# 2020年度多変数関数論冬セミナー

2020年12月17日〜19日オンライン開催

トップページ 参加者向け情報

2020年度の多変数関数論冬セミナーを次の要領で開催いたします。新型コロナウイルス感染防止のため、オンラインでの開催となります。

2020年12月17日（木）午後午前〜19日（土）正午頃

Zoomを利用したオンラインでの開催

## プログラムProgram

### 12月17日（木）Thursday, December 17

10:00–11:00 Masataka Iwai (Osaka City University & Kyoto University)
On asymptotic base loci of relative anti-canonical divisors

この講演では，$-K_{X/Y}$ から定まる“asymptotic base loci”という集合を用いて，上記の結果が巨大や擬有効へ拡張できることを紹介する．また「$-K_{X/Y}$ から定まる“asymptotic base loci”が，空集合か水平方向に存在するかのどちらかである」ことも紹介する．この研究は大阪大学の江尻祥氏と東北大学の松村慎一氏との共同研究である．

11:20–12:20 Genki Hosono (Tohoku University)

Recent result of Deng–Wang–Zhang–Zhou shows that plurisubharamonicity can be characterized in terms of $L^2$-extensions. After that, similar results involving $L^2$-estimates for $\bar{\partial}$-equation are obtained.

This kind of results can also be applied to (possibly singular) Hermitian vector bundles. Since the curvature current may not be well-defined for singular metrics, $L^2$-theoretic characterization can be useful for analyzing such metrics.

In this talk, I will explain the background of these theories, key results, and ideas of their proofs.

13:40–14:40 Takahiro Aoi (Osaka University)

Bando–Kobayashi proved that if a Fano manifold admits a certain smooth divisor which has a Ricci-positive Kähler Einstein metric, then its complement admits a complete Ricci-flat Kähler metric. For polarized manifolds, we can consider a problem related to a scalar curvature version of this result. In this talk, I explain my result of the study of this problem. In order to prove this, we solve a degenerate complex Monge–Ampère equation and glue plurisubharmonic functions by the regularized maximum function.

15:00–16:00 Saiei-Jaeyeong Matsubara-Heo (Kobe University)

We consider a period integral of a particular rank one connection on the $n$-point configuration space on Riemann sphere $\mathcal{M}_{0,n}$. This is presumably a natural generalization of the classical beta function. Combinatorics of a compactification of the real part $\mathcal{M}_{0,n}(R)$ gives rise to an interesting formula of a homology intersection number. Details are available in arXiv:2010.14142.

### 12月18日（金）Friday, December 18

10:00–11:00 Kaoru Sano (Doshisha University)
Zariski density of points with the maximal arithmetic degree

11:20–12:20 Tomoyuki Hisamoto (Tokyo Metropolitan University)

We first explain that the H-functional is minimized by the Kähler-Einstein metric, if it exists. Using multiplier ideal sheaves we show that the Kähler-Ricci flow minimizes the H-functional in general. From the work of Dervan-Székelyhidi the lower bound is given by the supremum of certain stability threshold. Our argument gives complex analytic proof of their identity.

13:40–14:40 Toshi Sugiyama (Gifu Pharmaceutical University)

15:00–16:00 Đình Tuân Huỳnh (Chinese Academy of Sciences & Hue University)

Recently, Dinh–Sibony introduced the notion of density currents associated to a family $\{T_i\}_{i=1}^q$ of finite positive closed currents on a compact Kähler manifold. In the case where such density current is uniquely determined, it could be used to define a suitable wedge product of $T_i$, called Dinh‐Sibony's product. The notion of density currents extends the classical theory of intersection for positive closed currents, and has several deep applications on complex dynamical systems.

In this talk, we will report our recent work on studying the Monge‐Ampère operator and comparing it with Dinh–Sibony's product defined via density currents. We show that if $u$ is a plurisubharmonic (p.s.h) function belonging to the Błocki–Cegrell class, which is the largest subset of p.s.h functions on some domain in $\mathbb{C}^n$ where one can define a Monge–Ampère operator that coincides with the usual one for smooth p.s.h functions and which is continuous under decreasing sequences, then the Dinh–Sibony $n$-fold self-product of $\mathrm{dd}^c u$ exists and coincides with $(\mathrm{dd}^c u)^n$. This means that the domain of definition of the Monge–Ampère operator defined via Dinh–Sibony's product contains the Błocki–Cegrell class. We also give a brief presentation to the notion of relative non-pluripolar product introduced by Duc–Viet Vu and some of its applications.

This talk is based on the recent joint work with Lucas Kaufmann (NUS) and Duc-Viet Vu (Köln).

### 12月19日（土）Saturday, December 19

10:00–11:00 Yusaku Tiba (Ochanomizu University)

Let $X$ be a Stein manifold and let $F \to X$ be a holomorphic vector bundle. Let $\varphi$ be an exhaustive plurisubharmonic function on $X$. We show a relation between the cohomology groups of $F$ on a neighborhood of the support of $dd^c \varphi$ and those on $X$. As a consequence, we give a variant of Lefschetz hyperplane theorem on a Stein manifold, that is, the lower cohomology groups of a neighborhood of the support of $dd^c \varphi$ is determined by the cohomology groups of $X$. We also show that the similar statements hold on a projective algebraic manifold.

11:20–12:20 Joe Kamimoto (Kyushu University)

The purpose of this talk is to investigate the geometric properties of real hypersurfaces of D'Angelo infinite type in $\mathbb{C}^n$. In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra, which is an important concept in singularity theory. More precisely, equivalence conditions are given in the case of some model hypersurfaces.