Wing-Keung To

Volumes of complex analytic subvarieties of Hermitian symmetric spaces

We give lower bounds of volumes of k-dimensional complex analytic subvarieties of certain naturally defined domains in n-dimensional complex space forms of constant (positive, zero, or negative) holomorphic sectional curvature. For each 1 ≤ k ≤ n, the lower bounds are sharp in the sense that these bounds are attained by k-dimensional complete totally geodesic complex submanifolds. Such lower bounds are obtained by constructing singular potential functions corresponding to blow-ups of the Kähler metrics involved. Similar lower bounds are also obtained in the case of Hermitian symmetric spaces of noncompact type. In this case, the lower bounds are sharp for those values of k at which the Hermitian symmetric space contains k-dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature.

Effective Isometric Embeddings for Certain Hermitian Holomorphic Line Bundles

We consider bihomogeneous polynomials on complex Euclidean spaces that are positive outside the origin and obtain effective estimates on certain modifications needed to turn them into squares of norms of vector-valued polynomials on complex Euclidean space. The corresponding results for hypersurfaces in complex Euclidean spaces are also proved. The results can be considered as Hermitian analogues of Hilberts seventeenth problem on representing a positive definite quadratic form on

On Seshadri Constants of Canonical Bundles of Compact Complex Hyperbolic Spaces

\mathbb{R}^n

Injectivity radius and gonality of a compact Riemann surface

as a sum of squares of rational functions. They can also be regarded as effective estimates on the power of a Hermitian line bundle required for isometric projective embedding. Further applications are discussed.

Buser-Sarnak invariant and projective normality of abelian varieties

Upper and lower bounds for the Seshadri constants of canonical bundles of compact hyperbolic spaces are given in terms of metric invariants. The lower bound is obtained by carrying out the symplectic blow-up construction for the Poincaré metric, and the upper bound is obtained by a convexity-type argument.

Effective bounds on holomorphic mappings into complex hyperbolic manifolds

We obtain a sharp lower bound for the volumes of purely

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

1

Uniform boundedness of level structures on abelian varieties over complex function fields

-dimensional complex analytic subvarieties in a geodesic tubular neighborhood of the diagonal of the Cartesian product of a compact Riemann surface with itself. This leads to a lower bound of the Seshadri number of the canonical line bundle of the Cartesian product with respect to the diagonal. As a consequence, we obtain an upper bound for the hyperbolic injectivity radii of compact Riemann surfaces of a fixed gonality. In particular, we obtain the limiting behavior of the gonalities of a tower of compact Riemann surfaces. We also give an application of our results to an invariant related to the ample cone of the symmetric product of a Riemann surface.

Total geodesy of proper holomorphic immersions between complex hyperbolic space forms of finite volume

We show that a general n-dimensional polarized abelian variety (A,L) of a given polarization type and satisfying \(h^0(A,L)\geq \frac{8^n}{2}. \frac{n^n}{n !}\) is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely onedimensional complex analytic subvariety in a geodesic tubular neighborhood of a subtorus of a compact complex torus.

Effective Pólya semi-positivity for non-negative polynomials on the simplex

We give an explicit upper bound on the number of non-constant holomorphic maps from a quasi-projective manifold into a complex hyperbolic manifold of finite volume. This gives an effective version of the results of Sunada and Noguchi.