ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Fri, 30 Sep 2022 23:49:40 GMT2022-09-30T23:49:40Z50221- Green's Functions for Quasi-Hyperbolic Metrics on Degenerating Riemann Surfaces with a Separating Nodehttps://scholarbank.nus.edu.sg/handle/10635/103355Title: Green's Functions for Quasi-Hyperbolic Metrics on Degenerating Riemann Surfaces with a Separating Node
Authors: To, W.-K.; Weng, L.
Abstract: In 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.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033551999-01-01T00:00:00Z
- Curvature of the L2-metric on the direct image of a family of Hermitian-Einstein vector bundleshttps://scholarbank.nus.edu.sg/handle/10635/103091Title: Curvature of the L2-metric on the direct image of a family of Hermitian-Einstein vector bundles
Authors: To, W.-K.; Weng, L.
Abstract: For a holomorphic family of simple Hermitian-Einstein holomorphic vector bundles over a compact Kähler manifold, the locally free part of the associated direct image sheaf over the parameter space forms a holomorphic vector bundle, and it is endowed with a Hermitian metric given by the L2 pairing using the Hermitian-Einstein metrics. Our main result in this paper is to compute the curvature of the L2-metric. In the case of a family of Hermitian holomorphic line bundles with fixed positive first Chern form and under certain curvature conditions, we show that the L2-metric is conformally equivalent to a Hermitian-Einstein metric. As applications, this proves the semi-stability of certain Picard bundles, and it leads to an alternative proof of a theorem of Kempf.
Mon, 01 Jun 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030911998-06-01T00:00:00Z
- Distribution of zeros of sections of canonical line bundles over towers of covershttps://scholarbank.nus.edu.sg/handle/10635/103147Title: Distribution of zeros of sections of canonical line bundles over towers of covers
Authors: To, W.-K.
Abstract: The 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.
Sun, 01 Apr 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031472001-04-01T00:00:00Z
- Weighted L2-cohomology of bounded domains with smooth compact quotientshttps://scholarbank.nus.edu.sg/handle/10635/104470Title: Weighted L2-cohomology of bounded domains with smooth compact quotients
Authors: To, W.-K.
Thu, 01 Jan 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044701998-01-01T00:00:00Z
- Volumes of complex analytic subvarieties of Hermitian symmetric spaceshttps://scholarbank.nus.edu.sg/handle/10635/104455Title: Volumes of complex analytic subvarieties of Hermitian symmetric spaces
Authors: Hwang, J.-M.; To, W.-K.
Abstract: 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.
Sun, 01 Dec 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044552002-12-01T00:00:00Z
- The asymptotic behavior of Green's functions for quasi-hyperbolic metrics on degenerating Riemann surfaceshttps://scholarbank.nus.edu.sg/handle/10635/104252Title: The asymptotic behavior of Green's functions for quasi-hyperbolic metrics on degenerating Riemann surfaces
Authors: To, W.-K.; Weng, L.
Abstract: In this article, we consider a family of compact Riemann surfaces of genus q ≥ 2 degenerating to a Riemann surface of genus q-1 with a non-separating node. We show that the Green's functions associated to a continuous family of quasi-hyperbolic metrics on such degenerating Riemann surfaces simply degenerate to that on the smooth part of the noded Riemann surface.
Fri, 01 Aug 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042521997-08-01T00:00:00Z
- Syzygies of compact complex hyperbolic manifoldshttps://scholarbank.nus.edu.sg/handle/10635/104244Title: Syzygies of compact complex hyperbolic manifolds
Authors: Hwang, J.-M.; To, W.-K.
Abstract: We 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.
Tue, 01 Jan 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042442013-01-01T00:00:00Z
- Admissible Hermitian metrics on families of line bundles over certain degenerating Riemann surfaceshttps://scholarbank.nus.edu.sg/handle/10635/102793Title: Admissible Hermitian metrics on families of line bundles over certain degenerating Riemann surfaces
Authors: To, W.-K.; Weng, L.
Abstract: We show that a family of line bundles of degree zero over a plumbing family of Riemann surfaces with a separating (resp. non-separating) node p admits a nice (resp. almost nice) family of flat p-singular Hermitian metrics. As a consequence, we give necessary and sufficient conditions for a family of line bundles over such families of Riemann surfaces to admit an (almost) nice family of p-singular Hermitian metrics which are admissible with respect to the canonical/hyperbolic (1,1)-forms on the Riemann surfaces.
Thu, 01 Feb 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027932001-02-01T00:00:00Z
- On Seshadri Constants of Canonical Bundles of Compact Complex Hyperbolic Spaceshttps://scholarbank.nus.edu.sg/handle/10635/103754Title: On Seshadri Constants of Canonical Bundles of Compact Complex Hyperbolic Spaces
Authors: Hwang, J.-M.; To, W.-K.
Abstract: Upper and lower bounds for the Seshadri constants of canonical bundles of compact complex 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.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037541999-01-01T00:00:00Z
- On Seshadri constants of canonical bundles of compact quotients of bounded symmetric domainshttps://scholarbank.nus.edu.sg/handle/10635/103755Title: On Seshadri constants of canonical bundles of compact quotients of bounded symmetric domains
Authors: Hwang, J.-M.; To, W.-K.
Abstract: We 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.
Sat, 01 Jan 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037552000-01-01T00:00:00Z
- Small Eigenvalues of the Laplacians on Vector Bundles over Towers of Covershttps://scholarbank.nus.edu.sg/handle/10635/104136Title: Small Eigenvalues of the Laplacians on Vector Bundles over Towers of Covers
Authors: To, W.-K.
Abstract: Under 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.
Wed, 01 Jan 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041361997-01-01T00:00:00Z
- The asymptotic behavior of the Takhtajan-Zograf metrichttps://scholarbank.nus.edu.sg/handle/10635/104254Title: The asymptotic behavior of the Takhtajan-Zograf metric
Authors: Obitsu, K.; To, W.-K.; Weng, L.
Abstract: We obtain the asymptotic behavior of the Takhtajan-Zograf metric on the Teichmüller space of punctured Riemann surfaces. © 2008 Springer-Verlag.
Sat, 01 Nov 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042542008-11-01T00:00:00Z
- The asymptotic behavior of the Takhtajan-Zograf metrichttps://scholarbank.nus.edu.sg/handle/10635/53214Title: The asymptotic behavior of the Takhtajan-Zograf metric
Authors: Obitsu, K.; To, W.-K.; Weng, L.
Abstract: We obtain the asymptotic behavior of the Takhtajan-Zograf metric on the Teichmüller space of punctured Riemann surfaces. © 2008 Springer-Verlag.
Sat, 01 Nov 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/532142008-11-01T00:00:00Z
- Effective bounds on holomorphic mappings into complex hyperbolic manifoldshttps://scholarbank.nus.edu.sg/handle/10635/103172Title: Effective bounds on holomorphic mappings into complex hyperbolic manifolds
Authors: Hwang, J.-M.; To, W.-K.
Abstract: We give an explicit upper bound on the number of non-constant holomorphic maps from a quasiprojective manifold into a complex hyperbolic manifold of finite volume. This gives an effective version of the results of Sunada and Noguchi. © 2006 London Mathematical Society.
Fri, 01 Dec 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031722006-12-01T00:00:00Z
- Total geodesy of proper holomorphic immersions between complex hyperbolic space forms of finite volumehttps://scholarbank.nus.edu.sg/handle/10635/104391Title: Total geodesy of proper holomorphic immersions between complex hyperbolic space forms of finite volume
Authors: To, W.-K.
Wed, 01 Sep 1993 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043911993-09-01T00:00:00Z
- L 2-metrics, projective flatness and families of polarized abelian varietieshttps://scholarbank.nus.edu.sg/handle/10635/103468Title: L 2-metrics, projective flatness and families of polarized abelian varieties
Authors: To, W.-K.; Weng, L.
Abstract: We compute the curvature of the L 2-metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kähler manifolds. As an application, we show that the L 2-metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kähler manifold, this leads to the poly-stability of the direct image with respect to any Kähler form on the parameter space.
Thu, 01 Jul 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034682004-07-01T00:00:00Z
- Kähler metrics of negative holomorphic bisectional curvature on Kodaira surfaceshttps://scholarbank.nus.edu.sg/handle/10635/103461Title: Kähler metrics of negative holomorphic bisectional curvature on Kodaira surfaces
Authors: To, W.-K.; Yeung, S.-K.
Abstract: We construct a Kähler metric of negative holomorphic bisectional curvature on any compact complex surface that admits a Kodaira fibration. This is achieved by considering the curvature of the Weil-Petersson metric on an associated Teichmüller space of punctured Riemann surfaces. © 2011 London Mathematical Society.
Wed, 01 Jun 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034612011-06-01T00:00:00Z
- Uniform boundedness of level structures on abelian varieties over complex function fieldshttps://scholarbank.nus.edu.sg/handle/10635/104424Title: Uniform boundedness of level structures on abelian varieties over complex function fields
Authors: Hwang, J.-M.; To, W.-K.
Abstract: Let X = Ω/ Γ be a smooth quotient of a bounded symmetric domain Ω by an arithmetic subgroup Γ ⊂ Aut(Ω). We prove the following generalization of Nadel's result: for any non-negative integer g, there exists a finite étale cover Xg = Ω/ Γ(g) of X determined by a subgroup Γ(g) ⊂ Γ depending only on g, such that for any compact Riemann surface R of genus g and any non-constant holomorphic map f: R → X*g from R into the Satake-Baily-Borel compactification X*g of Xg, the image f (R) lies in the boundary ∂Xg:= X*g\ Xg. Nadel proved it for g = 0 or 1. Moreover, for any positive integer n and any non-negative integer g ≥ 0, we show that there exists a positive number a(n, g) depending only on n and g with the following property: a principally polarized non-isotrivial n-dimensional abelian variety over a complex function field of genus g does not have a level-N structure for N ≥ a(n, g). This was proved by Nadel for g = 0 or 1, and by Noguchi for arbitrary g under the additional hypothesis that the abelian variety has non-empty singular fibers.
Thu, 01 Jun 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044242006-06-01T00:00:00Z
- Effective Pólya semi-positivity for non-negative polynomials on the simplexhttps://scholarbank.nus.edu.sg/handle/10635/103177Title: Effective Pólya semi-positivity for non-negative polynomials on the simplex
Authors: Mok, H.-N.; To, W.-K.
Abstract: We 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.
Fri, 01 Aug 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031772008-08-01T00:00:00Z
- Injectivity radius and gonality of a compact Riemann surfacehttps://scholarbank.nus.edu.sg/handle/10635/103430Title: Injectivity radius and gonality of a compact Riemann surface
Authors: Hwang, J.-M.; To, W.-K.
Abstract: We 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.
Wed, 01 Feb 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034302012-02-01T00:00:00Z
- Effective isometric embeddings for certain Hermitian holomorphic line bundleshttps://scholarbank.nus.edu.sg/handle/10635/103175Title: Effective isometric embeddings for certain Hermitian holomorphic line bundles
Authors: To, W.-K.; Yeung, S.-K.
Abstract: 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 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.
Thu, 01 Jun 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031752006-06-01T00:00:00Z
- Effective Ljasiewicz inequality for arithmetically defined varieties and a geometric application to bihomogeneous polynomialshttps://scholarbank.nus.edu.sg/handle/10635/103176Title: Effective Ljasiewicz inequality for arithmetically defined varieties and a geometric application to bihomogeneous polynomials
Authors: To, W.-K.; Yeung, S.-K.
Abstract: We establish two versions of effective Lojasiewicz inequality for arithmetically defined affine varieties. As an application, we consider bihomogeneous polynomials on the complex Euclidean space which are positive along the affine cone of an arithmetically defined projective variety, and we obtain effective estimates on certain modifications needed to turn them into sums of squares of pointwise norms of homogeneous polynomials. The latter can be interpreted as an effective result on isometric embeddings for the associated indefinite Hermitian holomorphic line bundles. © 2009 World Scientific Publishing Company.
Wed, 01 Jul 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031762009-07-01T00:00:00Z