Differential Equations on Complex Projective Hypersurfaces of Low Dimension

Let $n=2,3,4,5$ and let $X$ be a smooth complex projective hypersurface of $\mathbb P^{n+1}$. In this paper we find an effective lower bound for the degree of $X$, such that every holomorphic entire curve in $X$ must satisfy an algebraic differential equation of order $k=n=\dim X$, and also similar bounds for order $k>n$. Moreover, for every integer $n\ge 2$, we show that there are no such algebraic differential equations of order $k<n$ for a smooth hypersurface in $\mathbb P^{n+1}$.Comment: Final version, some minor changes according to referee's suggestions, to appear in Compositio Mathematic

Segre forms and Kobayashi-L\"ubke inequality

Starting from the description of Segre forms as direct images of (powers of) the first Chern form of the (anti)tautological line bundle on the projectivized bundle of a holomorphic hermitian vector bundle, we derive a version of the pointwise Kobayashi-L\"ubke inequality.Comment: 14 pages, no figures, comments are really very welcome! v2: Title changed, and added how to formally derive our main inequality from the Kobayashi-L\"ubke one following the suggestions of an anonymous referee. v3: some other minor changes and a remark added in the introduction following referee's suggestions. Final version, to appear on Math.

About the hyperbolicity of complete intersections

This note is an extended version of a thirty minutes talk given at the "XIX Congresso dell'Unione Matematica Italiana", held in Bologna from September 12th to September 17th, 2011. This was essentially a survey talk about connections between Kobayashi hyperbolicity properties and positivity properties of the canonical bundle of projective algebraic varieties.Comment: 9 pages, no figures, to appear on Boll. Unione Mat. Ita

Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle

We show that if a compact complex manifold admits a K\"ahler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.Comment: 12 pages, no figures, final version, to appear on J. Differential Geo

A survey on hyperbolicity of projective hypersurfaces

These are lecture notes of a course held at IMPA, Rio de Janiero, in september 2010: the purpose was to present recent results on Kobayashi hyperbolicity in complex geometry. Our ultimate goal is to describe the results obtained on questions related to the geometry of entire curves traced in generic complex projective hypersurfaces of high degree. For the convenience of the reader, this survey tries to be as self contained as possible.Comment: 108 pages, 2 figure

Existence of global invariant jet differentials on projective hypersurfaces of high degree

Let $X\subset\mathbb P^{n+1}$ be a smooth complex projective hypersurface. In this paper we show that, if the degree of $X$ is large enough, then there exist global sections of the bundle of invariant jet differentials of order $n$ on $X$, vanishing on an ample divisor. We also prove a logarithmic version, effective in low dimension, for the log-pair $(\mathbb P^n,D)$, where $D$ is a smooth irreducible divisor of high degree. Moreover, these result are sharp, \emph{i.e.} one cannot have such jet differentials of order less than $n$.Comment: Final version, to appear in Math. An

Smooth metrics on jet bundles and applications

Following a suggestion made by J.-P. Demailly, for each $k\ge 1$, we endow, by an induction process, the $k$-th (anti)tautological line bundle $\mathcal O_{X_k}(1)$ of an arbitrary complex directed manifold $(X,V)$ with a natural smooth hermitian metric. Then, we compute recursively the Chern curvature form for this metric, and we show that it depends (asymptotically -- in a sense to be specified later) only on the curvature of $V$ and on the structure of the fibration $X_k\to X$. When $X$ is a surface and $V=T_X$, we give explicit formulae to write down the above curvature as a product of matrices. As an application, we obtain a new proof of the existence of global invariant jet differentials vanishing on an ample divisor, for $X$ a minimal surface of general type whose Chern classes satisfy certain inequalities, without using a strong vanishing theorem of Bogomolov.Comment: 24 pages, no figures, comments are welcom

Rational curves on fibered Calabi-Yau manifolds

We show that a smooth projective complex manifold of dimension greater than two endowed with an elliptic fiber space structure and with finite fundamental group always contains a rational curve, provided its canonical bundle is relatively trivial. As an application of this result, we prove that any Calabi-Yau manifold that admits a fibration onto a curve whose general fibers are abelian varieties always contains a rational curve