Holomorphic sectional curvature of complex Finsler manifolds

In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identically zero holomorphic sectional curvature has negative Kodaira dimension under an extra condition.Comment: 20 pages, revised version, to appear in The Journal of Geometric Analysi

Norm estimates and asymptotic faithfulness of the quantum SU(n)SU(n) representations of the mapping class groups

We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier in [1] using Toeplitz operators of compact K\"ahler manifolds and in [10] using skein theory.Comment: Geometriae Dedicata (online), 10 pages, minor change

Positivity of Schur forms for strongly decomposably positive vector bundles

In this paper, we define two types of strongly decomposable positivity, which serve as generalizations of (dual) Nakano positivity and are stronger than the decomposable positivity introduced by S. Finski. We provide the criteria for strongly decomposable positivity of type I and type II and prove that the Schur forms of a strongly decomposable positive vector bundle of type I are weakly positive, while the Schur forms of a strongly decomposable positive vector bundle of type II are positive. These answer a question of Griffiths affirmatively for strongly decomposably positive vector bundles. Consequently, we present an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which was initially proven by S. Finski.Comment: 31 pages, 1 figure, final version, to appear in Forum of Mathematics, Sigm

Geodesic-Einstein metrics and nonlinear stabilities

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.Comment: 21 pages, the final version, to appear in Transactions of the American Mathematical Societ

On local stabilities of pp-K\"ahler structures

By use of a natural extension map and a power series method, we obtain a local stability theorem for p-K\"ahler structures with the $(p,p+1)$-th mild $\partial\bar\partial$-lemma under small differentiable deformations.Comment: Several typos have been fixed. Final version to appear in Compositio Mathematica. arXiv admin note: text overlap with arXiv:1609.0563