2018年度秋季総合分科会(於:岡山大学)

SUMMARY: On the moduli space \(\mathcal {M}_g\) of closed Riemann surfaces of genus \(g\geqq 2\) we define the function \(r_{\max }\) which maps each surface to its maximal injectivity radius. A surface attaining the maximum of \(r_{\max }\) is called an extremal surface. There are finitely many extremal surfaces in \(\mathcal {M}_g\) for every \(g\). A Riemann surface is said to be symmetric if it admits an anti-conformal involution. It is known that the set of all symmetric Riemann surfaces in \(\mathcal {M}_g\) is connected. A study of symmetric Riemann surfaces is related to that of Klein surfaces. Extremality is defined for closed Klein surfaces as well as for closed Riemann surfaces. In this talk we study some properties of extremal Riemann surfaces with respect to symmetricity and complex doubles of extremal Klein surfaces.

SUMMARY: We will report recent developments on the studies of normal two-dimensional complex singularities by means of resolution of singularities. According to special conditions, we can construct several resolution of singularites which reflects its own characteris properties.

Main Theme is the identification of the maximal ideal cycle by the Artin fundamental cycle.

This is one of famous and old theme in this field. Here, we will discuss the following special cases.

(1) Case with \(C^*\)-action, and more generally, normal two-dimensional singularities with star-shaped resolution

(2) About the condition of the normalized tangent cone; if it is reduced, this is a Kodaira singularity, we also general case in the relation with this class

(3) Case with star-shaped resolution where the central curve is a nonsingular rational curve. This is a spcical case of (1). We can talk about more precise characterization about Z=M

(4) Case of the form \(z^2 - f(x,y) = 0\). We can characterize the situation with \(Z^2 = -1\). As a ressult, we obatin the characterization of \(f(x,y)\) with M = Z in terms of Puiseux pairs.

Here many parts are based on the joint work with Tadashi Tomaru.

SUMMARY: In this talk, we will present the basic idea and results in connection with Complex Analysis from the book manuscript of division by zero calculus.

SUMMARY: From the viewpoint of the division by zero (0/0=1/0=z/0=0) and the division by zero calculus, we will show that the very beautiful theorem by descartes on three touching circles is valid for lines and points for circles except for one case. However, for the exceptional case, we can obtain interesting results from the division by zero calculus.

SUMMARY: Let \(X\) be a hyperbolic punctured sphere with \(n\) punctures. We will show that the number of possible partitions of the set of punctures with certain modulus greater than a universal constant is at most \(n-3\) and this number is the best possible. We also establish an analog of a collar lemma in hyperbolic geometry.

SUMMARY: We consider some generalizations of span and period circles related to period matrix. They are characterized by behavior spaces. By variational formulas of these quantities, as those of Hamano, but in the view of holomorphic quasiconformal deformation, sup or sub harmonicity of varius spans are given.

SUMMARY: Let \(R\) be an open Riemann surface of genus \(g\, (0 < g < \infty )\) and \(\chi = \{A_j, B_j \}_{j=1}^g\) be a fixed canonical homology basis of \(R\) modulo dividing cycles. Let \({\mathit \Sigma }^t(R)\) be the space of holomorphic hydorodynamic differentials on \(R\), \(t \in (-1, 1]\) being a parameter. The conventionally used subspace \(\{ \phi _j \mbox { whose $A_k$-period is equal to $\delta _{jk} \,(j,k=1,2,\ldots ,g)$} \}\) of \({\mathit \Sigma }^t(R)\) is obviously incomplete, for \(\dim _{\mathbb R} {\mathit \Sigma }^t(R) = 2g.\) To get rid of this insufficiency we take, in addition to \(\{\phi _j \}_{j=1}^g \) as above, \(\phi _{g+j} \in {\mathit \Sigma }^t(R), j=1,2,\ldots , g\) whose \(A_k\)-period is equal to \(i \delta _{jk}\). The matrix formed by the \(B_k\)-periods of \(\{\phi _j \}_{j=1}^{2g}\) is studied and some applications will be exhibited.

SUMMARY: The Fermat type functional equations \((*)\ f^n+g^n+h^n=1\) are considered in the complex plane. Alternative proofs for the known results for entire and meromorphic solutions to \((*)\) are given. Moreover, some conditions on degrees of polynomial solutions are given.

SUMMARY: It is proved that the Myrberg limit set of a non-elementary Kleinian group is contained in the horospheric limit set of any non-trivial normal subgroup.

SUMMARY: In this talk, we give two sided estimates for positive solutions of a semilinear elliptic equation with zero Dirichlet boundary value.

SUMMARY: It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. In this talk, the case of specific (non-real analytic) smooth functions is precisely investigated. Indeed, asymptotic limits of the respective local zeta functions at some singularities in one direction are explicitly computed. Surprisingly, it follows from these behaviors that these local zeta functions have singularities different from poles.

SUMMARY: It has been announced in the previous talk that local zeta functions associated with some specific smooth functions have singularities different from poles. In connection with the investigation of such singularities, we consider the meromorphy of local zeta functions in the case of smooth models represented as the summation of a monomial term and some flat terms. Then we obtain certain regions depending only on the above monomial terms to which local zeta functions can be meromorphically continued, and their poles are contained in arithmetic progressions constructed of negative rational numbers. From the results of the previous talk, it is expected that our result relating to the above regions is optimal for the model cases in some sense.

SUMMARY: We investigate the behavior of fiber Julia sets for polynomial skew products with super-saddle fixed points. It turns out that the Lavaurs map is identically equal to zero. We will show that the fiber Julia sets converge to the fiber Julia–Lavaurs set defined by the zero Lavaurs map.

SUMMARY: In this talk, we show the Jacobi inversion formulae of a compact Riemann surface \(X\) of genus \(g\) via the Weierstrass normal form (WNF). As a curve given by the WNF naturally appears in Weierstrass’ elliptic function theory, its generalization to a general curve also behaves naturally, which was proposed by Weierstrass. Using the WNF, we give the explicit expressions of meromorphic functions of \(S^k X\) \((k<g)\) in terms of theta function for the related Jacobi variety.

SUMMARY: In this talk, we show the behaviors of \(\sigma \) function and its related variables of an affine curve \(X_s\) given by \(y^3=x(x-s)(x-b_1)(x-b_2)\) for a limit \(s \to 0\). Since \(X_0\) is singular, its normalized curve \(\widehat X_0\) naturally appears. We have investigated the \(\sigma \) functions of both \(X_s\) (\(s\neq 0\)) and \(\widehat X_0\) for a decade. Using these results, we report the behaviors.

SUMMARY: We study the Fujita-type conjecture proposed by Popa and Schnell. We obtain an effective bound on the global generation of direct images of pluri-adjoint line bundles on the regular locus. We also obtain an effective bound on the generic global generation for a Kawamata log canonical \(\mathbb {Q}\)-pair. We use analytic methods such as \(L^2\) estimates, \(L^2\) extensions and injective theorems of cohomology groups.

SUMMARY: We study a singular Hermitian metric of a vector bundle. we prove the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analogy of a multiplier ideal sheaf, is coherent under some assumptions. Moreover, we prove a Nadel–Nakano type vanishing theorem of a vector bundle with a singular Hermitian metric.

SUMMARY: We give complex geometric descriptions of the notions of algebraic geometric positivity of torsion-free coherent sheaves, such as dd-ample at some point and weakly positive, by using singular Hermitian metrics.

SUMMARY: Let \(M\) be a three dimensional strictly pseudoconvex CR manifold. By refining Matsumoto’s construction, we construct a one parameter family of ACH metrics \(g^\lambda _{IJ}\ (\lambda \in \mathbb {R})\) on \(M\times [0, \infty )_\rho \), which solve the Einstein equation to infinite order. When \(\lambda =0\), the metric \(g^0_{IJ}\) is also self-dual to infinite order. As an application, we give another proof of the fact that a three dimensional CR manifold admits CR invariant powers of the sublaplacian of all orders, which was shown by Gover–Graham.

SUMMARY: An example is given of a hyperconvex manifold without non-constant bounded holomorphic functions, which is realized as a domain with real-analytic Levi-flat boundary in a projective surface.

SUMMARY: Let \(D\) be a pseudoconvex domain in \(\mathbb {C}^{n}\) for \(n \geq 4\). Let \(\varphi \) be an exhaustive plurisubharmonic function on \(D\). Our main theorem is the following: The direct limit of the cohomology of open sets which contain the support of \(i \partial \overline {\partial }\varphi \) is equal to the cohomology of \(D\) in low degrees. This theorem may be regarded as a pseudoconvex counterpart of the Lefschetz hyperplane theorem.

SUMMARY: Applying an \(L^2\) extension theorem, an alternate proof of the following is given. Theorem (T. Nishino, 1969) Let \(X\) be a two dimensional Stein manifold and let \(\pi \) be a holomorphic submersion from \(X\) onto the unit disc \(\mathbb {D}=\{t\in \mathbb {C};|t|<1\}\). Assume that every fiber of \(\pi \) is homomorphically equivalent to \(\mathbb {C}\). Then \(\pi \) is homomorphically equivalent to the projection \(\mathbb {C}\times \mathbb {D}\to \mathbb {D}\).

SUMMARY: We define the geometric simpleness for toroidal groups. We give an example of quasi-abelian variety which is geometrically simple, but not simple. We show that any quasi-abelian variety is isogenous to a product of geometrically simple quasi-abelian varieties. We also show that the \(\mathbb Q\)-extension of the ring of all endomorphisms of a geometrically simple quasi-abelian variety is a division algebra over \(\mathbb Q\).

SUMMARY: Let \(M\) be a connected Stein manifold of dimension \(N\) and let \(D\) be a Fock–Bargmann–Hartogs domain in \(\mathbb {C}^N\). In this talk, we announce the following result: If the identity component of \({\mathop {\rm Aut}}(M)\) is isomorphic to \({\mathop {\rm Aut}}(D)\) as topological groups, then \(M\) is biholomorphically equivalent to \(D\). As a consequence of this, we obtain a fundamental result on the topological group structure of \({\mathop {\rm Aut}}(D)\).

SUMMARY: In geometry of complex bounded domains, it is an interesting theme to characterize locally the domains with some global characteristic. In this talk, we give a semi-local characterization of homogeneous bounded domains. As applications, we obtain a characterization of bounded symmetric domains as well as a characterization of certain nonsymmetric homogeneous bounded domains.

SUMMARY: Let \(F\) be an open Riemann surface which admits Green’s function on \(F, \Delta _1\) the minimal Martin boundary of \(F, \) and \(D\) a non-compact and regular subdomain of \(F\) whose relative boundary \(\partial D\) of \(D\) is not compact. \(\Delta _1 (D) : = \{\zeta \in \Delta _1 ~|~ F\setminus D \mbox {is minimally thin at } \zeta \}. \) Suppose that every difference between two non-negative harmonic functions on \(D\) vanishing \(\partial D\) has the same minimal fine limit at every point \(\Delta _1 (D). \) Then, we prove that \(\Delta _1 (D)\) consists of only one point.

SUMMARY: In this talk, we give a simple proof for the boundary Schwarz lemma for pluriharmonic mappings between Euclidean unit balls. We also give some generalization to \(C^1\)-mappings between domains with smooth boundaries.

SUMMARY: In this talk, we prove a Schwarz lemma at the boundary for holomorphic self-mappings \(f\) of finite dimensional irreducible bounded symmetric domains without assuming the boundary regularity of \(f\). Our result generalizes the previous results obtained for holomorphic self-mappings \(f\) of the Euclidean unit ball, or of the classical Cartan domains of type I and of type II which are smooth up to the boundary.

SUMMARY: In this talk, we first generalize the boundary Schwarz lemma for holomorphic mappings \(f\) to infinite dimension by assuming the existence of the radial limit \(\lim _{r\to 1-0}Df(rz_0)z\) for each \(z\in H_1\). Next, by applying the boundary Schwarz lemma for holomorphic mappings between the Euclidean unit balls, we obtain two distortion theorems for various subclasses of the class of normalized starlike mappings.

SUMMARY: In this talk, we obtain a boundary rigidity theorem for holomorphic self-mappings of Hilbert balls in the case that there exists an interior fixed point.