Extension of holomorphic canonical forms on complete d-bounded Kahler manifolds

On compact holomorphically pseudosymmetric Kählerian manifolds

For compact Kählerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric Kählerian manifolds. These examples show that the compactness assumption cannot be omitted in the above stated theorem. Recently, the first examples of compact, simply connected essentially holomorphically pseudosymmet…

Closed almost-Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are Kähler

We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature $k\geq 0$ with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for $k<0$ if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern-Weil theory implies useful integral formulas, which are then combined with results from Seiberg--Witten theory.

The Kähler-Ricci flow, holomorphic vector fields and Fano bundles

We study the behavior of the Kähler-Ricci flow on compact manifolds developing finite-time singularities, in particular, when the flow contracts exceptional divisors or collapses the Fano fibers of a holomorphic fiber bundle. We present a technique using holomorphic vector fields to prove estimates related to the work of Song-Weinkove and Fu-Zhang.

The Holomorphic Sectional Curvature of General Natural KÄhler Structures on Cotangent Bundles

We study the conditions under which a Kählerian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ has constant holomorphic sectional curvature. We obtain that a certain parameter involved in the condition for $(T^*M,G,J)$ to be a Kählerian manifold, is expressed as a rational function of the other two, their derivatives, the constant sectional curvature of the base manifold $(M,g)$, and the constant holomorphic sectional curvature of the general natural Kählerian structure $(G,J)$.

Holomorphic submersions of locally conformally Kähler manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold X admits a holomorphic submersion with positive dimensional fibers at least one of which is of Kähler type, then X is globally conformally Kähler or biholomorphic, up to finite covers, to a Vaisman manifold (i.e. a mapping torus over a circle, with Sasakian fibre). As a consequence, we show that the product between a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.

具有测地全纯梯度的Kähler流形

Kähler manifolds with geodesic holomorphic gradients

本文分类了容许满足附加可积性条件的非平凡实全纯测地梯度向量场的紧致Kähler流形，证明它们均双全纯同构于复射影空间丛。

Numerically flat holomorphic bundles over non Kähler manifolds

In this paper, we study numerically flat holomorphic vector bundles over a compact non-Kähler manifold $(X, ω)$ with the Hermitian metric $ω$ satisfying the Gauduchon and Astheno-Kähler conditions. We prove that numerically flatness is equivalent to numerically effectiveness with vanishing first Chern number, semistablity with vanishing first and second Chern numbers, approximate Hermitian flatness and the existence of a filtration whose quotients are Hermitian flat. This gives an affirmative answer to the question proposed by Demailly, Peternell and Schneider.

Kähler流形上全纯函数维数估计的刚性

The rigidity of dimension estimate for holomorphic functions on Kähler manifolds

研究Kähler流形上全纯函数多项式增长的维数估计刚性，确定最大与第二大地维差距及最大体积增长维数，证明流形与复欧几里得空间双全形等价。

Locally conformally Kähler manifolds with holomorphic Lee field

We prove that a compact lcK manifold with holomorphic Lee vector field is Vaisman provided that either the Lee field has constant norm or the metric is Gauduchon (i.e., the Lee field is divergence-free). We also give examples of compact lcK manifolds with holomorphic Lee vector field which are not Vaisman.

具有常Levi-Civita或Bismut全纯截面曲率的局部共形Kähler流形

Locally conformally Kähler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature

本文针对具有常Levi-Civita或Bismut全纯截面曲率的局部共形Kähler流形，运用已有技术解答了相关类似问题，推进了非Kähler几何中相关猜想的研究。

Filtered ends, proper holomorphic mappings of Kahler manifolds to Riemann surfaces, and Kahler groups

The main goal of this paper is to prove that a connected bounded geometry complete Kahler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. This also provides a different proof of the theorem of Gromov and Schoen that, for a connected compact Kahler manifold whose fundamental group admits a proper amalgamated product decomposition, some finite unramified cover admits a surjective holomorphic mapping onto a curve of genus at least 2. The main results are also used to show that any properly ascending HNN extension with finitely generated base…

Anti-holomorphic semi-invariant submersions from Kählerian manifolds

We study anti-holomorphic semi-invariant submersions from Kählerian manifolds onto Riemannian manifolds. We prove that all distributions which are involved in the definition of the submersion are integrable. We also prove that the O'Neill's tensor $\mathcal{T}$ vanishes on the invariant vertical distribution. We give necessary and sufficient conditions for totally geodesicness and harmonicity of this type submersions. Moreover, we investigate the several curvatures of the total manifold and fibers and give a characterization theorem.

Kähler-Ricci收缩子上具有多项式增长的全纯函数环的有限生成性

Finite generation of the ring of holomorphic functions with polynomial growth on the Kähler-Ricci shrinker

该研究针对非紧梯度收缩Kähler-Ricci孤立子，在标量曲率满足温和假设的条件下，证明其上多项式增长全纯函数环OP(X)有限生成，部分证实了Munteanu和Wang的相关猜想。

Biharmonic holomorphic maps and conformally Kahler geometry

We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the Lichnerowicz theorem on harmonic maps. These third-order non-linear conditions are shown to greatly simplify on l.c.K. manifolds and construction methods and examples are given in all dimensions.

非Kaehler的Calabi-Yau流形与全纯几何结构

Non-Kähler Calabi-Yau manifolds and holomorphic geometric structures

本文研究了非Kaehler紧复流形上全纯几何结构的性质，证明了Vaisman Calabi-Yau流形上所有全纯类型全纯几何结构都是局部同构的，并且如果几何结构是刚性的，则Vaisman流形必须是一个Kodaira流形。

Liouville theorems and a Schwarz Lemma for holomorphic mappings between Kähler manifolds

We derive some consequences of the Liouville theorem for plurisubharmonic functions of L.-F. Tam and the author. The first result provides a nonlinear version of the complex splitting theorem (which splits off a factor of $\mathbb{C}$ isometrically from the simply-connected Kähler manifold with nonnegative bisectional curvature and a linear growth holomorphic function) of L.-F. Tam and the author. The second set of results concerns the so-called $k$-hyperbolicity and its connection with the negativity of the $k$-scalar curvature (when $k=1$ they are the negativity of holomorphic sectional curv…

Holomorphic sectional curvature, nefness and Miyaoka-Yau type inequality

On a compact Kähler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kähler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kähler manifold of almost nonpositive holomorphic sectional curvature has a nef canonical line bundle, contains no rational curves and satisfies some Miyaoka-Yau type inequalities. In the course of the discussions, we attach a real value to any fixed Kähler class which, up to a constant factor depending only on the dimension of manifold,…

A Liouville property of holomorphic maps

In this article, we prove a Liouville property of holomorphic maps from a complete Kahler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kahler manifold with a certain assumption on the sectional curvature.

Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach

Recently, Wu-Yau and Tosatti-Yang established the connection between the negativity of holomorphic sectional curvatures and the positivity of canonical bundles for compact Kähler manifolds. In this short note, we give anothe proof of their theorems by using the Kähler-Ricci flow.

Hirzebruch manifolds and positive holomorphic sectional curvature

This paper is the first step in a systematic project to study examples of Kähler manifolds with positive holomorphic sectional curvature ($H > 0$). Previously Hitchin proved that any compact Kähler surface with $H>0$ must be rational and he constructed such examples on Hirzebruch surfaces $M_{2, k}=\mathbb{P}(H^{k}\oplus 1_{\mathbb{CP}^1})$. We generalize Hitchin's construction and prove that any Hirzebruch manifold $M_{n, k}=\mathbb{P}(H^{k}\oplus 1_{\mathbb{CP}^{n-1}})$ admits a Kähler metric of $H>0$ in each of its Kähler classes. We demonstrate that the pinching behaviors of holomorphic se…

关于Kähler流形的Ricci对称性

On the Ricci symmetries of a Kähler manifold

本文研究Kähler流形的Ricci张量对称性，探讨Kähler-Einstein空间及其泛化，揭示holomorphic平面在Kähler几何中的作用，提出新的Einstein流形判据。

Holomorphic spinors and the Dirac equation

A closed spin Kähler manifold of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator is characterized by holomorphic spinors. It is shown that on any spin Kähler-Einstein manifold each holomorphic spinor is a finite sum of eigenspinors of the square of the Dirac operator. Vanishing theorems for holomorphic spinors are proved.

Holomorphic submersions onto Kähler or balanced manifolds

We study many properties concerning weak Kählerianity on compact complex manifolds which admits a holomorphic submersion onto a Kähler or a balanced manifold. We get generalizations of some results of Harvey and Lawson (the Kähler case), Michelson (the balanced case), Popovici (the sG case) and others.

A note on para-holomorphic Riemannian Einstein manifolds

The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. Firstly, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using an one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kähler Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits…

Equivariant strongly projectively flat maps of compact homogeneous Kähler manifolds

In this paper we define strongly projectively flatness of holomorphic maps into the complex Grassmannian manifold, which is a kind of generalization of holomorphic maps into the complex projective space and prove a rigidity of equivariant strongly projectively flat maps of compact simply connected homogeneous Kähler manifolds.

Holomorphic isometry from a Kahler manifold into a product of complex projective manifolds

We study the global property of local holomorphic isometric mappings from a class of Kahler manifolds into a product of projective algebraic manifolds with induced Fubini-Study metrics, where isometric factors are allowed to be negative.

一类向量丛全空间上的负曲率Kähler度量

Negatively curved Kähler metrics on total spaces of a class of vector bundles

本文针对一类向量丛全空间，在其配备非正曲率Kähler度量的条件下，证明其上存在丰富的负全纯双截曲率完备Kähler度量，并给出全纯函数维数估计与全纯映射Liouville定理。

Gromov-Hausdorff 极限与全纯等距

Gromov-Hausdorff limits and Holomorphic isometries

本文研究了固定 Kähler 环境空间中 Kähler 子流形序列的带点 Gromov-Hausdorff 收敛性，证明标量曲率下界保证收敛至平滑 Kähler 流形，并据此证明非紧完备 Kähler 流形不存在到复射影空间的全纯等距。

Conformally Einstein-Maxwell Kähler metrics and structure of the automorphism group

Let $(M,g)$ be a compact Kähler manifold and $f$ a positive smooth function such that its Hamiltonian vector field $K = J\mathrm{grad}_g f$ for the Kähler form $ω_g$ is a holomorphic Killing vector field. We say that the pair $(g,f)$ is conformally Einstein-Maxwell Kähler metric if the conformal metric $\tilde g = f^{-2}g$ has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell Kähler manifolds, extending the Lichnerowicz-Matsushima Theorem for constant scalar curvature Kähler manifold…

关于正全纯截面曲率的凯勒流形直径刚性

On diameter rigidity for Kähler manifolds with positive holomorphic sectional curvature

本文研究了正全纯截面曲率的凯勒流形的直径刚性性质。

On the axiom of spheres in Kahler geometry

It is known, that if a 2m-dimensional Kahler manifold satisfies the axiom of holomorphic 2n-spheres (1<n<m) or the axiom of antiholomorphic n-spheres (2<n), it is of constant holomorphic sectional curvature. In this paper the same result is obtained under weaker assumptions.

A Dajczer-Rodriguez Type Cylinder Theorem for Real Kahler Submanifolds

In 1991, Dajczer and Rodriguez proved in [10] that a complete minimal real Kahler submanifold of codimension 2, if with complex dimension > 2, would be either holomorphic, or a cylinder, or complex ruled. In this article, we generalize their result to real analytic complete real Kahler submanifolds of codimension 4. The conclusion is that such the submanifold, if with complex dimension > 4, would be either partially holomorphic, or a cylinder, or a twisted cylinder in the sense that the complex relative nullity foliation is contained in a strictly larger holomorphic foliation, whose leaves are…

Holomorphic functions on certain Kähler manifolds

We investigate Liouville theorems and dimension estimates for the space of exponentially growing holomorphic functions on complete Kähler manifolds. While our work is motivated by the study of gradient Ricci solitons in the theory of Ricci flow, the most general results we prove here do not require any knowledge of curvature.

Poisson structures and generalized Kahler structures

Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler structures and provide non-trivial examples of generalized Kahler submanifolds arising as holomorphic Poisson submanifolds. We also obtain unobstructed deformations of bi-Hermitian structures constructed from Poisson structures.

Asymptotics of the Yang-Mills Flow for Holomorphic Vector Bundles Over Kähler Manifolds: The Canonical Structure of the Limit

In the following article we study the limiting properties of the Yang-Mills flow associated to a holomorphic vector bundle E over an arbitrary compact Kähler manifold (X,ω). In particular we show that the flow is determined at infinity by the holomorphic structure of E. Namely, if we fix an integrable unitary reference connection A_0 defining the holomorphic structure, then the Yang-Mills flow with initial condition A_0, converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle E_{\infty}, which is isomorphic to…

Existence of a complete holomorphic vector field via the Kähler-Einstein metric

In this paper, we study the existence of a complete holomorphic vector fields on a strongly pseudoconvex complex manifold admitting a negatively curved complete Kähler-Einstein metric and a discrete sequence of automorphisms. Using the method of potential scaling, we will show that there is a potential function of the Kähler-Einstein metric whose differential has a constant length. Then we will construct a complete holomorphic vector field from the gradient vector field of the potential function.

Schwarz lemma from a Kähler manifold into a complex Finsler manifold

Suppose that $M$ is a Kähler manifold with a pole such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from below. Suppose that $N$ is a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant. In this paper, we establish a Schwarz lemma for holomorphic mappings $f$ form $M$ into $N$. As applications, we obtain a Liouville type rigidity result for holomorphic mappings $f$ from $M$ into $N$, as well as a rigidity theorem for bimeromorp…

Weak transcendental holomorphic Morse inequalities on compact Kähler manifolds

Transcendental holomorphic Morse inequalities aim at characterizing the positivity of transcendental cohomology classes of type $(1,1)$. In this paper, we prove a weak version of Demailly's conjecture on transcendental Morse inequalities on compact Kähler manifolds. And as a consequence, we partially improve a result of Boucksom-Demailly-Paun-Peternell.

Three Circle Theorems on Kähler manifolds and applications

The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions are derived. Among applications, we derive sharp dimension estimates (with rigidity) of holomorphic functions with polynomial growth when the holomorphic sectional curvature is nonnegative. When the bisectional curvature is nonnegative, this was due to Ni. Also we study holomorphi…

Globally conformally Kähler Einstein metrics on certain holomorphic bundles

The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally Kähler Einstein metrics on certain Hermitian holomorphic vector bundles and their subbundles over complete Kähler-Einstein manifolds. In special cases, we give the explicit expressions of of these metrics. These examples show that there is a compact Kähler manifold $M$ and its subvariety $N$ whose codimension is greater than 1 such that there is a complete conformally Kä…

Locally conformally Kahler manifolds admitting a holomorphic conformal flow

A manifold $M$ is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. Let $M$ be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on its covering. We show that $M$ admits an automorphic potential. This version was added 10 years after publication to correct some errors in the original.

A Schwarz lemma for weakly Kähler-Finsler manifolds

In this paper, we first establish several theorems about the estimation of distance function on real and strongly convex complex Finsler manifolds and then obtain a Schwarz lemma from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold. As applications, we prove that a holomorphic mapping from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold is necessary constant under an extra condition. In particular, we prove that a holomorphic mapping from a complex Minkowski space into a strongly pseud…

Hodge theory of holomorphic vector bundle on compact Kähler hyperbolic manifold

Let $E$ be a holomorphic vector bundle over a compact Kähler manifold $(X,ω)$ with negative sectional curvature $sec\leq -K<0$, $Δ_{E}$ be the Chern connection on $E$. In this article we show that if $C:=|[Λ,iΘ(E)]|\leq c_{n}K$, then $(X,E)$ satisfy a family of Chern number inequalities. The main idea in our proof is study the $L^{2}$ $\bar{\partial}_{\tilde{E}}$-harmonic forms on lifting bundle $\tilde{E}$ over the universal covering space $\tilde{X}$. We also observe that there is a closely relationship between the eigenvalue of the Laplace-Beltrami operator $Δ_{\bar{\partial}_{\tilde{E}}}$ …

Gradient Kahler Ricci Solitons

Some observations about the local and global generality of gradient Kahler Ricci solitons are made, including the existence of a canonically associated holomorphic volume form and vector field, the local generality of solutions with a prescribed holomorphic volume form and vector field, and the existence of Poincare coordinates in the case that the Ricci curvature is positive and the vector field has a fixed point.

Holomorphic affine connections on non-Kähler manifolds

Our aim here is to investigate the holomorphic geometric structures on compact complex manifolds which may not be Kähler. We prove that holomorphic geometric structures of affine type on compact Calabi-Yau manifolds with polystable tangent bundle (with respect to some Gauduchon metric on it) are locally homogeneous. In particular, if the geometric structure is rigid in Gromov's sense, then the fundamental group of the manifold must be infinite. We also prove that compact complex manifolds of algebraic dimension one bearing a holomorphic Riemannian metric must have infinite fundamental group.

典范丛的正性及具有拟负全纯截面曲率的 Hermitian 度量

Positivity of the canonical bundle and Hermitian metrics with quasi-negative holomorphic sectional curvature

本文证明紧致 Kähler 流形若存在关于 Chern 联络的拟负全纯截面曲率的 Hermitian 度量，则其典范丛丰沛，且该度量无需为 Kähler 度量，从而验证了 Yang-Zheng 拟负猜想的 Kähler 情形。

具有常Chern全纯截面曲率的紧致局部共形Kähler流形

Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature

本文证明了紧致局部共形Kähler流形上常全纯截面曲率猜想的Chern版本，通过曲率恒等式和Bochner-Kähler刚性，表明度量必为Kähler且为复空间形式。

Chern-Ricci curvatures, holomorphic sectional curvature and Hermitian metrics

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature, and complete non-Kähler metric with zero holomorphic sectional curvature and nonvanishing curvature tensor.

Kobayashi-Hitchin correspondence of generalized holomorphic vector bundles over generalized Kahler manifolds of symplectic type

In the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type was introduced from the moment map framework. In this paper we establish a Kobayashi-Hitchin correspondence, that is, the equivalence of the existence of an Einstein-Hermitian metric and $ψ$-polystability of a generalized holomorphic vector bundle over a compact generalized Kahler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles and we obtain $ψ$-stable Poisson mo…

$L^2$-extension theorems for jet sections of nef holomorphic vector bundles on compact Kähler manifolds and rational homogeneous manifolds, I

In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tangent bundles are rational homogeneous spaces.

Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains

In this Note we establish a relation between sections in globally generated holomorphic vector bundles on Kähler manifolds, isotropic with respect to a non-degenerate quadratic form, and totally geodesic foliations on Euclidean open domains. We find a geometric condition for a totally geodesic foliation to originate in a holomorphic vector bundle. For codimension-two foliations, this description recovers of P. Baird and J. C. Wood. The universal objects that play a key role are the orthogonal Grassmannians.

完备凯勒流形上的最小次数、体积增长与曲率衰减

Minimal Degrees, Volume Growth, and Curvature Decay on Complete Kähler Manifolds

本文研究了完备凯勒流形上非负双曲率的最小次数、体积增长与曲率衰减之间的关系，通过多项式增长的全纯函数和全纯体积形式建立了精确联系，并描述了凯勒-里流形的李雅普诺夫渐进行为，统一了刘和曹-李-Tam的两种证明方法，解决了杨的两个猜想。

紧致凯勒双曲流形的全纯形变

Holomorphic Deformations of Compact Kähler Hyperbolic Manifolds

本文研究紧致凯勒双曲流形的形变问题，通过提出凯勒双曲性的修正版本，为探讨 Gromov 经典凯勒双曲性的形变开性迈出第一步。

Holomorphic family of strongly pseudoconvex domains in a Kähler manifold

Let $p:X\rightarrow Y$ be a surjective holomorphic mapping between Kähler manifolds. Let $D$ be a bounded smooth domain in $X$ such that every generic fiber $D_y:=D\cap p^{-1}(y)$ for $y\in Y$ is a strongly pseudoconvex domain in $X_y:=p^{-1}(y)$, which admits the complete Kähler-Einstein metric. This family of Kähler-Einstein metrics induces a smooth $(1,1)$-form $ρ$ on $D$. In this paper, we prove that $ρ$ is positive-definite on $D$ if $D$ is strongly pseudoconvex. We also discuss the extensioin of $ρ$ as a positive current across singular fibers.