2018年度年会(於:東京大学)

SUMMARY: We consider random holomorphic dynamical systems. There are many new phenomena caused by the effect of randomness in random dynamical systems which cannot hold in deterministic dynamical systems. Such phenomena are called randomness-induced phenomena (or noise-induced phenomena). In this talk, we see some randomness-induced phenomena in random holomorphic dynamical systems and some applications of such phenomena to random relaxed Newton’s methods.

SUMMARY: Let \(X\) be a complex manifold and \(Y\subset X\) be a compact complex submanifold of \(X\). Our main interest is in the complex analytical structure of a tubular neighborhood of \(Y\). A motivation comes from a study of Hermitian metrics on a holomorphic line bundle \(L\) on \(X\), especially when \(L\) has a semi-positivity property in numerical (i.e. intersection-theoretical) sense, called “numerically effective (nef)”. We will explain the relationship between the study of neighborhoods of complex submanifolds and of numerically effective line bundles. We will also talk on other applications of the study of neighborhoods of complex submanifolds.

SUMMARY: Let \(\mathcal H\) be the class of functions \(f(z)\) which are analytic in the open unit disk \(\mathbb U\). Also let \(\mathcal A\) be the subclass of functions \(f(z)\) in \(\mathcal H\) with \(f(0) = 0\) and \(f'(0) = 1\). The object of the present talk is to show some properties of functions \(f(z)\) in \(\mathcal A\) concerning with univalences for Alexander type integrals.

SUMMARY: In this talk, we consider the polyharmonic Bergman spaces on the unit ball. As previous results, we obtain the estimates for the reproducing kernel of the polyharmonic Bergman space. By using the estimates for this kernel, we give a characterization of bouneded positive Toeplitz operators on the polyharmonic Bergman spaces.

SUMMARY: Hiroki Sato defined the Jørgensen number of a two-generator Kleinian group as a generalization of Jørgensen’s inequality. Oichi–Sato asked the following natural problem: for any real number \(r \geq 1\), when is there a Kleinian group whose Jørgensen number is equal to \(r\)? In this talk, we will give a complete solution for this realization problem.

SUMMARY: This talk will present the author’s recent results on the construction of a “canonical” Laplacian on circle packing fractals invariant under the action of certain Kleinian groups and on the asymptotic behavior of its eigenvalues. In the simplest case of the Apollonian gasket, Teplyaev (2004) constructed a Laplacian with respect to which the coordinate functions on the gasket are harmonic, and the author has recently proved its uniqueness and discovered an explicit expression of it in terms of the circle packing structure of the gasket, which immediately extends to general circle packing fractals and defines (a candidate of) a “canonical” Laplacian on them. Then the author has further proved Weyl’s asymptotic formula for the eigenvalues of this Laplacian, when the circle packing fractal is the limit set of certain Kleinian groups.

SUMMARY: By glueing the ideal hyperbolic Coxeter 4-pyramids, we construct new infinite series of non-simple ideal hyperbolic Coxeter 4-polytopes. In this way, we provide a first example of such a non-compact infinite polyhedral series and prove that their growth rates are Perron numbers.

SUMMARY: If \(p \ge 2\) and a Riemann surface \(R\) satisfies Lehner’s condition, then the \(p\)-integrable Teichmüller space \(T^p(R)\) has a complex Banach manifold structure modeled on \(p\)-integrable harmonic Beltrami differentials on \(R\). When \(p = 2\), then \(T^2(R)\) has a complex Hermitian metric, which is called the Weil–Petersson metric. It was shown that this metric is Kähler and has the negative holomorphic sectional curvature and negative Ricci curvature.

In this talk, we construct the \(p\)-Weil–Petersson metric on \(T^p(R)\) similarly to the Weil–Petersson metric on \(T^2(R)\). In particular, we will say that this metric is smooth and strongly pseudoconvex when \(p\) is an even number.

SUMMARY: For a (Gromov) hyperbolic group \(G\), which is a generalization of free groups (of finite rank) and surface groups, we can define the space \(GC(G)\) of geodesic currents on \(G\) and the space \(SC(G)\) of subset currents on \(G\). The space \(GC(G)\) is proved by Bonahon to be the completion of the set of conjugacy classes of cyclic subgroups of \(G\) with positive real weight. The space \(SC(G)\) is expected to be the completion of the set of conjugacy classes of quasi-convex subgroups of \(G\) with positive real weight, which is still an open problem in general. In the case that \(G\) is a free group, Kapovich–Nagnibeda solved the problem. We solve the problem in the case of a surface group.

SUMMARY: We investigate singular Hermitian metrics on vector bundles, especially strictly Griffiths positive ones. \(L^2\) esitimates and vanishing theorems usually require an assumption that vector bundles are Nakano positive, however there is no general definition of the Nakano positivity in the singular settings. In this talk, we show some \(L^2\) estimates and vanishing theromes by assuming that the vector bundle is strictly Griffiths positive and the base manifold is projective.

SUMMARY: We give a \(L^2\)-extension theorem of jets with a sharp constant using the method of Berndtsson–Lempert. We explain the result of a jet \(L^2\)-extension theorem obtained by McNeal–Varolin. We also present a method for sharper estimates.

SUMMARY: Let \(R\) be a marked open Riemann surface of finite genus. If there exists a conformal embedding of \(R\) into a closed Riemann surface \(\widetilde R\) of the same genus with prescribed homological types of surfaces, \(\widetilde R\) is called a closing of \(R\). A closing of \(R\) induces the Riemann’s period matrix \(T\) of \(\widetilde R\). Shiba–Yamaguchi investigated the set of all closings of \(R\), and showed that each diagonal element of \(T\) is a closed disk \(\mathfrak {M}\) in the upper half plane.

We shall study variation of the period matrices \(T(t)\) of the closings of an open Riemann surface \(R(t)\) with complex parameter \(t\), and show the rigidity of hyperbolic diameter of \(\mathfrak {M}(t)\) under the pseudoconvex variation of \(R(t)\).

SUMMARY: In this talk, we announce that a localization principle for biholomorphic mappings between equidimensional Fock–Bargmann–Hartogs domains holds. As an application of this, we can show that any proper holomorphic mapping between two equidimensional Fock–Bargmann–Hartogs domains satisfying some condition is necessarily a biholomorphic mapping.

SUMMARY: Let Z be the Artin fundamental cycle of a resolution of normal complex singularity of the form \(z^2 - f(x,y) = 0\). Once studied by Laufer around 1980’s, \(z^2= y(x^4+y^6)\) is famous as the example of the case \(Z^2=-1\) where the maximal ideal cycle \(M\) does not equals \(Z\) in all the resolution of singularities. We characterize the condition \(Z^2 = -1\) completely by means of equisingular class of \(f(x,y) = 0\) the numerical characters including a Puiseux pairs and other invariants. For the equality \(Z = M\) problem, we can extend Dixon’s theorems to the complete criterion about \(Z = M\) in terms of numerical characters of \(f(x,y) =0\).

SUMMARY: The notion of integral closure of an ideal is a key concept in commutative algebra and in singularity theory. Integral numbers, the degrees of integral dependence relations, are also of considerable importance. Effective methods for computing integral numbers w.r.t. an ideal are required. In this talk, first we give an algorithm for computing integral numbers w.r.t. an ideal in a ring of convergent power series. Second, we report some integral numbers of Malgrange’s singularities and Skoda–Briançon’s singularities.