Let M be a compact complex manifold and let ( L, H ) be a holomorphic Hermitian line bundle overM... more Let M be a compact complex manifold and let ( L, H ) be a holomorphic Hermitian line bundle overM such that the curvature form of h is nondegenerate and splits into the difference2C 2 of two semipositive forms2C and 2 whose null spaces define mutually transverse holomorphic foliations F and FC, respectively. Then Lm admits, for sufficiently largem 2 N, C1 sections whose ratio embeds M into CP N holomorphically (resp. antiholomorphically) along FC (resp. alongF ). Introduction In the theory of complex manifolds, geometric structures de fined by the subbundles of tangent bundles are basic in analyzing submanifolds and h olomorphic maps. Foliations is one of such structures. In [6], and embedding theorem was established for those mani folds equipped with two mutually transverse holomorphic foliations. Namely, i t was proved that, given a compact complex manifoldM with holomorphic foliationsFC and F such that the tangent bundle ofM is the direct sum of those of FC andF , M is embeddable...
Let D be a bounded pseudoconvex domain in CΓ, and let KD (z, w) be the Bergman kernel function of... more Let D be a bounded pseudoconvex domain in CΓ, and let KD (z, w) be the Bergman kernel function of D. The boundary behavior of KD(z, w), or that of KD (z, z), has attracted a lot of attention because it is closely related to the pseudoconformal geometry of D and dD. For instance, the Bergman metric dd\ogKD(z, z) is invariant under any biholomorphic transformation of D, and the growth-rate of KD (z, z) near a given boundary point x is controlled by the behavior of the Levi-form of dD near x. Roughly speaking, the rank of the Levi-form at x is a pseudoconformal invariant that measures the growth of KD(z, z) near x, or in other words it measures how much room is left for L holomorphic functions to live near x (cf. [Ho], [D], [0-1], [D-H-0]). As is well known, very deep analysis is possible for KD(z, w) in case dD is C°° and strongly pseudoconvex, or more generally of finite type, and as a result one extend Caratheodory's theorem to several complex variables by this approach (cf. [F]...
Let M be a compact complex manifold and let L → M be a holomorphic line bundle whose curvature fo... more Let M be a compact complex manifold and let L → M be a holomorphic line bundle whose curvature form is everywhere of signature (s + , s −). Under some conditions on the curvature form of L, it is shown that K ⊗ L ⊗m admits, for some K and sufficiently large m ∈ N, C ∞ sections t 0 ,. .. , t N such that the ratio (t 0 : • • • : t N) embeds M holomorphically in s + variables and antiholomorphically in s − variables. The result extends the Kodaira's embedding theorem as well as a result of Matsushima for complex tori.
It is proved that, if is a proper holomorphic map with one-dimensional fibers and a covering map,... more It is proved that, if is a proper holomorphic map with one-dimensional fibers and a covering map, a point t ∈ T has a neighbourhood U such that is holomorphically convex if and only if is holomorphically convex.
Some complex n-tori admit holomorphic foliations of codimension one besides the flat ones. It wil... more Some complex n-tori admit holomorphic foliations of codimension one besides the flat ones. It will be shown that such nonlinear foliations, possibly with singularities, can be reduced to those on 2-tori under some topological conditions. A crucial step is an application of the Hodge theory on pseudoconvex manifolds.
We show that the Bergman metric of a bounded homogeneous domain has a potential function whose gr... more We show that the Bergman metric of a bounded homogeneous domain has a potential function whose gradient has a constant norm with respect to the Bergman metric, and further that this constant is independent of the choice of such a potential function.
Let D be a bounded pseudoconvex domain in Cn, and let KD (z, w) be the Bergman kernel function of... more Let D be a bounded pseudoconvex domain in Cn, and let KD (z, w) be the Bergman kernel function of D. The boundary behavior of KD (z, w), or that of KD (z, z), has attracted a lot of attention because it is closely related to the pseudoconformal geometry of D and ∂D.
A real hypersurface M in a complex manifold X is said to be Levi flat if it separates X locally i... more A real hypersurface M in a complex manifold X is said to be Levi flat if it separates X locally into two Stein pieces. It is proved that there exist no real analytic Levi flat hypersurfaces in ℙ2.
It is well known since long time that quasiconformally different finite Riemann surfaces give ris... more It is well known since long time that quasiconformally different finite Riemann surfaces give rise to biholomorphically nonequivalent Teichmüller spaces except for a few obvious cases (cf. [R], [E-K]). This is deduced as an application of Royden’s theorem asserting that the Teichmüller metric is equal to the Kobayashi metric. For the case of infinite Riemann surfaces, however, it is still unknown whether or not the corresponding result holds, although it has been shown by F. Gardiner [G] that Royden’s theorem is also valid for the infinite dimensional Teichmüller spaces. On the other hand, recent activity of several mathematicians shows that the infinite dimensional Teichmüller spaces are interesting objects of complex analytic geometry (cf. [Kru], [T], [N], [E-K-K]). Therefore, based on the generalized form of Royden’s theorem, one might well look for further insight into Teichmüller spaces by studying the above mentioned nonequivalence question.
Publications of the Research Institute for Mathematical Sciences, 2012
An extension of Kodaira's embedding theorem is proved for compact smoothly foliated manifolds wit... more An extension of Kodaira's embedding theorem is proved for compact smoothly foliated manifolds with complex leaves. Existence of a leafwise positive line bundle is shown to be sufficient for C k projective embeddability for all k < ∞.
As an application of a sharp L2 extension theorem for holomorphic functions in Guan and Zhou, a s... more As an application of a sharp L2 extension theorem for holomorphic functions in Guan and Zhou, a stability theorem for the boundary asymptotics of the Bergman kernel is proved. An alternate proof of the extension theorem is given, too. It is a simplified proof in the sense that it is free from ordinary differential equations.
It is proved that the C 2-smoothly bounded pseudoconvex domains in P n admit bounded plurisubharm... more It is proved that the C 2-smoothly bounded pseudoconvex domains in P n admit bounded plurisubharmonic exhaustion functions. Further arguments are given concerning the question of existence of strictly plurisubharmonic functions on neighbourhoods of real hypersurfaces in F n. Let Ω (c M be a pseudoconvex domain in a Kahler manifold M. When M is P fc , Takeuchi [T], showed that the function-log<5^ is strictly plurisubharmonic (p.s.h.) in Ω. Here SQ denotes the distance to the boundary for the standard Kahler metric on Ψ k. The result was extended by Elencwajg [E] to the case where M is Kahler with strictly positive holomorphic bisectional curvature. See also Suzuki [Su] and Green-Wu [G.W]. Based on their result we show that if Ω <ε M is pseudoconvex with C 2 boundary, then there is a bounded strictly p.s.h. function on Ω. When M-C k the question was solved by Diedrich-Fornaess [D.F]. For a survey in this case see [S]. We give an example of a compact Kahler manifold M, containing a Stein domain Ω d M, with smooth boundary, however given any neighborhood U of 9Ω, there is no nonconstant bounded p.s.h. function on U Π Ω. We show next that the existence of a strictly p.s.h. function near dΩ is equivalent to the nonexistence of a positive current T of bidimension (1,1) supported on dΩ and satisfying the equation ddT = 0. This result is inspired by a duality argument due to Sullivan [Su]. §1. Plurisubharmonic exhaustion function on smoothly bounded domains Let (M,ω) be a Kahler manifold. Let Ω d M be a pseudoconvex domain with smooth boundary. We consider first the question of existence of a strictly plurisubharmonic bounded exhaustion function for Ω.
It is shown that any compact Levi flat manifold admitting a positive line bundle is embeddable in... more It is shown that any compact Levi flat manifold admitting a positive line bundle is embeddable into ℙ n by a CR mapping with an arbitrarily high, though finite, order of regularity.
A general extension theorem for L 2 holomorphic bundle-valued top forms is formulated. Although i... more A general extension theorem for L 2 holomorphic bundle-valued top forms is formulated. Although its proof is based on a principle similar to Ohsawa-Takegoshi's extension theorem, it explains previous L 2 extendability results systematically and bridges extension theory and division theory.
Publications of the Research Institute for Mathematical Sciences, 1992
§ 0. Statement of the Results Let DdC n be a bounded domain with C°°-smooth pseudoconvex boundary... more § 0. Statement of the Results Let DdC n be a bounded domain with C°°-smooth pseudoconvex boundary, and letp^dD be any point. By a two-sided bumping family of D at p we mean a family of smoothly bounded pseudoconvex domains {A}-i^<i satisfying the following properties : 2) 3) {dD t } _!<,<! is a C°°-family of real hypersurfaces in C n , 4) for any neighborhood U of p in C n there exists a ^0<0 such that A 0 VD-, 0 c[/. Remark. A two-sided bumping family of D at p exists, of course, if dD is strictly pseudoconvex at p. Recently, it was shown by Cho [Ch], that such a family also exists, if dD is of finite type at p. By a peak function at p we mean a continuous function / on D which is holomorphic on D and satisfies f(p) = 1 and | /(z) | < 1 on D\{p}. By ^(z, w) we denote the Bergman kernel function of D and we put K D (z) = K D (z, z). Finally, we write ds 2 D for the Bergman metric of D. Our goal is to prove continuity results in the parameter t of a bumping family for the Bergman kernel, the Bergman metric and related functions. Namely, we will show the following theorems :
Let M be a compact complex manifold and let ( L, H ) be a holomorphic Hermitian line bundle overM... more Let M be a compact complex manifold and let ( L, H ) be a holomorphic Hermitian line bundle overM such that the curvature form of h is nondegenerate and splits into the difference2C 2 of two semipositive forms2C and 2 whose null spaces define mutually transverse holomorphic foliations F and FC, respectively. Then Lm admits, for sufficiently largem 2 N, C1 sections whose ratio embeds M into CP N holomorphically (resp. antiholomorphically) along FC (resp. alongF ). Introduction In the theory of complex manifolds, geometric structures de fined by the subbundles of tangent bundles are basic in analyzing submanifolds and h olomorphic maps. Foliations is one of such structures. In [6], and embedding theorem was established for those mani folds equipped with two mutually transverse holomorphic foliations. Namely, i t was proved that, given a compact complex manifoldM with holomorphic foliationsFC and F such that the tangent bundle ofM is the direct sum of those of FC andF , M is embeddable...
Let D be a bounded pseudoconvex domain in CΓ, and let KD (z, w) be the Bergman kernel function of... more Let D be a bounded pseudoconvex domain in CΓ, and let KD (z, w) be the Bergman kernel function of D. The boundary behavior of KD(z, w), or that of KD (z, z), has attracted a lot of attention because it is closely related to the pseudoconformal geometry of D and dD. For instance, the Bergman metric dd\ogKD(z, z) is invariant under any biholomorphic transformation of D, and the growth-rate of KD (z, z) near a given boundary point x is controlled by the behavior of the Levi-form of dD near x. Roughly speaking, the rank of the Levi-form at x is a pseudoconformal invariant that measures the growth of KD(z, z) near x, or in other words it measures how much room is left for L holomorphic functions to live near x (cf. [Ho], [D], [0-1], [D-H-0]). As is well known, very deep analysis is possible for KD(z, w) in case dD is C°° and strongly pseudoconvex, or more generally of finite type, and as a result one extend Caratheodory's theorem to several complex variables by this approach (cf. [F]...
Let M be a compact complex manifold and let L → M be a holomorphic line bundle whose curvature fo... more Let M be a compact complex manifold and let L → M be a holomorphic line bundle whose curvature form is everywhere of signature (s + , s −). Under some conditions on the curvature form of L, it is shown that K ⊗ L ⊗m admits, for some K and sufficiently large m ∈ N, C ∞ sections t 0 ,. .. , t N such that the ratio (t 0 : • • • : t N) embeds M holomorphically in s + variables and antiholomorphically in s − variables. The result extends the Kodaira's embedding theorem as well as a result of Matsushima for complex tori.
It is proved that, if is a proper holomorphic map with one-dimensional fibers and a covering map,... more It is proved that, if is a proper holomorphic map with one-dimensional fibers and a covering map, a point t ∈ T has a neighbourhood U such that is holomorphically convex if and only if is holomorphically convex.
Some complex n-tori admit holomorphic foliations of codimension one besides the flat ones. It wil... more Some complex n-tori admit holomorphic foliations of codimension one besides the flat ones. It will be shown that such nonlinear foliations, possibly with singularities, can be reduced to those on 2-tori under some topological conditions. A crucial step is an application of the Hodge theory on pseudoconvex manifolds.
We show that the Bergman metric of a bounded homogeneous domain has a potential function whose gr... more We show that the Bergman metric of a bounded homogeneous domain has a potential function whose gradient has a constant norm with respect to the Bergman metric, and further that this constant is independent of the choice of such a potential function.
Let D be a bounded pseudoconvex domain in Cn, and let KD (z, w) be the Bergman kernel function of... more Let D be a bounded pseudoconvex domain in Cn, and let KD (z, w) be the Bergman kernel function of D. The boundary behavior of KD (z, w), or that of KD (z, z), has attracted a lot of attention because it is closely related to the pseudoconformal geometry of D and ∂D.
A real hypersurface M in a complex manifold X is said to be Levi flat if it separates X locally i... more A real hypersurface M in a complex manifold X is said to be Levi flat if it separates X locally into two Stein pieces. It is proved that there exist no real analytic Levi flat hypersurfaces in ℙ2.
It is well known since long time that quasiconformally different finite Riemann surfaces give ris... more It is well known since long time that quasiconformally different finite Riemann surfaces give rise to biholomorphically nonequivalent Teichmüller spaces except for a few obvious cases (cf. [R], [E-K]). This is deduced as an application of Royden’s theorem asserting that the Teichmüller metric is equal to the Kobayashi metric. For the case of infinite Riemann surfaces, however, it is still unknown whether or not the corresponding result holds, although it has been shown by F. Gardiner [G] that Royden’s theorem is also valid for the infinite dimensional Teichmüller spaces. On the other hand, recent activity of several mathematicians shows that the infinite dimensional Teichmüller spaces are interesting objects of complex analytic geometry (cf. [Kru], [T], [N], [E-K-K]). Therefore, based on the generalized form of Royden’s theorem, one might well look for further insight into Teichmüller spaces by studying the above mentioned nonequivalence question.
Publications of the Research Institute for Mathematical Sciences, 2012
An extension of Kodaira's embedding theorem is proved for compact smoothly foliated manifolds wit... more An extension of Kodaira's embedding theorem is proved for compact smoothly foliated manifolds with complex leaves. Existence of a leafwise positive line bundle is shown to be sufficient for C k projective embeddability for all k < ∞.
As an application of a sharp L2 extension theorem for holomorphic functions in Guan and Zhou, a s... more As an application of a sharp L2 extension theorem for holomorphic functions in Guan and Zhou, a stability theorem for the boundary asymptotics of the Bergman kernel is proved. An alternate proof of the extension theorem is given, too. It is a simplified proof in the sense that it is free from ordinary differential equations.
It is proved that the C 2-smoothly bounded pseudoconvex domains in P n admit bounded plurisubharm... more It is proved that the C 2-smoothly bounded pseudoconvex domains in P n admit bounded plurisubharmonic exhaustion functions. Further arguments are given concerning the question of existence of strictly plurisubharmonic functions on neighbourhoods of real hypersurfaces in F n. Let Ω (c M be a pseudoconvex domain in a Kahler manifold M. When M is P fc , Takeuchi [T], showed that the function-log<5^ is strictly plurisubharmonic (p.s.h.) in Ω. Here SQ denotes the distance to the boundary for the standard Kahler metric on Ψ k. The result was extended by Elencwajg [E] to the case where M is Kahler with strictly positive holomorphic bisectional curvature. See also Suzuki [Su] and Green-Wu [G.W]. Based on their result we show that if Ω <ε M is pseudoconvex with C 2 boundary, then there is a bounded strictly p.s.h. function on Ω. When M-C k the question was solved by Diedrich-Fornaess [D.F]. For a survey in this case see [S]. We give an example of a compact Kahler manifold M, containing a Stein domain Ω d M, with smooth boundary, however given any neighborhood U of 9Ω, there is no nonconstant bounded p.s.h. function on U Π Ω. We show next that the existence of a strictly p.s.h. function near dΩ is equivalent to the nonexistence of a positive current T of bidimension (1,1) supported on dΩ and satisfying the equation ddT = 0. This result is inspired by a duality argument due to Sullivan [Su]. §1. Plurisubharmonic exhaustion function on smoothly bounded domains Let (M,ω) be a Kahler manifold. Let Ω d M be a pseudoconvex domain with smooth boundary. We consider first the question of existence of a strictly plurisubharmonic bounded exhaustion function for Ω.
It is shown that any compact Levi flat manifold admitting a positive line bundle is embeddable in... more It is shown that any compact Levi flat manifold admitting a positive line bundle is embeddable into ℙ n by a CR mapping with an arbitrarily high, though finite, order of regularity.
A general extension theorem for L 2 holomorphic bundle-valued top forms is formulated. Although i... more A general extension theorem for L 2 holomorphic bundle-valued top forms is formulated. Although its proof is based on a principle similar to Ohsawa-Takegoshi's extension theorem, it explains previous L 2 extendability results systematically and bridges extension theory and division theory.
Publications of the Research Institute for Mathematical Sciences, 1992
§ 0. Statement of the Results Let DdC n be a bounded domain with C°°-smooth pseudoconvex boundary... more § 0. Statement of the Results Let DdC n be a bounded domain with C°°-smooth pseudoconvex boundary, and letp^dD be any point. By a two-sided bumping family of D at p we mean a family of smoothly bounded pseudoconvex domains {A}-i^<i satisfying the following properties : 2) 3) {dD t } _!<,<! is a C°°-family of real hypersurfaces in C n , 4) for any neighborhood U of p in C n there exists a ^0<0 such that A 0 VD-, 0 c[/. Remark. A two-sided bumping family of D at p exists, of course, if dD is strictly pseudoconvex at p. Recently, it was shown by Cho [Ch], that such a family also exists, if dD is of finite type at p. By a peak function at p we mean a continuous function / on D which is holomorphic on D and satisfies f(p) = 1 and | /(z) | < 1 on D\{p}. By ^(z, w) we denote the Bergman kernel function of D and we put K D (z) = K D (z, z). Finally, we write ds 2 D for the Bergman metric of D. Our goal is to prove continuity results in the parameter t of a bumping family for the Bergman kernel, the Bergman metric and related functions. Namely, we will show the following theorems :
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