Seminar on Geometric Complex Analysis

Date, time & place Monday 10:30 - 12:00 128Room #128 (Graduate School of Math. Sci. Bldg.) Kengo Hirachi, Shigeharu Takayama, Ryosuke Nomura

Seminar information archive

2022/07/11

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Yoshihiko Matsumoto (Osaka University)
The CR Killing operator and Bernstein-Gelfand-Gelfand construction in CR geometry (Japanese)
[ Abstract ]
In this talk, I introduce the CR Killing operator associated with compatible almost CR structures on contact manifolds, which describes trivial infinitesimal deformations generated by contact Hamiltonian vector fields, and discuss how it can also be reconstructed by the Bernstein-Gelfand-Gelfand construction in the general theory of parabolic geometries. The “modified” adjoint tractor connection defined by Cap (2008) plays a crucial role. If time permits, I’d also like to discuss what this observation might mean in relation to asymptotically complex hyperbolic Einstein metrics, which are bulk geometric structures for compatible almost CR structures at infinity.
[ Reference URL ]
https://forms.gle/hYT2hVhDE3q1wDSh6

2022/07/04

10:30-12:00   Online
Katsutoshi Yamanoi (Osaka University)
Bloch's principle for holomorphic maps into subvarieties of semi-abelian varieties (Japanese)
[ Abstract ]
We discuss a generalization of the logarithmic Bloch-Ochiai theorem about entire curves in subvarieties of semi-abelian varieties, in terms of sequences of holomorphic maps from the unit disc.

This generalization implies, among other things, that subvarieties of log general type in semi-abelian varieties are pseudo-Kobayashi hyperbolic.

As another application, we discuss an improvement of a classical theorem due to Cartan in 1920's about the system of nowhere vanishing holomorphic functions on the unit disc satisfying Borel's identity.
[ Reference URL ]
https://forms.gle/hYT2hVhDE3q1wDSh6

2022/06/20

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Taiji Marugame (The University of Electro-Communications)
Constructions of CR GJMS operators in dimension three (Japanese)
[ Abstract ]
CR GJMS operators are invariant differential operators on CR manifolds whose leading parts are powers of the sublaplacian. Such operators can be constructed by Fefferman's ambient metric or the Cheng-Yau metric, but the construction is obstructed at a finite order due to the ambiguity of these metrics. Gover-Graham constructed some higher order CR GJMS operators by using tractor calculus and BGG constructions.  In particular, they showed that three dimensional CR manifolds admit CR GJMS operators of all orders. In this talk, we give proofs to this fact in two different ways. One is by the use of self-dual Einstein ACH metric and the other is by the Graham-Hirachi inhomogeneous ambient metric adapted to the Fefferman conformal structure. We also state a conjecture on the relationship between these two metrics.
[ Reference URL ]
https://forms.gle/hYT2hVhDE3q1wDSh6

2022/05/30

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Yusaku Tiba (Ochanomizu University)
Asymptotic estimates of holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds (Japanese)
[ Abstract ]
In this talk, we study an asymptotic estimate of holomorphic sections of a positive line bundle. Let $M$ be a complex manifold and $L$ be a positive line bundle over $M$ with a Hermitian metric $h$ whose Chern form is a Kähler form $\omega$. Let $X \subset M$ be a Lagrangian submanifold of $(M, \omega)$. When $X$ satisfies the Bohr-Sommerfeld condition, we prove a submean value theorem for holomorphic sections and we give an asymptotic estimate of $\inf_{x \in X}|f(x)|_{h^k}$ for $f \in H^0(M, L^k)$. This estimate provides an analog result about the leading term of the asymptotic series expansion formula of the Bergman kernel function.
[ Reference URL ]
https://forms.gle/hYT2hVhDE3q1wDSh6

2022/04/18

10:30-12:00   Online
Takeo Ohasawa (Nagoya University)
Approximation and bundle convexity on complex manifolds of pseudo convex type (Japanese)
[ Abstract ]
An approximation theorem will be proved for the space of holomorphic sections of vector bundles on certain Zariski open sets of weakly 1-complete manifolds. As an existence result on such manifolds, a solution of the bundle-valued version of the Levi problem will be given by a variant of a method of Hoermander.
[ Reference URL ]
https://forms.gle/hYT2hVhDE3q1wDSh6

2022/01/24

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Junjiro Noguchi (The University of Tokyo)
Analytic Ax-Schanuel Theorem for semi-abelian varieties and Nevanlinna theory (Japanese)
[ Abstract ]
The present study is motivated by $\textit{Schanuel Conjecture}$, which in particular implies the algebraic independence of $e$ and $\pi$. Our aim is to explore, as a transcendental functional analogue of Schanuel Conjecture, the value distribution theory (Nevanlinna theory) of the entire curve $\widehat{\mathrm{ex}}_A f:=(\exp_Af,f):\mathbf{C} \to A \times \mathrm{Lie}(A)$ associated with an entire curve $f: \mathbf{C} \to \mathrm{Lie}(A)$, where $\exp_A:\mathrm{Lie}(A)\to A$ is an exponential map of a semi-abelian variety $A$.

We firstly give a Nevanlinna theoretic proof to the $\textit{analytic Ax-Schanuel Theorem}$ for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series $\mathbf{C}[[t]]$ (Ax-Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that in the case of $A=(\mathbf{C}^*)^n$ and $\mathrm{Lie}((\mathbf{C}^*)^n)=\mathbf{C}^n$, the elements of the vector-valued function $f(z)-f(0)$ are $\mathbf{Q}$-linearly independent. Then by the method of Nevanlinna theory (the Log Bloch-Ochiai Theorem), we prove that $\mathrm{tr.deg}_\mathbf{C}\, \widehat{\mathrm{ex}}_A f \geq n+ 1.$

Secondly, we prove a $\textit{Second Main Theorem}$ for $\widehat{\mathrm{ex}}_A f$ and an algebraic divisor $D$ on $A \times \mathrm{Lie}(A)$ with compactifications $\bar D \subset \bar A \times \overline{\mathrm{Lie}(A)}$ such that
$T_{\widehat{\mathrm{ex}}_Af}(r, L({\bar D})) \leq N_1 (r, (\widehat{\mathrm{ex}}_A f)^* D)+ \varepsilon T_{\exp_Af}(r)+O(\log r) ~~ ||_\varepsilon.$
We will also deal with the intersections of $\widehat{\mathrm{ex}}_Af$ with higher codimensional algebraic cycles of $A \times \mathrm{Lie}(A)$ as well as the case of higher jets.
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/12/13

10:30-12:00   Online
Masaya Kawamura (National Institute of Technology)
A generalized Hermitian curvature flow on almost Hermitian manifolds (Japanese)
[ Abstract ]
It is well-known that the Uniformization theorem (any Riemannian metric on a closed 2-manifold is conformal to one of constant curvature) can be proven by using the Ricci flow. J. Streets and G. Tian questioned whether or not a geometric flow can be used to classify non-Kähler complex surfaces as in the case of the Ricci flow. Also they asked if it is possible to prove classification results in higher dimensions by using geometric flows in non-Kähler Hermitian geometry. Streets and Tian considered that these flows should be close to the Kähler-Ricci flow as much as possible. From this point of view, they introduced a geometric flow called the Hermitian curvature flow (HCF) which evolves an initial Hermitian metric in the direction of a Ricci-type tensor of the Chern connection modified with some lower order torsion terms. Streets and Tian also introduced another geometric flow, which is called the pluriclosed flow (PCF), by choosing torsion terms to preserve the pluriclosed condition on Hermitian metrics. Y. Ustinovskiy studied a particular version of the HCF over a compact Hermitian manifold. Ustinovskiy proved that if the initial metric has Griffiths positive (non-negative) Chern curvature, then this property is preserved along the flow.

In recent years, some results concerning geometric flows on complex manifolds have been extended to the almost complex setting. For instance, L. Vezzoni defined a new Hermitian curvature flow on almost Hermitian manifolds for generalizing some studies on the HCF and the Hermitian Hilbert functional. And J. Chu, V. Tosatti and B. Weinkove considered parabolic Monge-Ampère equation on almost Hermitian manifolds, which is equivalent to the almost complex Chern-Ricci flow. T. Zheng characterized the maximal existence time for a solution to the almost complex Chern-Ricci flow.

In this talk, we consider a generalized Hermitian curvature flow in almost Hermitian geometry and introduce that it has some properties such as the long-time existence obstruction, the uniform equivalence between its solution and an almost Hermitian metric, and the preservation result along the flow.
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/11/29

10:30-12:00   Online
Akira Kitaoka (The University of Tokyo)
レンズ空間上のRay-Singer捩率とRumin複体のラプラシアン (Japanese)
[ Abstract ]
Rumin複体は、接触多様体に関するBernstein-Gelfand-Gelfand複体(BGG複体)である。BGG複体は、放物型幾何やフィルター付き多様体に対して構成される複体であり、BGG複体のコホモロジーはde Rhamコホモロジーに一致するという事が挙げられる。また、Rumin複体はsub-Riemmann極限を考えた際に自然に現れるという性質を持つ。

De Rham複体を使って定義した概念をRumin複体に置き換えるとどうなるのか、ということを考える。本講演では、この考えを解析的捩率に適応した場合を話す。レンズ空間上のユニモジュラーなホロのミーから誘導される平坦ベクトル束に対して、Rumin複体の解析的捩率の値が、Betti数とRay-Singer捩率を用いて表されることを報告する。
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/11/15

10:30-12:00   Online
Katsusuke Nabeshima (Tokyo University of Science)
Computing logarithmic vector fields along an isolated singularity and Bruce-Roberts Milnor ideals (Japanese)
[ Abstract ]
The concept of logarithmic vector fields along a hypersurface, introduced by K. Saito (1980), is of considerable importance in singularity theory.
Logarithmic vector fields have been extensively studied and utilized by several researchers. A. G. Aleksandrov (1986) and J. Wahl (1983) considered quasihomogeneous complete intersection cases and gave independently, among other things, a closed formula of generators of logarithmic vector fields. However, there is no closed formula for generators of logarithmic vector fields, even for semi-quasihomogeneous hypersurface isolated singularity cases. Many problems related with logarithmic vector fields remain still unsolved, especially for non-quasihomogeneous cases.
Bruce-Roberts Milnor number was introduced in 1988 by J. W. Bruce and R. M. Roberts as a generalization of the Milnor number, a multiplicity of an isolated critical point of a holomorphic function germ. This number is defined for a critical point of a holomorphic function on a singular variety in terms of logarithmic vector fields. Recently, Bruce-Robert Milnor numbers are investigated by several researchers. However, many problems related with Bruce-Roberts Milnor numbers remain unsolved.
In this talk, we consider logarithmic vector fields along a hypersurface with an isolated singularity. We present methods to study complex analytic properties of logarithmic vector fields and illustrate an algorithm for computing logarithmic vector fields. As an application of logarithmic vector fields, we consider Bruce-Roberts Milnor numbers in the context of symbolic computation.
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/10/11

10:30-12:00   Online
Takahiro Aoi (Abuno High School)
cscK計量に付随する完備スカラー平坦Kähler計量について (Japanese)
[ Abstract ]

[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/07/19

10:30-12:00   Online
Makoto Abe (Hiroshima University)
$\mathbb{C}^n$上の不分岐Riemann領域に対する中間的擬凸性 (Japanese)
[ Abstract ]
The talk is based on a joint work with T. Shima and S. Sugiyama.
We characterize the intermediate pseudoconvexity for unramified Riemann domains over $\mathbb{C}^n$ by the continuity property which holds for a class of maps whose projections to $\mathbb{C}^n$ are families of unidirectionally parameterized intermediate dimensional analytic balls written by polynomials of degree $\le 2$.
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/07/12

10:30-12:00   Online
Katsuhiko Matsuzaki (Waseda University)
Parametrization of Weil-Petersson curves on the plane (Japanese)
[ Abstract ]
A Weil-Petersson curve is the image of the real line by a quasiconformal homeomorphism of the plane whose complex dilatation is square integrable with respect to the hyperbolic metrics on the upper and the lower half-planes. We consider two parameter spaces of all such curves and show that they are biholomorphically equivalent. As a consequence, we prove that the variant of the Beurling-Ahlfors quasiconformal extension defined by using the heat kernel for the convolution yields a global real-analytic section for the Teichmueller projection to the Weil-Petersson Teichmueller space. This is a joint work with Huaying Wei.
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/07/05

10:30-12:00   Online
Nitta Yasufumi (Tokyo University of Science)
Several stronger concepts of relative K-stability for polarized toric manifolds (Japanese)
[ Abstract ]
We study relations between algebro-geometric stabilities for polarized toric manifolds. In this talk, we introduce several strengthenings of relative K-stability such as uniform stability and K-stability tested by more objects than test configurations, and show that these approaches are all equivalent. As a consequence, we solve a uniform version of the Yau-Tian-Donaldson conjecture for Calabi's extremal Kähler metrics in the toric setting. This talk is based on a joint work with Shunsuke Saito.
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/06/28

10:30-12:00   Online
Yûsuke Okuyama (Kyoto Institute of Technology)
Orevkov's theorem, Bézout's theorem, and the converse of Brolin's theorem (Japanese)
[ Abstract ]
The converse of Brolin's theorem was a problem on characterizing polynomials among rational functions (on the complex projective line) in terms of the equilibrium measures canonically associated to rational functions. We would talk about a history on the studies of this problem, its optimal solution, and a proof outline. The proof is reduced to Bézout's theorem from algebraic geometry, thanks to Orevkov's irreducibility theorem on polynomial lemniscates. This talk is based on joint works with Małgorzata Stawiska (Mathematical Reviews).
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/06/14

10:30-12:00   Online
Takayuki Koike (Osaka City University)
Projective K3 surfaces containing Levi-flat hypersurfaces (Japanese)
[ Abstract ]
In May 2017, I reported on the gluing construction of a K3 surface at Seminar on Geometric Complex Analysis.
Here, by the gluing construction of a K3 surface, I mean the construction of a K3 surface by holomorphically gluing two open complex surfaces which are the complements of tubular neighborhoods of elliptic curves included in the blow-ups of the projective planes by nine points.
As of 2017, it was an open problem whether a projective K3 surface can be obtained by the gluing construction. Recently, I and Takato Uehara found a very concrete way to construct a projective K3 surface by the gluing method. As a corollary, we obtained the existence of non-Kummer projective K3 surface with compact Levi-flat hypersurfaces.
In this talk, I will explain the detail of the concrete gluing construction of such a K3 surface.
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/06/07

10:30-12:00   Online
Kurando Baba (Tokyo University of Science)
Calabi-Yau structure and Bargmann type transformation on the Cayley projective plane (Japanese)
[ Abstract ]
In this talk, I would like to discuss a problem of the geometric quantization for the Cayley projective plane. Our purposes are to show the existence of a Calabi-Yau structure on the punctured cotangent bundle of the Cayley projective plane, and to construct a Bargmann type transformation between a space of holomorphic functions on the bundle and the $L_2$-space on the Cayley projective space. The transformation gives a quantization of the geodesic flow in terms of one parameter group of elliptic Fourier integral operators. This talk is based on a joint work with Kenro Furutani (Osaka City University Advanced Mathematical Institute): arXiv:2101.07505.
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/05/31

10:30-12:00   Online
Yuya Takeuchi (Tsukuba University)
Nonnegativity of the CR Paneitz operator for embeddable CR manifolds (Japanese)
[ Abstract ]
The CR Paneitz operator, which is a fourth-order CR invariant differential operator, plays a crucial role in three-dimensional CR geometry; it is deeply connected to global embeddability and the CR positive mass theorem. In this talk, I will show that the CR Paneitz operator is nonnegative for embeddable CR manifolds. I will also apply this result to some problems in CR geometry. In particular, I will give an affirmative solution to the CR Yamabe problem for embeddable CR manifolds.
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/05/24

10:30-12:00   Online
Atsushi Hayashimoto (Nagano National College of Technology)
Cartan-Hartogs領域の固有正則写像 (Japanese)
[ Abstract ]
2つの球の間の固有正則写像は自己同型写像である。球を別の領域にしたらどうなるかを調べたい。球の一般化として複素擬楕円体や有界対称領域が考えられる。これら2つの領域を合わせた領域としてHua領域がある。これは有界対称領域の上に複素擬楕円体が乗っているような領域である。Hua領域の一番簡単な場合としてCartan-Hartogs領域があり、これらの間の固有正則写像の分類問題を考える。分類すると本質的には１種類の写像しかないことが分かる。ここでは2つの多項式写像が自己同型写像の差を省いて一致すれば、Isotoropy写像の差を省いて一致することを使う。
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/05/10

10:30-12:00   Online
Naohiko Kasuya (Hokkaido University)

[ Abstract ]

[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/04/26

10:30-12:00   Online
Jun O'Hara (Chiba University)

[ Abstract ]
$M$を多様体、$z$を複素数とし、$M$の二点間の距離の$z$乗を積空間$M\times M$上積分したものを考えると、$z$の実部が大きいところで$z$の正則関数になる。解析接続により複素平面上の有理関数で1位の極のみ持つものが得られる。この有理型関数、特にその留数の性質を紹介する。具体的には、メビウス不変性、留数と似た量（曲面のWillmoreエネルギー、4次元多様体のGraham-Wittenエネルギー、積分幾何で出てくる内在的体積、ラプラシアンのスペクトルなど）との比較、有理型関数・留数による多様体の同定問題などを扱う。

[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/04/19

10:30-12:00   Online
Shouhei Ma (Tokyo Institute of Technology)
カスプと有理同値 (Japanese)
[ Abstract ]

1970年代にマニンとドリンフェルトは合同モジュラー曲線の２つのカスプの差がピカール群において有限位数であることを発見した。

[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/01/25

10:30-12:00   Online
Young-Jun Choi (Pusan National University)
Existence of a complete holomorphic vector field via the Kähler-Einstein metric
[ Abstract ]
A fundamental problem in Several Complex Variables is to classify bounded pseudoconvex domains in the complex Euclidean space with a noncompact automorphism group, especially with a compact quotient. In the results of Wong-Rosay and Frankel, they make use of the "Scaling method'' for obtaining an 1-parameter family of automorphisms, which generates a holomorphic vector field.
In this talk, we discuss the existence of a nowhere vanishing complete holomorphic vector filed on a strongly pseudoconvex manifold admtting a negatively curved Kähler-Einstein metric and discrete sequence of automorphisms by introducing the scaling method on potentials of the Kähler-Einstein metric.
[ Reference URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2021/01/18

10:30-12:00   Online
HAMANO Sachiko (Osaka City University)
The hydrodynamic period matrices and closings of an open Riemann surface of finite genus
[ Abstract ]
A closing of an open Riemann srface $R$ of finite genus is a shorter name of a closed Riemann surface of the same genus into which $R$ can be embedded by a homology type preserving conformal mapping. We observe the Riemann period matrices of all closings of $R$ in the Siegel upper half space. It is known that every hydrodynamic differential on $R$ yields a closing of $R$ called a hydrodynamic closing. (A hydrodynamic differential is a holomorphic which describes a steady flow on $R$ of an ideal fluid.) We study the period matices induced by hydrodynamic closings of $R$. This is a joint work with Masakazu Shiba.
[ Reference URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2020/12/21

10:30-12:00   Online
Martin Sera (KUAS)
On a mixed Monge-Ampère operator for quasiplurisubharmonic functions
[ Abstract ]
This reports on a joint work with R. Lärkäng and E. Wulcan. We consider mixed Monge-Ampère products of quasiplurisubharmonic functions with analytic singularities (introduced in a previous work with H. Raufi additionally). These products have the advantage that they preserve mass (a property which is missing for non-pluripolar products).
The main result of the work presented here is that such Monge-Ampère products can be regularized as explicit one parameter limits of mixed Monge-Ampère products of smooth functions, generalizing a result of Andersson-Błocki-Wulcan. We will explain how the theory of residue currents, going back to Coleff-Herrera, Passare and others, plays an important role in the proof.
As a consequence, we get an approximation of Chern and Segre currents of certain singular hermitian metrics on vector bundles by smooth forms in the corresponding Chern and Segre classes.
[ Reference URL ]
https://zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

2020/12/14

10:30-12:00   Online