Wing Keung To
Email Address
mattowk@nus.edu.sg
22 results
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Publication Injectivity radius and gonality of a compact Riemann surface(2012-02) Hwang, J.-M.; To, W.-K.; MATHEMATICSWe obtain a sharp lower bound for the volumes of purely 1-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. © 2012 by The Johns Hopkins University Press.Publication On Seshadri constants of canonical bundles of compact quotients of bounded symmetric domains(2000) Hwang, J.-M.; To, W.-K.; MATHEMATICSWe give lower and upper bounds for Seshadri constants of the canonical line bundles of smooth compact quotients of bounded symmetric domains in terms of metric invariants.Publication Distribution of zeros of sections of canonical line bundles over towers of covers(2001-04) To, W.-K.; MATHEMATICSThe paper obtains the limiting behaviour of the expectations of the zeros of holomorphic sections of the canonical line bundles over a tower of covers of a compact complex manifold.Publication Syzygies of compact complex hyperbolic manifolds(2013) Hwang, J.-M.; To, W.-K.; MATHEMATICSWe give a sufficient condition for the pluri-canonical bundles of a compact complex hyperbolic manifold to satisfy the property (Np) on linear syzygies in terms of the hyperbolic injectivity radius. In the process, we obtain sharp lower bounds for the volumes of one-dimensional complex analytic subvarieties in geodesic tubular neighborhoods of the Cartesian self-product of a compact complex hyperbolic manifold.Publication Effective Pólya semi-positivity for non-negative polynomials on the simplex(2008-08) Mok, H.-N.; To, W.-K.; MATHEMATICSWe consider homogeneous polynomials f ∈ R [x1, ..., xn] which are non-negative on the standard simplex in Rn, and we obtain sufficient conditions for such an f to be Pólya semi-positive, that is, all the coefficients of (x1 + ⋯ + xn)N f are non-negative for all sufficiently large positive integers N. Such sufficient conditions are expressed in terms of the vanishing orders of the monomial terms of f along the faces of the simplex. Our result also gives effective estimates on N under such conditions. Moreover, we also show that any Pólya semi-positive polynomial necessarily satisfies a slightly weaker condition. In particular, our results lead to a simple characterization of the Pólya semi-positive polynomials in the low dimensional case when n ≤ 3 as well as the case (in any dimension) when the zero set of the polynomial in the simplex consists of a finite number of points. We also discuss an application to the representations of non-homogeneous polynomials which are non-negative on a general simplex. © 2008 Elsevier Inc. All rights reserved.Publication Volumes of complex analytic subvarieties of Hermitian symmetric spaces(2002-12) Hwang, J.-M.; To, W.-K.; MATHEMATICSWe 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.Publication Small Eigenvalues of the Laplacians on Vector Bundles over Towers of Covers(1997) To, W.-K.; MATHEMATICSUnder the conditions that a compact Riemannian manifold is of sufficiently pinched negative sectional curvature and that a smooth Hermitian vector bundle over the manifold is also of sufficiently small curvature, we prove some pinching results on the asymptotic behavior of the numbers of small eigenvalues of the Laplacians on the induced Hermitian vector bundles over a tower of covers of the manifold. In the process we also obtain interesting results on the non-existence of square integrable 'almost harmonic' vector bundle-valued forms omitting the middle degree(s) on the universal cover.Publication Green's Functions for Quasi-Hyperbolic Metrics on Degenerating Riemann Surfaces with a Separating Node(1999) To, W.-K.; Weng, L.; MATHEMATICSIn this article, we consider a family of compact Riemann surfaces of genus q ≥ 2 degenerating to a Riemann surface with a separating node and many non-separating nodes. We obtain the asymptotic behavior of Green's functions associated to a continuous family of quasi-hyperbolic metrics on such degenerating Riemann surfaces.Publication Effective isometric embeddings for certain Hermitian holomorphic line bundles(2006-06) To, W.-K.; Yeung, S.-K.; MATHEMATICSWe 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 Hilbert's seventeenth problem on representing a positive definite quadratic form on ℝn 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. © 2006 London Mathematical Society.Publication Weighted L2-cohomology of bounded domains with smooth compact quotients(1998) To, W.-K.; MATHEMATICS
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