2020年度多変数関数論冬セミナー

射影代数多様体間の正則写像 $f\colon X \rightarrow Y$ について，その相対的反標準因子 $-K_{X/Y}$ は豊富などの正値性を持ちづらいことがわかっている．例えば，Kollar-宮岡-森から「$-K_{X/Y}$ が豊富ならば，$Y$ の次元が0になる」ことが知られており，Cao-Horingなどから「$-K_{X/Y}$ がネフならば，$f$ が解析的ファイバー束になる」ことが知られている．

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．

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.

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.

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.

代数体上の代数多様体とその自己射の反復合成に対し、有理点の高さの漸近的な指数増大度を測る量である算術次数が川口–Silvermanにより導入された。算術次数は力学系の幾何的な不変量である力学系次数以下であることが松澤により知られている。

今回、一般に算術次数が力学系次数に一致する有理点がZariski稠密にあることを代数閉体上で示し、また幾らかの場合には、固定した代数体上で同様のことを示すことに成功した。

本講演では、この結果について解説を行う。本研究は柴田崇広氏との共同研究である。

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.

複素1変数の $d$ 次多項式 $f(z)$ は複素平面からそれ自身への正則写像であるが，その固定点におけるmultiplier（微分係数）たち（或いは，ほぼ同じことであるが，固定点におけるholomorphic index（正則指数）たち）がどのような値であるかは，$f(z)$ の反復合成の性質を調べる複素力学系において非常に重要である。

講演者は，$d$ 個の複素数を先に与えたときに，これらを固定点の正則指数にもつような複素1変数 $d$ 次多項式 $f(z)$ が，アファイン共役のレベルで何個あるのかを，重複固定点をもたない場合において完全に調べ上げた。そのために用いた主な道具は，分岐被覆の次数と intersection multiplicity の関係，および射影空間における交点理論である。また，調べ上げた結果得られた式は，何らかの inclusion-exclusion formula であるようにも見える。その後，重複固定点をもつ場合においても，generic な結果を得ることができた。本講演では，これらの結果を紹介し，時間の許す範囲でその証明の概要を述べる。

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).

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.

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.