Seminar on Geometric Complex Analysis
Seminar information archive ~02/15|Next seminar|Future seminars 02/16~
Date, time & place | Monday 10:30 - 12:00 128Room #128 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | Kengo Hirachi, Shigeharu Takayama |
Seminar information archive
2020/05/18
10:30-12:00 Online
KASUYA Hisashi (Osaka University)
Higgs bundles and flat connections over compact Sasakian manifolds
https://forms.gle/vSFPoVR6ugrkTGhX7
KASUYA Hisashi (Osaka University)
Higgs bundles and flat connections over compact Sasakian manifolds
[ Abstract ]
It is known that on a compact Kähler manifold, there is a correspondence between semisimple flat vector bundles and polystable higgs bundles with vanishing Chern classes via harmonic metrics (Simpson-Corlette). The purpose of this talk is to give the Sasakian (odd dimensional analogue of Kähler geometry) version of this correspondence. We prove that on a compact Sasakian manifold, there is an correspondence between semisimple flat vector bundles and the polystable basic Higgs bundles with vanishing basic Chern classes. (Joint work with Indranil Biswas, arXiv:1905.06178)
[ Reference URL ]It is known that on a compact Kähler manifold, there is a correspondence between semisimple flat vector bundles and polystable higgs bundles with vanishing Chern classes via harmonic metrics (Simpson-Corlette). The purpose of this talk is to give the Sasakian (odd dimensional analogue of Kähler geometry) version of this correspondence. We prove that on a compact Sasakian manifold, there is an correspondence between semisimple flat vector bundles and the polystable basic Higgs bundles with vanishing basic Chern classes. (Joint work with Indranil Biswas, arXiv:1905.06178)
https://forms.gle/vSFPoVR6ugrkTGhX7
2020/02/17
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Toshiki Mabuchi (Osaka Univ.)
Precompactness of the moduli space of pseudo-normed graded algebras
Toshiki Mabuchi (Osaka Univ.)
Precompactness of the moduli space of pseudo-normed graded algebras
[ Abstract ]
Graded algebras (such as canonical rings) coming from the spaces of sections of polarized algebraic varieties are studied by many mathematicians. On the other hand, the pseudo-norm project proposed by S.-T. Yau and C.-Y. Chi gives us a new differential geometric aspect of the Torelli type theorem.
In this talk, we give the details of how the geometry of pseudo-normed graded algebras allows us to obtain a natural compactification of the moduli space of pseudo-normed graded algebras.
(1) For a sequence of pseudo-normed graded algebras (of the same type), the above precompactness gives us some limit different from the Gromov-Hausdorff limit in Riemannian geometry.
(2) As an example of our construction, we have the Deligne-Mumford compactification, in which the notion of the orthogonal direct sum of pseudo-normed spaces comes up naturally. We also have a higher dimensional analogue by using weight filtration.
Graded algebras (such as canonical rings) coming from the spaces of sections of polarized algebraic varieties are studied by many mathematicians. On the other hand, the pseudo-norm project proposed by S.-T. Yau and C.-Y. Chi gives us a new differential geometric aspect of the Torelli type theorem.
In this talk, we give the details of how the geometry of pseudo-normed graded algebras allows us to obtain a natural compactification of the moduli space of pseudo-normed graded algebras.
(1) For a sequence of pseudo-normed graded algebras (of the same type), the above precompactness gives us some limit different from the Gromov-Hausdorff limit in Riemannian geometry.
(2) As an example of our construction, we have the Deligne-Mumford compactification, in which the notion of the orthogonal direct sum of pseudo-normed spaces comes up naturally. We also have a higher dimensional analogue by using weight filtration.
2020/01/27
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Hajime Tsuji (Sophia Univ.)
Canonical measure and it’s applications
Hajime Tsuji (Sophia Univ.)
Canonical measure and it’s applications
[ Abstract ]
The canonical measure is a natural generalization of K\”ahler-Einstein metrics to the case of projective manifolds with nonnegative Kodaira dimension. In this talk we consider the variation of canonical measures under projective deformations and give some applications.
The canonical measure is a natural generalization of K\”ahler-Einstein metrics to the case of projective manifolds with nonnegative Kodaira dimension. In this talk we consider the variation of canonical measures under projective deformations and give some applications.
2020/01/20
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Masanori Adachi (Shizuoka Univ.)
Diederich-Fornaess and Steinness indices for abstract CR manifolds
Masanori Adachi (Shizuoka Univ.)
Diederich-Fornaess and Steinness indices for abstract CR manifolds
[ Abstract ]
The Diederich-Fornaes and Steinness indices are estimated for weakly pseudoconvex domains in complex manifolds in terms of the D'Angelo 1-form of the boundary CR manifolds. In particular, CR invariance of these indices is shown when the domain is Takeuchi 1-convex. This is a joint work with Jihun Yum (Pusan National University).
The Diederich-Fornaes and Steinness indices are estimated for weakly pseudoconvex domains in complex manifolds in terms of the D'Angelo 1-form of the boundary CR manifolds. In particular, CR invariance of these indices is shown when the domain is Takeuchi 1-convex. This is a joint work with Jihun Yum (Pusan National University).
2019/12/16
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Genki Hosono (Tohoku Univ.)
A simplified proof of the optimal L^2 extension theorem and its application (Japanese)
Genki Hosono (Tohoku Univ.)
A simplified proof of the optimal L^2 extension theorem and its application (Japanese)
[ Abstract ]
I will explain a simplified proof of an optimal version of the Ohsawa-Takegoshi L^2-extension theorem. In the proof, I use a method of Berndtsson-Lempert and skip some argument by the method of McNeal-Varolin. As an application, I will explain a result on extensions from possibly non-reduced varieties.
I will explain a simplified proof of an optimal version of the Ohsawa-Takegoshi L^2-extension theorem. In the proof, I use a method of Berndtsson-Lempert and skip some argument by the method of McNeal-Varolin. As an application, I will explain a result on extensions from possibly non-reduced varieties.
2019/12/11
16:00-17:00 Room #156 (Graduate School of Math. Sci. Bldg.)
Joel Merker (Paris Sud)
Einstein-Weyl structures (English)
Joel Merker (Paris Sud)
Einstein-Weyl structures (English)
[ Abstract ]
On a conformal 3D manifold with electromagnetic field, Einstein-Weyl equations are the counterpart of Einstein's classical field equations. In 1943, Elie Cartan showed, using abstract arguments, that the general solution depends on 4 functions of 2 variables. I will present families of explicit solutions depending on 9 functions of 1 variable, much beyond what was known before. Such solutions are generic in the sense that the Cotton tensor is nonzero. This is joint work with Pawel Nurowski.
On a conformal 3D manifold with electromagnetic field, Einstein-Weyl equations are the counterpart of Einstein's classical field equations. In 1943, Elie Cartan showed, using abstract arguments, that the general solution depends on 4 functions of 2 variables. I will present families of explicit solutions depending on 9 functions of 1 variable, much beyond what was known before. Such solutions are generic in the sense that the Cotton tensor is nonzero. This is joint work with Pawel Nurowski.
2019/12/09
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Akira Kitaoka (The Univ. of Tokyo)
Analytic torsions associated with the Rumin complex on contact spheres (Japanese)
Akira Kitaoka (The Univ. of Tokyo)
Analytic torsions associated with the Rumin complex on contact spheres (Japanese)
[ Abstract ]
The Rumin complex, which is defined on contact manifolds, is a resolution of the constant sheaf of $\mathbb{R}$ given by a subquotient of the de Rham complex. In this talk, we explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.
The Rumin complex, which is defined on contact manifolds, is a resolution of the constant sheaf of $\mathbb{R}$ given by a subquotient of the de Rham complex. In this talk, we explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.
2019/12/02
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Nobuhiro Honda (Tokyo Tech.)
Toward classification of Moishezon twistor spaces
Nobuhiro Honda (Tokyo Tech.)
Toward classification of Moishezon twistor spaces
[ Abstract ]
Twistor spaces are complex 3-folds which arise from 4-dimensional conformal geometry. These spaces always have negative Kodaira dimension, and most of them are known to be non-Kahler. But there are a plenty of compact twistor spaces which are Moishezon variety. The topology of such spaces is strongly constrained, and it seems not hopeless to obtain a classification and explicit description of them. I will talk about results in such a direction, which classify such spaces under a simple assumption. No example seems to be known which does not satisfy that assumption.
Twistor spaces are complex 3-folds which arise from 4-dimensional conformal geometry. These spaces always have negative Kodaira dimension, and most of them are known to be non-Kahler. But there are a plenty of compact twistor spaces which are Moishezon variety. The topology of such spaces is strongly constrained, and it seems not hopeless to obtain a classification and explicit description of them. I will talk about results in such a direction, which classify such spaces under a simple assumption. No example seems to be known which does not satisfy that assumption.
2019/11/18
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Ken-ichi Yoshikawa (Kyoto Univ.)
j-invariant and Borcherds Phi-function (Japanese)
Ken-ichi Yoshikawa (Kyoto Univ.)
j-invariant and Borcherds Phi-function (Japanese)
[ Abstract ]
The j-invariant is a modular function on the complex upper half plane inducing an isomorphism between the moduli space of elliptic curves and the complex plane. Besides the j-invariant itself, the difference of j-invariants has also attracted some mathematicians. In this talk, I will explain a factorization of the difference of j-invariants in terms of Borcherds Phi-function, the automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor. This is a joint work with Shu Kawaguchi and Shigeru Mukai.
The j-invariant is a modular function on the complex upper half plane inducing an isomorphism between the moduli space of elliptic curves and the complex plane. Besides the j-invariant itself, the difference of j-invariants has also attracted some mathematicians. In this talk, I will explain a factorization of the difference of j-invariants in terms of Borcherds Phi-function, the automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor. This is a joint work with Shu Kawaguchi and Shigeru Mukai.
2019/10/28
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Junjiro Noguchi (Univ. of Tokyo)
On Kiyoshi Oka's unpublished papers 1943 (Japanese)
Junjiro Noguchi (Univ. of Tokyo)
On Kiyoshi Oka's unpublished papers 1943 (Japanese)
2019/10/21
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Yoshihiko Matsumoto (Osaka Univ.)
Canonical almost complex structures on ACH Einstein manifolds
Yoshihiko Matsumoto (Osaka Univ.)
Canonical almost complex structures on ACH Einstein manifolds
[ Abstract ]
Einstein ACH (asymptotically complex hyperbolic) manifolds are seen as a device that establishes a correspondence between CR geometry on the boundary and Riemannian geometry in “the bulk.” This talk concerns an idea of enriching the geometric structure of the bulk by adding some almost complex structure compatible with the metric. I will introduce an energy functional of almost complex structures and discuss an existence result of critical points when the given ACH Einstein metric is a small perturbation of the Cheng-Yau complete K?hler-Einstein metric on a bounded strictly pseudoconvex domain. The renormalized Chern-Gauss-Bonnet formula is also planned to be discussed.
Einstein ACH (asymptotically complex hyperbolic) manifolds are seen as a device that establishes a correspondence between CR geometry on the boundary and Riemannian geometry in “the bulk.” This talk concerns an idea of enriching the geometric structure of the bulk by adding some almost complex structure compatible with the metric. I will introduce an energy functional of almost complex structures and discuss an existence result of critical points when the given ACH Einstein metric is a small perturbation of the Cheng-Yau complete K?hler-Einstein metric on a bounded strictly pseudoconvex domain. The renormalized Chern-Gauss-Bonnet formula is also planned to be discussed.
2019/10/07
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Yusaku Chiba (Ochanomizu Univ.)
Cohomology of vector bundles and non-pluriharmonic loci (Japanese)
Yusaku Chiba (Ochanomizu Univ.)
Cohomology of vector bundles and non-pluriharmonic loci (Japanese)
[ Abstract ]
We study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem. We especially study the examples of non-pluriharmonic loci in smooth toric varieties. I would like to explain the relation of non-pluriharmonic loci and polytopes.
We study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem. We especially study the examples of non-pluriharmonic loci in smooth toric varieties. I would like to explain the relation of non-pluriharmonic loci and polytopes.
2019/09/30
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Sachiko Hamano (Osaka City Univ.)
Rigidity of the directional moduli on pseudoconvex domains fibered by open Riemann surfaces
Sachiko Hamano (Osaka City Univ.)
Rigidity of the directional moduli on pseudoconvex domains fibered by open Riemann surfaces
[ Abstract ]
G. Schmieder-M. Shiba observed conformal embeddings of a fixed open Riemann surface of positive finite genus into closed Riemann surfaces of the same genus, and they showed the range of each diagonal element of the period matrices. Now we shall consider a smooth deformation of open Riemann surfaces with a complex parameter. In this talk, we show the rigidity of directional moduli induced by elements of the period matrices on pseudoconvex domains fibered by open Riemann surfaces of the same topological type.
G. Schmieder-M. Shiba observed conformal embeddings of a fixed open Riemann surface of positive finite genus into closed Riemann surfaces of the same genus, and they showed the range of each diagonal element of the period matrices. Now we shall consider a smooth deformation of open Riemann surfaces with a complex parameter. In this talk, we show the rigidity of directional moduli induced by elements of the period matrices on pseudoconvex domains fibered by open Riemann surfaces of the same topological type.
2019/07/08
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Hiroshi Kaneko (Tokyo University of Science)
A Riemann-Roch theorem on a weighted infinite graph (Japanese)
Hiroshi Kaneko (Tokyo University of Science)
A Riemann-Roch theorem on a weighted infinite graph (Japanese)
[ Abstract ]
A Riemann-Roch theorem on a connected finite graph was initiated by M. Baker and S. Norine, where connected graph with finite vertices was investigated and unit weight was given on each edge and vertex of the graph. Since a counterpart of the lowest exponents of the complex variable in the Laurent series was proposed as divisor for the Riemann-Roch theorem on graph, its relationships with tropical geometry were highlighted earlier than other complex analytical observations on graphs. On the other hand, M. Baker and F. Shokrieh revealed tight relationships between chip-firing games and potential theory on graphs, by characterizing reduced divisors on graphs as the solution to an energy minimization problem. The objective of this talk is to establish a Riemann-Roch theorem on an edge-weighted infinite graph. We introduce vertex weight assigned by the given weights of adjacent edges other than the units for expression of divisors and assume finiteness of total mass of graph. This is a joint work with A. Atsuji.
A Riemann-Roch theorem on a connected finite graph was initiated by M. Baker and S. Norine, where connected graph with finite vertices was investigated and unit weight was given on each edge and vertex of the graph. Since a counterpart of the lowest exponents of the complex variable in the Laurent series was proposed as divisor for the Riemann-Roch theorem on graph, its relationships with tropical geometry were highlighted earlier than other complex analytical observations on graphs. On the other hand, M. Baker and F. Shokrieh revealed tight relationships between chip-firing games and potential theory on graphs, by characterizing reduced divisors on graphs as the solution to an energy minimization problem. The objective of this talk is to establish a Riemann-Roch theorem on an edge-weighted infinite graph. We introduce vertex weight assigned by the given weights of adjacent edges other than the units for expression of divisors and assume finiteness of total mass of graph. This is a joint work with A. Atsuji.
2019/07/01
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Yeping Zhang (Kyoto Univ.)
BCOV invariant and birational equivalence (English)
Yeping Zhang (Kyoto Univ.)
BCOV invariant and birational equivalence (English)
[ Abstract ]
Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called BCOV invariant. Now we consider a pair (X,Y), where X is a Kaehler manifold and $Y ¥subseteq X$ is a canonical divisor. In this talk, we extend the BCOV invariant to such pairs. The extended BCOV invariant is well-behaved under birational equivalence. We expect that these considerations may eventually lead to a positive answer to Yoshikawa's conjecture that the BCOV invariant for Calabi-Yau threefold is a birational invariant.
Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called BCOV invariant. Now we consider a pair (X,Y), where X is a Kaehler manifold and $Y ¥subseteq X$ is a canonical divisor. In this talk, we extend the BCOV invariant to such pairs. The extended BCOV invariant is well-behaved under birational equivalence. We expect that these considerations may eventually lead to a positive answer to Yoshikawa's conjecture that the BCOV invariant for Calabi-Yau threefold is a birational invariant.
2019/06/24
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Atsushi Yamamori (Kogakuin University)
A certain holomorphic invariant and its applications (Japanese)
Atsushi Yamamori (Kogakuin University)
A certain holomorphic invariant and its applications (Japanese)
[ Abstract ]
In this talk, we first explain a Bergman geometric proof of inequivalence of the unit ball and the bidisk. In this proof, the homogeneity of the domains plays a substantial role. We next explain a recent attempt to extend our method for non-homogeneous cases.
In this talk, we first explain a Bergman geometric proof of inequivalence of the unit ball and the bidisk. In this proof, the homogeneity of the domains plays a substantial role. We next explain a recent attempt to extend our method for non-homogeneous cases.
2019/06/17
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Andrei Pajitnov (Universite de Nantes)
Inoue surfaces and their generalizations (English)
Andrei Pajitnov (Universite de Nantes)
Inoue surfaces and their generalizations (English)
[ Abstract ]
In 1972 M. Inoue constructed complex non-algebraic surfaces that proved very important for classification of surfaces via the Enriques-Kodaira scheme. Inoue surface is the quotient of H ¥times C by action of a discreet group associated to a given matrix in SL(3, Z). In 2005 K. Oeljeklaus and M. Toma generalized Inoue’s construction to higher dimensions. Oeljeklaus-Toma manifold is the quotient of H^s ¥times C^n by action of a discreet group, associated to the maximal order of a given algebraic number field.
In this talk, I will give a brief overview of these works and related results. Then I will discuss a new generalization of Inoue surfaces to higher dimensions. The manifold in question is the quotient of H ¥times C^n by action of a discreet group associated to a given matrix in SL(2n+1, Z). This is joint work with Hisaaki Endo.
In 1972 M. Inoue constructed complex non-algebraic surfaces that proved very important for classification of surfaces via the Enriques-Kodaira scheme. Inoue surface is the quotient of H ¥times C by action of a discreet group associated to a given matrix in SL(3, Z). In 2005 K. Oeljeklaus and M. Toma generalized Inoue’s construction to higher dimensions. Oeljeklaus-Toma manifold is the quotient of H^s ¥times C^n by action of a discreet group, associated to the maximal order of a given algebraic number field.
In this talk, I will give a brief overview of these works and related results. Then I will discuss a new generalization of Inoue surfaces to higher dimensions. The manifold in question is the quotient of H ¥times C^n by action of a discreet group associated to a given matrix in SL(2n+1, Z). This is joint work with Hisaaki Endo.
2019/05/27
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Takayuki Koike (Osaka City Univ.)
Gluing construction of K3 surfaces (Japanese)
Takayuki Koike (Osaka City Univ.)
Gluing construction of K3 surfaces (Japanese)
[ Abstract ]
Arnol'd showed the uniqueness of the complex analytic structure of a small neighborhood of an elliptic curve embedded in a surface whose normal bundle satisfies "Diophantine condition" in the Picard variety. By applying this theorem, we construct a K3 surface by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of anti-canonical curves of blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the resulting K3 surface is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction. (Based on joint work with Takato Uehara)
Arnol'd showed the uniqueness of the complex analytic structure of a small neighborhood of an elliptic curve embedded in a surface whose normal bundle satisfies "Diophantine condition" in the Picard variety. By applying this theorem, we construct a K3 surface by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of anti-canonical curves of blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the resulting K3 surface is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction. (Based on joint work with Takato Uehara)
2019/05/20
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Tomohiro Okuma (Yamagata Univ.)
Cohomology and normal reduction numbers of normal surface singularities (Japanese)
Tomohiro Okuma (Yamagata Univ.)
Cohomology and normal reduction numbers of normal surface singularities (Japanese)
[ Abstract ]
The normal reduction number of a normal surface singularity relates the maximal degree of the generators of associated graded algebra for certain line bundles on resolution spaces. We show fundamental properties of this invariant and formulas for some special cases. This talk is based on the joint work with Kei-ichi Watanabe and Ken-ichi Yoshida.
The normal reduction number of a normal surface singularity relates the maximal degree of the generators of associated graded algebra for certain line bundles on resolution spaces. We show fundamental properties of this invariant and formulas for some special cases. This talk is based on the joint work with Kei-ichi Watanabe and Ken-ichi Yoshida.
2019/05/13
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Homare Tadano (Tokyo Univ. of Science)
Some Bonnet--Myers Type Theorems for Transverse Ricci Solitons on Complete Sasaki Manifolds (Japanese)
Homare Tadano (Tokyo Univ. of Science)
Some Bonnet--Myers Type Theorems for Transverse Ricci Solitons on Complete Sasaki Manifolds (Japanese)
[ Abstract ]
The aim of this talk is to discuss the compactness of complete Ricci solitons and its generalizations. Ricci solitons were introduced by R. Hamilton in 1982 and are natural generalizations of Einstein manifolds. They correspond to self-similar solutions to the Ricci flow and often arise as singularity models of the flow. The importance of Ricci solitons was demonstrated by G. Perelman, where they played crucial roles in his affirmative resolution of the Poincare conjecture.
In this talk, after we review basic facts on Ricci solitons, I would like to introduce some Bonnet--Myers type theorems for complete Ricci solitons. Our results generalize the previous Bonnet--Myers type theorems due to W. Ambrose (1957), J. Cheeger, M. Gromov, and M. Taylor (1982), M. Fernandez-Lopez and E. Garcia-Rio (2008), M. Limoncu (2010, 2012), Z. Qian (1997), Y. Soylu (2017), and G. Wei and W. Wylie (2009). Moreover, I would also like to extend such Bonnet--Myers type theorems to the case of transverse Ricci solitons on complete Sasaki manifolds. Our results generalize the previous Bonnet--Myers type theorems for complete Sasaki manifolds due to I. Hasegawa and M. Seino (1981) and Y. Nitta (2009).
The aim of this talk is to discuss the compactness of complete Ricci solitons and its generalizations. Ricci solitons were introduced by R. Hamilton in 1982 and are natural generalizations of Einstein manifolds. They correspond to self-similar solutions to the Ricci flow and often arise as singularity models of the flow. The importance of Ricci solitons was demonstrated by G. Perelman, where they played crucial roles in his affirmative resolution of the Poincare conjecture.
In this talk, after we review basic facts on Ricci solitons, I would like to introduce some Bonnet--Myers type theorems for complete Ricci solitons. Our results generalize the previous Bonnet--Myers type theorems due to W. Ambrose (1957), J. Cheeger, M. Gromov, and M. Taylor (1982), M. Fernandez-Lopez and E. Garcia-Rio (2008), M. Limoncu (2010, 2012), Z. Qian (1997), Y. Soylu (2017), and G. Wei and W. Wylie (2009). Moreover, I would also like to extend such Bonnet--Myers type theorems to the case of transverse Ricci solitons on complete Sasaki manifolds. Our results generalize the previous Bonnet--Myers type theorems for complete Sasaki manifolds due to I. Hasegawa and M. Seino (1981) and Y. Nitta (2009).
2019/04/22
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Tomoyuki Hisamoto (Nayoya Univ.)
Optimal destabilizer for a Fano manifold (Japanese)
Tomoyuki Hisamoto (Nayoya Univ.)
Optimal destabilizer for a Fano manifold (Japanese)
[ Abstract ]
Around 2005, S. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence of the normalized Donaldson-Futaki invariants.
For a Fano manifold we construct a sequence of multiplier ideal sheaves from a new geometric flow and answer to Donaldson's question.
Around 2005, S. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence of the normalized Donaldson-Futaki invariants.
For a Fano manifold we construct a sequence of multiplier ideal sheaves from a new geometric flow and answer to Donaldson's question.
2019/04/15
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Takeo Ohsawa (Nagoya Univ.)
(Japanese)
Takeo Ohsawa (Nagoya Univ.)
(Japanese)
2019/01/28
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Kentaro Ohno (University of Tokyo)
Minimizing CM degree and slope stability of projective varieties (JAPANESE)
Kentaro Ohno (University of Tokyo)
Minimizing CM degree and slope stability of projective varieties (JAPANESE)
[ Abstract ]
Chow-Mumford (CM) line bundle is considered to play an important role in moduli problem for K-stable Fano varieties. In this talk, we consider a minimization problem of the degree of the CM line bundle among all possible fillings of a polarized family over a punctured curve. We show that such minimization implies the slope semistability of the fiber if the central fiber is smooth.
Chow-Mumford (CM) line bundle is considered to play an important role in moduli problem for K-stable Fano varieties. In this talk, we consider a minimization problem of the degree of the CM line bundle among all possible fillings of a polarized family over a punctured curve. We show that such minimization implies the slope semistability of the fiber if the central fiber is smooth.
2019/01/21
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Nicholas James McCleerey (Northwestern University)
POLAR TRANSFORM AND LOCAL POSITIVITY FOR CURVES
(ENGLISH)
Nicholas James McCleerey (Northwestern University)
POLAR TRANSFORM AND LOCAL POSITIVITY FOR CURVES
(ENGLISH)
[ Abstract ]
Using the duality of positive cones, we show that applying the polar transform from convexanalysis to local positivity invariants for divisors gives interesting and new local positivity invariantsfor curves. These new invariants have nice properties similar to those for divisors. In particular, thisenables us to give a characterization of the divisorial components of the non-K¨ahler locus of a big class. This is joint worth with Jian Xiao.
Using the duality of positive cones, we show that applying the polar transform from convexanalysis to local positivity invariants for divisors gives interesting and new local positivity invariantsfor curves. These new invariants have nice properties similar to those for divisors. In particular, thisenables us to give a characterization of the divisorial components of the non-K¨ahler locus of a big class. This is joint worth with Jian Xiao.
2018/12/17
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Joe Kamimoto (Kyushu University)
Newton polyhedra and order of contact on real hypersurfaces (JAPANESE)
Joe Kamimoto (Kyushu University)
Newton polyhedra and order of contact on real hypersurfaces (JAPANESE)
[ Abstract ]
This talk will concern some issues on order of contact on real hypersurfaces, which was introduced by D'Angelo. To be more precise, a sufficient condition for the equality of regular type and singular type is given. This condition is written by using the Newton polyhedron of a defining function. Our result includes earlier known results concerning convex domains, pseudoconvex Reinhardt domains and pseudoconvex domains whose regular types are 4. Furthermore, under the above condition, the values of the types can be directly seen in a simple geometrical information from the Newton polyhedron.
The technique of using Newton polyhedra has many significant applications in singularity theory. In particular, this technique has been great success in the study of the Lojasiewicz exponent. Our study about the types is analogous to some works on the Lojasiewicz exponent.
This talk will concern some issues on order of contact on real hypersurfaces, which was introduced by D'Angelo. To be more precise, a sufficient condition for the equality of regular type and singular type is given. This condition is written by using the Newton polyhedron of a defining function. Our result includes earlier known results concerning convex domains, pseudoconvex Reinhardt domains and pseudoconvex domains whose regular types are 4. Furthermore, under the above condition, the values of the types can be directly seen in a simple geometrical information from the Newton polyhedron.
The technique of using Newton polyhedra has many significant applications in singularity theory. In particular, this technique has been great success in the study of the Lojasiewicz exponent. Our study about the types is analogous to some works on the Lojasiewicz exponent.