Finsler metrics and Kobayashi hyperbolicity of the moduli spaces of canonically polarized manifolds

@article{To2015FinslerMA,
  title={Finsler metrics and Kobayashi hyperbolicity of the moduli spaces of canonically polarized manifolds},
  author={Wing-Keung To and Sai-Kee Yeung},
  journal={Annals of Mathematics},
  year={2015},
  volume={181},
  pages={547-586}
}
We show that the base complex manifold of an eectively parametrized holomorphic family of compact canonically polarized complex manifolds admits a smooth invariant Finsler metric whose holomorphic sectional curvature is bounded above by a negative constant. As a consequence, we show that such a base manifold is Kobayashi hyperbolic. 
View via Publisher
annals.math.princeton.edu
43 Citations
Highly Influential Citations
5
Background Citations
15
Methods Citations
7
Results Citations
1

43 Citations

Augmented Weil–Petersson metrics on moduli spaces of polarized Ricci-flat Kähler manifolds and orbifolds

. We show that the base complex manifold of an effectively parametrized family of compact polarized Ricci-flat K¨ahler orbifolds, and in particular manifolds, admits a smooth augmented Weil-Petersson…

Holomorphic Sectional Curvature of Complex Finsler Manifolds

  • Xueyuan Wan
  • Mathematics
    Journal of geometric analysis
  • 2019
An inequality in terms of holomorphic sectional curvature of complex Finsler metrics is obtained and a Schwarz Lemma from a complete Riemannian manifold to a complex F Insler manifold is proved.

Moduli of canonically polarized manifolds, higher order Kodaira-Spencer maps, and an analogy to Calabi-Yau manifolds

A Finsler metric, whose holomorphic curvature is bounded from above by a negative constant, is constructed on the moduli stack of canonically polarized manifolds including singularities. Demailly's…

Nevanlinna Theory on Moduli Space and the big Picard Theorem

We show that the moduli stack Mh of polarized complex manifolds with Hilbert polynomial h is Borel hyperbolic in the following sense: if for some quasi-projective variety U there exists a…

Hyperbolicity of Moduli Spaces of Log-Canonically Polarized Manifolds

In recent years, there are quite a lot of interests and results related to hyperbolicity properties of the base spaces of various families of projective algebraic varieties. Not much is known for…

Curvature properties for moduli of canonically polarized manifolds—An analogy to moduli of Calabi–Yau manifolds

Hyperbolicity of coarse moduli spaces and isotriviality for certain families

In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli spaces of canonically polarized or polarized Calabi-Yau manifolds in the sense of complex $V$-spaces (a generalization of…

Hyperbolicity of the base of an admissible family of log canonical stable minimal models

We investigate the stratified hyperbolicity properties of Birkar's moduli stack of log canonical (lc) stable minimal models. The main technical result is a construction of Viehweg-Zuo's system of…

Hyperbolicity for Log Smooth Families with Maximal Variation

We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good…

Brody hyperbolicity of base spaces of certain families of varieties

We prove that quasi-projective base spaces of smooth families of minimal varieties of general type with maximal variation do not admit Zariski dense entire curves. We deduce the fact that moduli…

References

SHOWING 1-10 OF 29 REFERENCES

On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds

We show that the moduli stack Mh of canonically polarized complex manifolds with Hilbert polynomial h is Brody hyperbolic. Hence if Mh denotes the corresponding coarse moduli scheme, and if U → Mh is…

The Curvature of the Petersson-Weil Metric on the Moduli Space of Kähler-Einstein Manifolds

The Petersson-Weil metric is a main tool for investigating the geometry of moduli spaces. When A. Weil considered the classical Teichmuller space from the viewpoint of deformation theory, he…

Extremal bounded holomorphic functions and an embedding theorem for arithmetic varieties of rank ≥ 2

Let Ω be a bounded symmetric domain of rank ≥ 2, and Γ ⊂ Aut(Ω) be a torsion-free irreducible cocompact lattice, X := Ω/Γ. On the projective manifold X there is the canonical Kahler-Einstein metric,…

Quasi-projective moduli for polarized manifolds

  • E. Viehweg
  • Mathematics
    Ergebnisse der Mathematik und ihrer Grenzgebiete
  • 1995
This text discusses two subjects of quite different natures: construction methods for quotients of quasi-projective schemes either by group actions or by equivalence relations; and properties of…

Hyperbolic complex spaces

1. Distance Geometry.- 2. Schwarz Lemma and Negative Curvature.- 3. Intrinsic Distances.- 4. Intrinsic Distances for Domains.- 5. Holomorphic Maps into Hyperbolic Spaces.- 6. Extension and Finiteness…

On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*

Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the…

Positivity of relative canonical bundles and applications

Given a family $f:\mathcal{X} \to S$ of canonically polarized manifolds, the unique Kähler–Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle…

Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees

We prove that the defect vanishes for a holomorphic map f from the affine complex line to an abelian variety A and for an ample divisor D in A. The proof uses the translational invariance of the…

Curvature of the Weil-Petersson Metric in the Moduli Space of Compact Kähler-Einstein Manifolds of Negative First Chem Class

For compact Riemann surfaces of genus at least two, using Petersson’s Hermitian pairing for automorphic forms, Weil introduced a Hermitian metric for the Teiclmuller space, now known as the…

Intrinsic Metrics on Teichmüller Space

1. My purpose here is to describe some recent results about metrics on the Teichmuller space Tg9 that is on the space of moduli of compact Riemann surfaces of a given genus g > 1. If W is a…