A compact complex manifold is Kobayashi non-hyperbolic if there exists an entire curve on it. Using mirror symmetry we establish that there are (possibly singular) elliptic or rational curves on any Calabi-Yau manifold $X$, whose mirror dual $\check X$ exists and is not "Hodge degenerate", therefore proving that $X$ is Kobayashi non-hyperbolic. We are not aware of any higher dimensional simply connected Calabi-Yau manifolds that satisfy the "Hodge degenerate" condition.
We prove that every bounded strictly $J$-convex region equipped with the Kobayashi metric is hyperbolic in the sense of Gromov. We apply this result to the study of the dynamics of pseudo-holomorphic maps.
In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with $C^\infty$ boundary being of finite type in the sense of D'Angelo is equivalent to the Gromov hyperbolicity of the Kobayashi metric. We also show that bounded domains which are locally convexifiable and have finite type in the sense of D'Angelo have Gromov hyperbolic Kobayashi metric. The proofs use ideas from the theory of the Hilbert metric.
This survey article mainly addresses to graduate students and young researchers in complex geometry willing to enter the beautiful word of connections between curvature and Kobayashi hyperbolicity. It is a detailed account of a recent breakthrough by Wu and Yau, shortly after generalized by Tosatti and Yang (and others), which sits on the crossroad between complex differential geometry and Kobayashi hyperbolicity. More specifically, an old conjecture by Kobayashi, stated at the very beginning of the theory, predicts that a compact hyperbolic manifold should have ample canonical bundle. Now, on…
The classical Brody's theorem asserts the equivalence between two notions of hyperbolicity for compact complex spaces, one named after Kobayashi and one expressed in terms of lack of non constant holomorphic entire functions (compactness is only used to prove the harder implication). We extend this theorem to Deligne-Mumford analytic stacks, by first providing definitions of what we think of Kobayashi and Brody hyperbolicity for such objects and then proving the equivalence of these concepts under an assumption of compactness.
We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi distance. We also provide examples of bounded smooth convex domains that are not strongly pseudoconvex but are Gromov hyperbolic.
We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains.The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain $Ω$ in $\mathbb R^n$ with the Kobayashi distance of the tube domain over the domain $Ω$ is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tu…
Let $X$ be an algebraic variety over $\mathbb{C}$. We say that $X$ is Borel hyperbolic if, for every finite type reduced scheme $S$ over $\mathbb{C}$, every holomorphic map $S^{an}\to X^{an}$ is algebraic. We use a transcendental specialization technique to prove that $X$ is Borel hyperbolic if and only if, for every smooth affine curve $C$ over $\mathbb{C}$, every holomorphic map $C^{an}\to X^{an}$ is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.
The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture for any hyperkähler manifold that admits a deformation with two Lagrangian fibrations and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperkähler manifolds are semi-ample. We prove that the Kobayashi pseudom…
We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold $(M,J)$ admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in $M$ and to give a sufficient condition for the complete hyperbolicity of a domain in $(M,J)$. Finally we obtain the regularity up to the bounday of $J$-holomorphic discs attached to a totally real submanifold in $M$.
Demailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraic…
The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable holomorphic vector bundle over a compact Kaehler manifold is stable in the sense of Takemoto-Mumford if and only if the vector bundle admits a Hermitian-Einstein metric. Its manifold analogue known as Yau's conjecture, which originated from Calabi's conjecture, asks whether ``stability'' and ``existence of extremal metrics'' for polarized manifolds are equivalent. In this note, the recent progress of this subject, by Donaldson, Tian and our grou…
In this paper we study when the Kobayashi distance on a Kobayashi hyperbolic domain has certain visibility properties, with a focus on unbounded domains. "Visibility" in this context is reminiscent of visibility, seen in negatively curved Riemannian manifolds, in the sense of Eberlein-O'Neill. However, we do not assume that the domains studied are Cauchy-complete with respect to the Kobayashi distance, as this is hard to establish for domains in $\mathbb{C}^d$, $d \geq 2$. We study the various ways in which this property controls the boundary behavior of holomorphic maps. Among these results i…
We derive some consequences of the Liouville theorem for plurisubharmonic functions of L.-F. Tam and the author. The first result provides a nonlinear version of the complex splitting theorem (which splits off a factor of $\mathbb{C}$ isometrically from the simply-connected Kähler manifold with nonnegative bisectional curvature and a linear growth holomorphic function) of L.-F. Tam and the author. The second set of results concerns the so-called $k$-hyperbolicity and its connection with the negativity of the $k$-scalar curvature (when $k=1$ they are the negativity of holomorphic sectional curv…
The Kummer Construction of Calabi-Yau and Hyper-Kähler Metrics on the $K3$ Surface, and Large Families of Volume Non-collapsed Limiting Compact Hyper-Kähler Orbifolds
