Logarithmic Surfaces and Hyperbolicity

In 1981 J.Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate. In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: Theorem: In a logarithmic surface with logarithmic irregularity 2 and logarithmic Kodaira dimension 2, any Brody curve is algebraically degenerate. We also deal with the case of arbitrary logarithmic Kodaira dimension. As a corollary, we get hyperbolicity for such logarithmic surfaces not containing non-hyperbolic algebraic curves and having hyperbolically stratified boundary divisors. In particular we get the "best possible" result on algebraic degeneracy of Brody curves in the complex plane minus a curve consisting of three components, thus improving results of Dethloff-Schumacher-Wong from 1995.Comment: 34 pages. Final version, to appear in Annales Fourie

Estimates of the number of rational mappings from a fixed variety to varieties of general type

First we find effective bounds for the number of dominant rational maps $f:X \rightarrow Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type $\{A \cdot K_X^n\}^{\{B \cdot K_X^n\}^2}$, where $n=dimX$, $K_X$ is the canonical bundle of $X$ and $A,B$ are some constants, depending only on $n$. Then we show that for any variety $X$ there exist numbers $c(X)$ and $C(X)$ with the following properties: For any threefold $Y$ of general type the number of dominant rational maps $f:X \r Y$ is bounded above by $c(X)$. The number of threefolds $Y$, modulo birational equivalence, for which there exist dominant rational maps $f:X \r Y$, is bounded above by $C(X)$. If, moreover, $X$ is a threefold of general type, we prove that $c(X)$ and $C(X)$ only depend on the index $r_{X_c}$ of the canonical model $X_c$ of $X$ and on $K_{X_c}^3$.Comment: A revised version. The presentation of results and proofs has been improved. AMS-TeX, 19 page

Uniqueness Problem for Meromorphic Mappings with Truncated Multiplicities and Moving Targets

In this paper, using techniques of value distribution theory, we give a uniqueness theorem for meromorphic mappings of C^m into P^n with (3n+1) moving targets and truncated multiplicities.Comment: Final version, published in Nagoya J. Mat

Plane curves with a big fundamental group of the complement

Let C \s \pr^2 be an irreducible plane curve whose dual C^* \s \pr^{2*} is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincar\'e group \pi_1(\pr^2 \se C) contains a free group with two generators. If the geometric genus $g$ of $C$ is at least 2, then a subgroup of $G$ can be mapped epimorphically onto the fundamental group of the normalization of $C$, and the result follows. To handle the cases $g=0,1$, we construct universal families of immersed plane curves and their Picard bundles. This allows us to reduce the consideration to the case of Pl\"ucker curves. Such a curve $C$ can be regarded as a plane section of the corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying Zariski--Lefschetz type arguments we deduce the result from `the bigness' of the $d$-th braid group $B_{d,g}$ of the Riemann surface of $C$.Comment: 23 pages LaTeX. A revised version. The unnecessary restriction $d \ge 2g - 1$ of the previous version has been removed, and the main result has taken its final for

Logarithmic Jet Bundles and Applications

We generalize Demailly's construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. We establish this logarithmic generalization explicitly via coordinates, just as Noguchi's generalization of the jets used by Green-Griffiths. As a first application, we give a metric proof for the logarithmic version of Lang's conjecture concerning the hyperbolicity of complements of divisors in a semi-abelian variety as well as for the corresponding big Picard theorem.Comment: 49 pages, Late