache, Math. Z. 222, 7–57 (1996; Zbl 0949.14006)] and a vanishing theorem for the second cohomology of the symmetric powers of \(\Omega_X\). Using logarithmic differentials the authors show how to lift (at least asymptotically) the symmetric differentials to a minimal resolution of singularities \(Y \rightarrow X\). The theory of algebraic foliations can then be used to show that \(Y\), and hence \(X\), is algebraically hyperbolic. Note furthermore that \(Y\) can be realised as the central fibre of a family of smooth surfaces such that the general fibre is a smooth hypersurface of \(\mathbb P^3\). In contrast to the plurigenera of the surfaces which are invariant under deformation, the central fibre of this family has ‘many’ symmetric differentials, while the general fibre has none.

Reviewer: Andreas HĂ¶ring (Strasbourg)

##### MSC:

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |

32S20 | Global theory of complex singularities; cohomological properties |

14J29 | Surfaces of general type |

14J70 | Hypersurfaces and algebraic geometry |

14J17 | Singularities of surfaces or higher-dimensional varieties |

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\textit{F. Bogomolov} and \textit{B. De Oliveira}, J. Reine Angew. Math. 596, 89--101 (2006; Zbl 1108.14013)

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