Construction of constant scalar curvature K\"ahler cone metrics

TL;DR

This paper establishes existence results for constant scalar curvature K"ahler cone metrics on compact K"ahler manifolds with conic singularities, using a Fredholm alternative for the Lichnerowicz operator.

Contribution

It introduces a Fredholm alternative framework for the Lichnerowicz operator in conic settings and proves new existence results for K"ahler cone metrics under various conditions.

Findings

Existence of constant scalar curvature K"ahler cone metrics under small deformations.

Existence of such metrics on Fano manifolds.

Regularity of Hermitian-Einstein metrics on projectivised stable bundles.

Abstract

Over a compact K\"ahler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a K\"ahler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature K\"ahler metrics with conic singularities: existence result under small deformations of K\"ahler classes, existence result over a Fano manifold, existence result over certain ruled manifolds. In this last case, we consider the projectivisation of a parabolic stable holomorphic bundle. This leads us to prove that the existing Hermitian-Einstein metric on this bundle enjoys a regularity property along the divisor on the base.

Equations

$${\mathbb{L}\mathrm{ic}}_{\omega }(u)={△}_{\omega }^{2}u+u^{i\bar{j}}{R}_{i\bar{j}}(\omega )$$

$${\mathbb{L}\mathrm{ic}}_{\omega }(u)=f$$

$$0<\beta <\frac{1}{2};\alpha \beta <1-2\beta .$$

$$0<\beta <\min \left(\frac{1}{2},\left(1-\frac{1}{\lambda }\right)+\frac{n+1}{n}\min \left(\frac{1}{\lambda }\alpha (-K_X),\alpha ({L_D}_{|D}){|}_D\right)\right).$$

$$\Omega =[k{\pi }^{\ast }{\omega }_B+{\hat{\omega }}_E]$$

$$M:=X\setminus D.$$

$${\omega }_{cone}$$

$${\Gamma }_{1k}^1({\omega }_{cone})={\Gamma }_{11}^i({\omega }_{cone})={\Gamma }_{jk}^1({\omega }_{cone})={\Gamma }_{1k}^i({\omega }_{cone})={\Gamma }_{jk}^i({\omega }_{cone})=0,$$

$$W(z^1,\cdots ,z^n):=(w^1=|z^1{|}^{\beta -1}{z}^1,z^2,\cdots ,z^n).$$

$$f(0,z^2,\cdots ,z^n)=0.$$

$$\{\begin{array}{rlr}& \sigma (\frac{\partial }{\partial z^i},\frac{\partial }{\partial z^{\bar{j}}})\in C^{,\alpha ,\beta },& |z^1{|}^{2-2\beta }\sigma (\frac{\partial }{\partial z^1},\frac{\partial }{\partial z^{\bar{1}}})\in C^{,\alpha ,\beta },\\ & |z^1{|}^{1-\beta }\sigma (\frac{\partial }{\partial z^1},\frac{\partial }{\partial z^{\bar{j}}})\in C^{,\alpha ,\beta },& |z^1{|}^{1-\beta }\sigma (\frac{\partial }{\partial z^i},\frac{\partial }{\partial z^{\bar{1}}})\in C^{,\alpha ,\beta }.\end{array}$$

$$C^{2,\alpha ,\beta }=\{u|u,\partial u,\sqrt{-1}\partial \bar{\partial }u\in C^{,\alpha ,\beta }\}.$$

$$\{\begin{array}{rl}& {\nabla }_i^{cone}{\nabla }_{\bar{l}}^{cone}{\nabla }_k^{cone}u,|z^1{|}^{3-3\beta }{\nabla }_1^{cone}{\nabla }_{\bar{1}}^{cone}{\nabla }_1^{cone}u\in C^{,\alpha ,\beta }(\tilde{U}),\\ & |z^1{|}^{1-\beta }{\nabla }_i^{cone}{\nabla }_{\bar{1}}^{cone}{\nabla }_k^{cone}u,|z^1{|}^{1-\beta }{\nabla }_1^{cone}{\nabla }_{\bar{l}}^{cone}{\nabla }_k^{cone}u\in C^{,\alpha ,\beta }(\tilde{U}),\\ & |z^1{|}^{2-2\beta }{\nabla }_i^{cone}{\nabla }_{\bar{1}}^{cone}{\nabla }_1^{cone}u,|z^1{|}^{2-2\beta }{\nabla }_1^{cone}{\nabla }_{\bar{l}}^{cone}{\nabla }_1^{cone}u\in C^{,\alpha ,\beta }(\tilde{U}).\end{array}$$

$$\displaystyle\left\{\begin{aligned} &|z^{1}|^{2-2\beta}\nabla^{cone}_{\bar{l}}\nabla^{cone}_{k}\nabla^{cone}_{\bar{j}}\nabla^{cone}_{i}u=\frac{\partial^{4}u}{\partial z^{\bar{j}}\partial z^{i}\partial z^{\bar{l}}\partial z^{k}}\in C^{,\alpha,\beta}(\tilde{U}),\\ &|z^{1}|^{2-2\beta}\nabla^{cone}_{\bar{l}}\nabla^{cone}_{1}\nabla^{cone}_{\bar{j}}\nabla^{cone}_{i}u=|z^{1}|^{1-\beta}\frac{\partial^{4}u}{\partial z^{\bar{j}}\partial z^{i}\partial z^{\bar{l}}\partial z^{1}}\in C^{,\alpha,\beta}(\tilde{U}),\\ &|z^{1}|^{2-2\beta}\nabla^{cone}_{\bar{1}}\nabla^{cone}_{1}\nabla^{cone}_{\bar{j}}\nabla^{cone}_{i}u=|z^{1}|^{2-2\beta}\frac{\partial^{4}u}{\partial z^{\bar{j}}\partial z^{i}\partial z^{\bar{1}}\partial z^{1}}\in C^{,\alpha,\beta}(\tilde{U}),\\ &|z^{1}|^{2-2\beta}\nabla^{cone}_{\bar{l}}\nabla^{cone}_{1}\nabla^{cone}_{\bar{j}}\nabla^{cone}_{1}u\\ &\hfill$\displaystyle=|z^{1}|^{2-2\beta}[\frac{\partial^{4} u}{\partial z^{\bar{l}}\partial z^{1}\partial z^{\bar{j}}\partial z^{1}}+\frac{1-\beta}{z^{1}}\frac{\partial^{3} u}{\partial z^{\bar{l}}\partial z^{1}\partial z^{\bar{j}}}]\in C^{,\alpha,\beta}(\tilde{U}),$\ignorespaces\\ &|z^{1}|^{3-3\beta}\nabla^{cone}_{\bar{1}}\nabla^{cone}_{1}\nabla^{cone}_{\bar{1}}\nabla^{cone}_{i}u\\ &\hfill$\displaystyle=|z^{1}|^{3-3\beta}[\frac{\partial^{4} u}{\partial z^{\bar{1}}\partial z^{1}\partial z^{\bar{1}}\partial z^{i}}+\frac{1-\beta}{z^{\bar{1}}}\frac{\partial^{3} u}{ \partial z^{1} \partial z^{\bar{1}}\partial z^{ i}}]\in C^{,\alpha,\beta}(\tilde{U}),$\ignorespaces\\ &|z^{1}|^{4-4\beta}\nabla^{cone}_{\bar{1}}\nabla^{cone}_{1}\nabla^{cone}_{\bar{1}}\nabla^{cone}_{1}u\\ &\hfill$\displaystyle=|z^{1}|^{4-4\beta}[\frac{\partial^{4} u}{\partial z^{\bar{1}}\partial z^{1}\partial z^{\bar{1}}\partial z^{1}} +\frac{1-\beta}{z^{\bar{1}}}\frac{\partial^{3} u}{\partial z^{1}\partial z^{\bar{1}} \partial z^{1}} +\frac{1-\beta}{z^{1}}\frac{\partial^{3} u}{\partial z^{1}\partial z^{\bar{1}} \partial z^{\bar{1}}}$\ignorespaces\\ &\hfill$\displaystyle+\frac{(1-\beta)^{2}}{|z^{1}|^{2}}\frac{\partial^{2} u}{\partial z^{1}\partial z^{\bar{1}}}]\in C^{,\alpha,\beta}(\tilde{U}).$\ignorespaces\end{aligned}\right.$$

$${\omega }_D={\omega }_0+\delta \frac{\sqrt{-1}}{2}\partial \bar{\partial }|s{|}_{h_{\Lambda }}^{2\beta }$$

$$\{\begin{array}{rl}& \frac{\partial g_{k\bar{l}}}{\partial z^i}\in C^{,\alpha ,\beta },|z^1{|}^{1-\beta }\frac{\partial g_{1\bar{l}}}{\partial z^i}\in C_{\{0\}}^{,\alpha ,\beta },|z^1{|}^{2-2\beta }\frac{\partial g_{1\bar{1}}}{\partial z^i}\in C^{,\alpha ,\beta },\\ & |z^1{|}^{2-2\beta }{\nabla }_1^{cone}{g}_{1\bar{l}}=|z^1{|}^{2-2\beta }[\frac{\partial g_{1\bar{l}}}{\partial z^1}+\frac{1-\beta }{z^1}{g}_{1\bar{l}}]\in C_{\{0\}}^{,\alpha ,\beta },\\ & |z^1{|}^{3-3\beta }{\nabla }_1^{cone}{g}_{1\bar{1}}=|z^1{|}^{3-3\beta }[\frac{\partial g_{1\bar{1}}}{\partial z^1}+\frac{1-\beta }{z^1}{g}_{1\bar{1}}]\in C_{\{0\}}^{,\alpha ,\beta }.\end{array}$$

$$\{\begin{array}{rl}& {\Gamma }_{jk}^i\in C^{,\alpha ,\beta },|z^1{|}^{1-\beta }{\Gamma }_{j1}^i\in C_{\{0\}}^{,\alpha ,\beta },|z^1{|}^{\beta -1}{\Gamma }_{jk}^1\in C_{\{0\}}^{,\alpha ,\beta },\\ & {\Gamma }_{j1}^1\in C^{,\alpha ,\beta },|z^1{|}^{2-2\beta }{\Gamma }_{11}^i\in C_{\{0\}}^{,\alpha ,\beta },\\ & |z^1{|}^{1-\beta }\left({\Gamma }_{11}^1+\frac{1-\beta }{z^1}\right)\in C_{\{0\}}^{,\alpha ,\beta }.\end{array}$$

$$\{\begin{array}{rl}& \frac{{\partial }^2{g}_{k\bar{l}}}{\partial z^i\partial z^{\bar{j}}}\in C^{,\alpha ,\beta },|z^1{|}^{1-\beta }\frac{{\partial }^2{g}_{1\bar{l}}}{\partial z^i\partial z^{\bar{j}}}\in C_{\{0\}}^{,\alpha ,\beta },\\ & |z^1{|}^{2-2\beta }\frac{{\partial }^2{g}_{1\bar{1}}}{\partial z^i\partial z^{\bar{j}}}\in C^{,\alpha ,\beta },\\ & |z^1{|}^{2-2\beta }{\nabla }_{\bar{j}}^{cone}{\nabla }_1^{cone}{g}_{1\bar{l}}\in C_{\{0\}}^{,\alpha ,\beta },\\ & |z^1{|}^{3-3\beta }{\nabla }_{\bar{1}}^{cone}{\nabla }_i^{cone}{g}_{1\bar{1}}\in C_{\{0\}}^{,\alpha ,\beta },\\ & |z^1{|}^{4-4\beta }{\nabla }_{\bar{1}}^{cone}{\nabla }_1^{cone}{g}_{1\bar{1}}\in C_{\{0\}}^{,\alpha ,\beta }.\end{array}$$

$$\{\begin{array}{rlr}& {\nabla }_k{\nabla }_{\bar{j}}{\nabla }_{i}u,& |z^1{|}^{3-3\beta }{\nabla }_1{\nabla }_{\bar{1}}{\nabla }_{1}u,\\ & |z^1{|}^{1-\beta }{\nabla }_k{\nabla }_{\bar{1}}{\nabla }_{i}u,& |z^1{|}^{1-\beta }{\nabla }_k{\nabla }_{\bar{j}}{\nabla }_{1}u,\\ & |z^1{|}^{2-2\beta }{\nabla }_k{\nabla }_{\bar{1}}{\nabla }_{1}u,& |z^1{|}^{2-2\beta }{\nabla }_1{\nabla }_{\bar{j}}{\nabla }_{1}u\end{array}$$

$$|u{|}_{C^{3,\alpha ,\beta }(X;\omega )}=$$

$$+\sum _{2\le i,j,l\le n}[$$

$$+|{\nabla }_k{\nabla }_{\bar{j}}{\nabla }_{1}u{|}_{C^{,\alpha ,\beta }(X)}+|{\nabla }_k{\nabla }_{\bar{1}}{\nabla }_{1}u{|}_{C^{,\alpha ,\beta }(X)}$$

$$+|{\nabla }_1{\nabla }_{\bar{j}}{\nabla }_{1}u{|}_{C^{,\alpha ,\beta }(X)}+|{\nabla }_1{\nabla }_{\bar{1}}{\nabla }_{1}u{|}_{C^{,\alpha ,\beta }(X)}].$$

$$\{\begin{array}{rl}& {\nabla }_{\bar{l}}{\nabla }_k{\nabla }_{\bar{j}}{\nabla }_{i}u,|z^1{|}^{1-\beta }{\nabla }_{\bar{l}}{\nabla }_1{\nabla }_{\bar{j}}{\nabla }_{i}u,\\ & |z^1{|}^{2-2\beta }{\nabla }_{\bar{1}}{\nabla }_1{\nabla }_{\bar{j}}{\nabla }_{i}u,|z^1{|}^{2-2\beta }{\nabla }_{\bar{l}}{\nabla }_1{\nabla }_{\bar{j}}{\nabla }_{1}u,\\ & |z^1{|}^{3-3\beta }{\nabla }_{\bar{1}}{\nabla }_1{\nabla }_{\bar{1}}{\nabla }_{i}u,|z^1{|}^{4-4\beta }{\nabla }_{\bar{1}}{\nabla }_1{\nabla }_{\bar{1}}{\nabla }_{1}u\end{array}$$

$$|u{|}_{C^{4,\alpha ,\beta }(X;\omega )}=$$

$$+\sum _{2\le i,j,k,l\le n}[$$

$$+|{\nabla }_{\bar{1}}{\nabla }_1{\nabla }_{\bar{j}}{\nabla }_{i}u{|}_{C^{,\alpha ,\beta }(X)}+|{\nabla }_{\bar{l}}{\nabla }_1{\nabla }_{\bar{j}}{\nabla }_{1}u{|}_{C^{,\alpha ,\beta }(X)}$$

$$+|{\nabla }_{\bar{1}}{\nabla }_1{\nabla }_{\bar{1}}{\nabla }_{i}u{|}_{C^{,\alpha ,\beta }(X)}+|{\nabla }_{\bar{1}}{\nabla }_1{\nabla }_{\bar{1}}{\nabla }_1{|}_{C^{,\alpha ,\beta }(X)}].$$

$${\omega }_{\phi }={\omega }_D+\sqrt{-1}\partial \bar{\partial }\phi .$$

$$\{\begin{array}{rl}& \frac{\partial g_{k\bar{l}}}{\partial z^i},|z^1{|}^{1-\beta }\frac{\partial g_{1\bar{l}}}{\partial z^i},|z^1{|}^{2-2\beta }\frac{\partial g_{1\bar{1}}}{\partial z^i},\\ & |z^1{|}^{2-2\beta }{\nabla }_1^{cone}{g}_{1\bar{l}},|z^1{|}^{3-3\beta }{\nabla }_1^{cone}{g}_{1\bar{1}};\end{array}$$

$$\{\begin{array}{rl}& {\Gamma }_{jk}^i,|z^1{|}^{1-\beta }{\Gamma }_{j1}^i,|z^1{|}^{\beta -1}{\Gamma }_{jk}^1,\\ & {\Gamma }_{j1}^1,|z^1{|}^{2-2\beta }{\Gamma }_{11}^i,|z^1{|}^{1-\beta }({\Gamma }_{11}^1+\frac{1-\beta }{z^1});\end{array}$$

$$\{\begin{array}{rl}& \frac{{\partial }^2{g}_{k\bar{l}}}{\partial z^i\partial z^{\bar{j}}},|z^1{|}^{1-\beta }\frac{{\partial }^2{g}_{1\bar{l}}}{\partial z^i\partial z^{\bar{j}}},|z^1{|}^{2-2\beta }\frac{{\partial }^2{g}_{1\bar{1}}}{\partial z^i\partial z^{\bar{j}}},\\ & |z^1{|}^{2-2\beta }{\nabla }_{\bar{j}}^{cone}{\nabla }_1^{cone}{g}_{1\bar{l}},|z^1{|}^{3-3\beta }{\nabla }_{\bar{1}}^{cone}{\nabla }_i^{cone}{g}_{1\bar{1}},\\ & |z^1{|}^{4-4\beta }{\nabla }_{\bar{1}}^{cone}{\nabla }_1^{cone}{g}_{1\bar{1}}.\end{array}$$

$$\{\begin{array}{rl}& \frac{\partial g_{k\bar{l}}}{\partial z^i},|z^1{|}^{1-\beta }\frac{\partial g_{1\bar{l}}}{\partial z^i},|z^1{|}^{2-2\beta }\frac{\partial g_{1\bar{1}}}{\partial z^i},\\ & |z^1{|}^{2-2\beta }{\nabla }_1^{cone}{g}_{1\bar{l}},|z^1{|}^{3-3\beta }{\nabla }_1^{cone}{g}_{1\bar{1}};\end{array}$$

Full text

1. Introduction

In this paper we investigate the construction of constant scalar curvature Kähler metrics (cscK in short) with conical singularities over a smooth compact Kähler manifold and provide several existence results. Starting with a model metric $\omega_{D}$ with conical singularity along a smooth divisor $D$ of the compact Kähler manifold $X$ (see definition in Section 2.2), we are interested in cscK cone Kähler metrics $\omega$, i.e metrics of the form $\omega=\omega_{D}+i\partial\bar{\partial}\varphi$ such that

• •

$\omega$ is a Kähler cone metric,

• •

$\omega$ has constant scalar curvature over the regular part $M:=X\setminus D$.

In our study, we will consider the linearization of the constant scalar curvature equation. This leads to consider the Lichnerowicz operator over functions $u$ defined by

[TABLE]

and the associated Lichnerowicz equation

[TABLE]

for $f\in C^{,\alpha,\beta}$ with $\int_{M}f\omega^{n}=0$, $n$ being the complex dimension of the manifold and $\omega$ defined as above. Note that our study will require to work with certain Hölder spaces adapted to the singularities, the spaces $C^{.,\alpha,\beta}$, that are described in details in Section 2.2. In particular, we say a Kähler potential $\varphi$ is $C^{2,\alpha,\beta}$ cscK cone potential (resp. $C^{4,\alpha,\beta}$ cscK cone potential) if $\omega=\omega_{D}+i\partial\bar{\partial}\varphi$ is a cscK cone metric and additionally $\varphi\in C^{2,\alpha,\beta}$ (resp. $C^{4,\alpha,\beta}$). In the sequel when we speak of cscK cone metric, its potential is at least $C^{2,\alpha,\beta}$ as in [LiL, Definition 2.9].

We will need a certain restriction on the cone angle $2\pi\beta$ and the Hölder exponent $\alpha$, namely that

[TABLE]

This restriction appeared in previous works, e.g. in [MR3144178, LZ2] and is required to have regularity results.

Our first theorem is an analytic result Fredholm alternative type. It provides a solution to the Lichnerowicz equation (1.1) over the Hölder spaces $C^{4,\alpha,\beta}$ for $C^{,\alpha,\beta}$ data.

Theorem 1** (Linear theory).**

Let $X$ be a compact Kähler manifold, $D\subset X$ a smooth divisor, $\omega$ a cscK cone metric with $C^{2,\alpha,\beta}$ potential such that the cone angle $2\pi\beta$ and the Hölder exponent $\alpha$ satisfy Condition (C). Assume that $f\in C^{,\alpha,\beta}$ with normalisation condition $\int_{M}f\omega^{n}=0$. Then one of the following holds:

• •

Either the Lichnerowicz equation ${{\mathbb{L}\mathrm{ic}}}_{\omega}(u)=f$ has a unique $C^{4,\alpha,\beta}$ solution.

• •

Or the kernel of ${{\mathbb{L}\mathrm{ic}}}_{\omega}(u)$ generates a holomorphic vector field tangent to $D$.

Note that the solution furnished by the theorem can be extended continuously to the whole manifold $X$.

As in the smooth situation (see for instance Lebrun-Simanca’s results in [MR1274118, Corollary 2]), this result provides an existence theorem by small deformations. To derive it, we just use the implicit function theorem together with the one-one correspondence between the kernel of the Lichnerowicz operator and the holomorphic vector fields on the manifold tangential to the divisor, see [LZ2, Section 4]. We introduce some notations. Set $Aut(X)$ the group of holomorphic transformations of $X$ given by diffeomorphisms of $X$ that preserve the complex structure. We consider the subgroup $Aut_{D}(X,[\omega])\subset Aut(X)$ as the identity component of the automorphisms group that preserve the Kähler class $[\omega]$ and fix the divisor $D$. Then, $Lie(Aut_{D}(X,[\omega]))$ consists in the Lie algebra of holomorphic vector fields tangential to $D$ with holomorphy potential. Recall that a holomorphy potential is a function whose complex gradient, with respect to the metric $\omega$ is a holomorphic vector field.

Corollary 1.1** (CscK cone metric by deformation).**

Consider $(X,\omega)$ compact Kähler manifold endowed with $\omega$ a cscK cone metric along $D\subset X$, smooth divisor, with angle $\beta$ satisfying Condition (C). Assume that the Lie algebra $Lie(Aut_{D}(X,[\omega_{B}]))$ is trivial. Then the set of all Kähler classes around $[\omega]$ containing a cscK metric with cone singularities is non-empty and open.

A direct application of this last result is the existence of cscK cone metrics close to Kähler-Einstein cone metrics on Fano manifolds. Before stating the result, we refer to [MR2484031, MR3107540] for a definition the $\alpha$-invariant for general polarization and its relation with log-canonical thresholds. The next corollary is obtained from the results of Berman [MR3107540] and Li-Sun [MR3248054, Corollaries 2.19 and 2.21] on existence of a Kähler-Einstein cone metric over a Fano manifold and the non existence of holomorphic vector field tangent to $D$ (when the parameter $\lambda$ below is greater or equal to $1$). The regularity of the Kähler-Einstein cone metric is also sufficient to apply Theorem 1 (the regularity issue is discussed in [LZ], see also references therein).

Corollary 1.2** (CscK cone metrics for Fano manifolds).**

Assume that $\Omega_{0}=c_{1}(-K_{X})$ and $D$ is a smooth divisor which is $\mathbb{Q}$-linearly equivalent to $-\lambda K_{X}$, where $\lambda\in\mathbb{Q}_{+}^{*}$. Denote $L_{D}$ the line bundle associated to $D$.

1. (i)

If $\lambda\geq 1$, then there is a constant $\delta$ such that in the nearby class $\Omega$ satisfying $|\Omega-\Omega_{0}|<\delta$ there exists a constant scalar curvature Kähler cone metric in $\Omega$ with cone angle $2\pi\beta$ satisfying

[TABLE] 2. (ii)

If $\frac{2n}{2n+1}<\lambda<1$, $Lie(Aut_{D}(X,\Omega_{0}))$ is trivial, then the same conclusion as in (i) holds for angle $2\pi\beta$ satisfying (1.2) and the extra condition $\beta>n\left(\frac{1}{\lambda}-1\right).$

Note that the upped bound in (1.2) may not be optimal but has the advantage of being effective and calculable, we also refer to [MR3470713] on that point.

Our next main result is a construction theorem of cscK metrics with cone singularities in Kähler classes (that may not be integral) over projective bundles, which generalizes the main result of [Hong2]. It is also an application of Theorem 1 but requires much more work. The notion of parabolic stability is explained in Sections 4.1 and 4.3.

Theorem 2** (CscK cone metric for projective bundles).**

*Let $B$ be a base compact Kähler manifold endowed with a cscK metric $\omega_{B}$ with cone singularities along $D\subset B$, smooth divisor with trivial Lie algebra $Lie(Aut_{D}(B,[\omega_{B}]))$. Assume the Hölder exponent $\alpha$ and the angle $2\pi\beta$ of $\omega_{B}$ satisfy Condition (C). Let $E$ be a parabolic stable vector bundle over $B$ with respect to $\omega_{B}$.

Then, for $k\in\mathbb{N}^{*}$ large enough, there exists a cscK metric with cone singularities on $X:=\mathbb{P}E^{*}$ in the class*

[TABLE]

where $\pi:X\rightarrow B$ and $\hat{\omega}_{E}$ represents the first Chern class of $\mathcal{O}_{\mathbb{P}E^{*}}(1)$. This cscK metric has its cone singularities along $\mathcal{D}:=\pi^{-1}(D)$ with $C^{4,\alpha,\beta}$ potential.

Remark 1.1*.*

In Theorem 2 the assumption on $E$ could be replaced by saying that it is an indecomposable holomorphic vector bundle equipped with a parabolic structure and a Hermitian-Einstein cone metric compatible with this structure, providing a purely differential geometric statement.

Let’s do now some brief comments. CscK cone metrics constitute a natural generalization of Kähler-Einstein metrics with conical singularities (see Section 2.4) . The importance of the notion of Kähler-Einstein cone metric is now well established from Chen-Donaldson-Sun’s breakthrough for the celebrated Yau-Tian-Donaldson conjecture, when one restricts attention to Fano manifolds and the anticanonical class, see for instance the pioneering paper [MR2975584] or the survey [MR3522175] and references therein. One may expect that the study of cscK cone metrics may lead to new progress on Yau-Tian-Donaldson conjecture for general polarizations or may have applications for construction of moduli spaces or Chern number inequalities. Furthermore, a logarithmic version of Yau-Tian-Donaldson conjecture is expected to be also true in the context of cscK cone metrics. Nevertheless, as far as we know, only very few examples of cscK cone metrics that are not Kähler-Einstein appeared in the literature. CscK cone metrics are far from being well understood and for instance uniqueness results have only appeared very recently, cf. [LZ2, LZ3].

From the point of view of existence, the case of curves has been studied by R.C. McOwen, M. Troyanov and F. Luo–G.Tian in the late eighties. In higher dimension, Y. Hashimoto [Hasfutaki] has recently obtained momentum-constructed cscK cone metrics on the projective completion of a pluricanonical line bundle over a product of Kähler-Einstein Fano manifolds. This enabled him to give first evidence of the log Yau-Tian-Donaldson conjecture. Note that his definition of Kähler cone metrics is more restricted than the general usual definition that we consider here. A more general setup has been studied in [Ke1] where it is shown morally that the notion of cscK cone metric is the most natural notion of Kähler metrics with special curvature properties for projective bundles over a curve, when the holomorphic bundle is irreducible and not Mumford stable (otherwise, the “right” notion would be the classical notions of smooth extremal/cscK metric). A related work for extremal Kähler cone metric, still on the projective completion of a line bundle over admissible manifolds, can also be found in [LiH].

Our results provide an effective method to construct plenty of cscK cone metrics on various manifolds and partially generalize previous results op. cit. We also expect that Theorem 1 will have many applications in a long range, including for studying log-K-stability (see Section 7).

We shall now explain the structure of the paper. In Section 2, we introduce the notion of metrics with singularities, together with the adapted Hölder spaces and recall some results about regularity of cscK cone metrics. Among other things we prove the vanishing of the log-Futaki invariant of a Kähler class endowed with a cscK cone metric. In Section 3, we introduce weighted Sobolev spaces and obtain Schauder type estimates for Laplacian equation associated to a Kähler cone metric under half angle condition (Proposition 3.5) or without half angle condition but with weaker regularity (Proposition 3.6). This allows us to see that a weak solution $u$ to the bi-Laplacian equation $\Delta^{2}u-K\Delta u=f$ is actually $C^{4,\alpha,\beta}$. Using this result and a continuity method, we are able to prove Theorem 1 by proving the key estimate (Theorem 3) showing closeness. In Section 4, we introduce the notion of Hermitian-Einstein cone metrics, that are hermitian metrics over a parabolic vector bundle that satisfy the Einstein equation (with respect to a Kähler cone metric) together with a certain regularity property. Theorem 4 shows that a parabolic stable vector bundle can be equipped with a Hermitian-Einstein cone metric, refining results of Simpson [Sim, MR1040197] and Li [LiJ]. Using this result, we adapt the work of Hong [Hong2, Hong3] for smooth cscK metrics to the conical setting and construct inductively almost cscK cone metrics (Proposition 5.2). Using now Theorem 1 and taking the adiabatic limit, we can deduce Theorem 2 in Section 5.5. In Section 6, we explain that the existence of a Kähler-Einstein cone metric on a manifold provides a Hermitian-Einstein cone metric on its tangent bundle, generalizing a well-known result in the smooth case. This could be used to provide extra concrete examples of applications of our Theorem 2. Eventually, in Section 7, we discuss natural generalizations of our work and some possible applications to other geometric questions. In the particular case of the projectivisation of a parabolic vector bundle over a curve, we formulate a conjecture between existence of cscK cone metric, log K-stability and parabolic stability.

Contents

1. 1 Introduction

2. 2 CscK metrics with cone singularities

3. 2.1 Hölder spaces in cone charts

4. 2.2 Model cone metric

5. 2.3 Second order elliptic equations with conical singularities

6. 2.4 Some properties of cscK cone metrics

7. 3 Lichnerowicz equations with conical singularities

8. 3.1 Sobolev spaces for cone metrics

9. 3.2 A partial $L^{p}$ estimate

10. 3.3 Weak solutions to bi-Laplacian equations: existence

11. 3.4 Weak solutions to bi-Laplacian equations: regularity

12. 3.5 Fredholm alternative for Lichnerowicz operator

13. 4 Hermitian-Einstein metrics with conical singularities

14. 4.1 Stable parabolic structures

15. 4.2 Hölder spaces for bundle endomorphisms

16. 4.3 Existence of Hermitian-Einstein cone metrics

17. 4.4 Parabolic stability and holomorphic vector fields

18. 5 Construction of cscK cone metrics over projective bundles

19. 5.1 Construction of background metrics

20. 5.2 Expansion of scalar curvature

21. 5.3 Approximate cscK cone metrics

22. 5.4 Decomposition of the holomorphic tangent bundle

23. 5.5 Proof of Theorem 2

24. 6 Kähler-Einstein cone metrics and Tangent bundle

25. 7 Further Applications and Remarks

26. 7.1 Simple normal crossings divisors

27. 7.2 Twisted conical path for cscK metric

28. 7.3 Other applications of Theorem 1

29. 7.4 Generalizations of Theorem 2

30. 7.5 Log K-stability

2. CscK metrics with cone singularities

Let $(X,\omega_{0})$ be a Kähler manifold. We denote $[\omega_{0}]$ the Kähler class containing the smooth Kähler metric $\omega_{0}$. We let $D$ be a smooth divisor in $X$ with $0<\beta<\frac{1}{2}$.

Given a point $p$ in $D$, let $\{z^{1},\dots z^{k}\}$ be the local defining functions of the hypersurfaces where $p$ locates label. The local chart $(U_{p},z^{i})$ centered at $p$ is called cone chart at $p$.

Definition 2.1**.**

A Kähler cone metric $\omega$ of cone angle $2\pi\beta$ along $D$, is a closed positive $(1,1)$ current, which is also a smooth Kähler metric on the regular part

[TABLE]

In a local cone chart $U_{p}$, the Kähler form is quasi-isometric to the standard cone flat metric, which is

[TABLE]

The standard cone metric has nice properties. The Christoffel symbols of the connection of $\omega_{cone}$ under the holomorphic coordinate $\{z^{1},\dots z^{n}\}$ are for all $2\leq i,j,k\leq n$,

[TABLE]

except $\Gamma^{1}_{11}(\omega_{cone})=-\frac{1-\beta}{z^{1}}.$ Also, the Riemannian curvature of $\omega_{cone}$ is identical to zero.

2.1. Hölder spaces in cone charts

In this section, we start by recalling the definition of Donaldson’s Hölder spaces [MR2975584], see also [MR3144178].

A quasi-isometric mapping $W$ is well defined in the cone chart $U_{p}$ as follows,

[TABLE]

We let $v(w^{1},\cdots,z^{n})=u(z^{1},\cdots,z^{n})$. A function $u(z):U_{p}\rightarrow\mathbb{R}$ is said to be $C^{,\alpha,\beta}$, if $v(w^{1},\cdots,z^{n})$ is a $C^{\alpha}$ Hölder function in the classical sense. The space $C^{,\alpha,\beta}_{\{0\}}$ contains all functions $f\in C^{,\alpha,\beta}$ such that

[TABLE]

In the cone charts $U_{p}$, the Hölder semi-norm $[u]_{C^{,\alpha,\beta}(U_{p})}$ is defined to be $[v]_{C^{\alpha}(W(U_{p}))}$ and then the Hölder norm $|u|_{C^{,\alpha,\beta}(U_{p})}$ is $\sup_{U_{p}}|u|+[u]_{C^{,\alpha,\beta}(U_{p})}$ in the usual sense. The global semi-norm or norm on the whole manifold $X$ is defined by using a partition of unity of $X$, since in the charts away from $D$, everything is defined in the classical sense. Together with the Hölder norm, $C^{,\alpha,\beta}$ becomes a Banach space. A $(1,1)$-form $\sigma$ is said to be $C^{,\alpha,\beta}$, if for any $2\leq i,j\leq n$,

[TABLE]

Similarly, we could define $C^{,\alpha,\beta}$ of higher order tensors. The Hölder space $C^{2,\alpha,\beta}$ is defined by

[TABLE]

Note that the spaces $C^{,\alpha,\beta}$ and $C^{2,\alpha,\beta}$ are independent of the choice of the background Kähler cone metrics which are equivalent. But we can see that the higher order spaces are more complicated, since the geometry of the background metric is involved. The Hölder space $C^{3,\alpha,\beta}$ and $C^{4,\alpha,\beta}$ are introduced in [MR3405866] and further detailed computations can be found in [LZ2]. The idea is that we first define the local model Hölder spaces in the cone charts, and then extend it to the whole manifold by using a global background Kähler cone metric.

We identify $\tilde{U}$ in the complex Euclidean space as the image of the cone chart $U\subset X$ under the cone chart, i.e a quasi-isometry $\rho:U\rightarrow\tilde{U}\subset\mathbb{C}^{n}$. We call $\tilde{U}$ an image cone chart. Then we have $C^{,\alpha,\beta}(\tilde{U})$ and $C^{2,\alpha,\beta}(\tilde{U})$ defined in $\tilde{U}$ as above with respect to $\omega_{cone}$.

Definition 2.2**.**

The Hölder space $C^{3,\alpha,\beta}(\tilde{U})$ is defined as the set of function $u\in C^{2,\alpha,\beta}(\tilde{U})$ such that its 3rd order covariant derivatives with respect to $\nabla^{cone}$ associated to the metric $\omega_{cone}$ are $C^{,\alpha,\beta}(\tilde{U})$ in an image cone chart $\tilde{U}$. More precisely, written down with respect to the standard cone metric $\omega_{cone}$, the following covariant derivatives are required to be $C^{,\alpha,\beta}(\tilde{U})$,

[TABLE]

The $C^{4,\alpha,\beta}(\tilde{U})$ is defined in the similar way as follows.

Definition 2.3**.**

We say a $C^{3,\alpha,\beta}(\tilde{U})$ function $u$ is $C^{4,\alpha,\beta}(\tilde{U})$ if it satisfies for any $2\leq i,j,k,l\leq n$, in the image cone chart $\tilde{U}$,

[TABLE]

We notice from the definitions above that when the same derivative $\partial z^{1}$ or $\partial z^{\bar{1}}$ appear twice, we need extra lower order derivatives to adjust the normal derivatives.

2.2. Model cone metric

Let $\mathcal{H}_{\beta}(X,D)$ be the space of Kähler cone metrics of cone angle $2\pi\beta$ along $D$ in the Kähler class $[\omega_{0}]$. We denote by $\mathcal{H}^{2,\alpha,\beta}(X,D)$ the space of $\omega_{0}$-plurisubharmonic functions which are $C^{2,\alpha,\beta}(X)$. Let $s$ be a section of associated line bundle of $D$. It is explained by Donaldson in [MR2975584] that for sufficiently small $\delta>0$,

[TABLE]

is a Kähler cone metric and independent of the choices of $\omega_{0}$, $h_{\Lambda}$, $\delta$ up to quasi-isometry. We call such $\omega_{D}$ the model metric with conical singularity. The model metric $\omega_{D}$ has rich geometric information, that we list now. The detailed computation could be found in [MR3144178, MR3405866].

Lemma 2.1**.**

Assume that $0<\beta<\frac{1}{2}$. The following properties of the model metric $\omega_{D}$ hold in the cone charts.

• •

For any $2\leq i,j,k\leq n$,

[TABLE]

• •

The Christoffel symbols of the connection of $\omega_{D}$ satisfy for any $2\leq i,j,k\leq n$,

[TABLE]

• •

For any $2\leq i,j,l\leq n$, the second order covariant derivatives of the model metric $\omega_{D}$ are

[TABLE]

We could define the $C^{3,\alpha,\beta}$ and $C^{4,\alpha,\beta}$ spaces with respect to $\omega_{D}$ on the whole manifold $X$ via replacing the metric $\omega_{cone}$ with $\omega_{D}$ in both definitions (2.4) and (2.5). In fact, we have a more general property.

Proposition 2.1**.**

Assume that $\omega$ is a Kähler cone metric and its connection satisfies (2.8). Then the local $C^{3,\alpha,\beta}(\tilde{U})$ function $u$ could be extended to be global. Precisely, its 3rd order covariant derivatives belong to $C^{,\alpha,\beta}(X)$, i.e. letting $\nabla$ denote the covariant derivative regarding to the Kähler cone metric $\omega$, for any $2\leq i,j,k,l\leq n$,

[TABLE]

belong to $C^{,\alpha,\beta}(X)$.

Thus we can define the $C^{3,\alpha,\beta}(X;\omega)$ norm of a function $u$ on the whole manifold $X$ as

[TABLE]

Proposition 2.2**.**

Assume that $\omega$ is a Kähler cone metric and satisfies (2.8), (2.7) and (2.9). Then the local $C^{4,\alpha,\beta}(\tilde{U})$ function $u$ could be extended to be global, i.e. its 4th order covariant derivatives are all $C^{,\alpha,\beta}(X)$, i.e. letting $\nabla$ denote the covariant derivative regarding to the Kähler cone metric $\omega$, for any $2\leq i,j,k,l\leq n$,

[TABLE]

belong to $C^{,\alpha,\beta}(X)$.

The $C^{4,\alpha,\beta}(X;\omega)$ norm of a function $u$ is defined in the same way

[TABLE]

The proofs of Propositions 2.2 and 2.1 are carried out in [LZ2]. Moreover both spaces $C^{3,\alpha,\beta}(X;\omega)$ and $C^{4,\alpha,\beta}(X;\omega)$ are Banach spaces, as proved in [LZ2, Section 5].

Remark 2.1*.*

It is natural to use the model metric $\omega_{D}$ in both Propositions 2.1 and 2.2, since it satisfies all required conditions. The same scheme of ideas allows to define without difficulty higher order function spaces $C^{k,\alpha,\beta}$ for any $k\geq 5$.

Next we consider the general Kähler cone metric

[TABLE]

When we have a Kähler potential $\varphi\in C^{4,\alpha,\beta}(X;\omega_{D})$, we get $\log\omega^{n}_{\varphi}\in C^{2,\alpha,\beta}(X)$, and the Ricci curvature $-\sqrt{-1}\partial\bar{\partial}\log\omega^{n}_{\varphi}\in C^{,\alpha,\beta}(X)$, according to the definition above and Lemma 2.1, under the restriction (C). Furthermore, we have the following information for the connection and curvature of $\omega_{\varphi}$.

Corollary 2.1**.**

Assume that the potential function $\varphi$ of a Kähler cone metric belongs to $C^{4,\alpha,\beta}(X;\omega_{D})$. Then the properties (2.7), (2.8) and (2.9) hold for $g_{\varphi}$. Actually, denoting for simplicity $g=g_{\varphi}$ as the Riemannian metric associated to $\omega_{\varphi}$, we have that all following elements are in $C^{,\alpha,\beta}$,

• •

for any $2\leq i,j,k\leq n$,

[TABLE]

• •

the Christoffel symbols of the connection of $g_{\varphi}$ for any $2\leq i,j,k\leq n$,

[TABLE]

• •

for any $2\leq i,j,l\leq n$, the 2nd order covariant derivatives of $g_{\varphi}$

[TABLE]

Proof.

We choose $\omega$ to be $\omega_{cone}$ in both Proposition 2.1 and Proposition 2.2. Then the conclusion follows by applying Lemma 2.1. ∎

In addition, if we weaken the condition on the cone potential $\varphi$, we have the following bound.

Corollary 2.2**.**

Assume that the potential function $\varphi$ of a Kähler cone metric belongs to $C^{3,\beta}(X;\omega_{D})$. Then the properties (2.7) and (2.8) hold for $g_{\varphi}$. Actually, denoting for simplicity $g=g_{\varphi}$ as the Riemannian metric associated to $\omega_{\varphi}$, we have that all following elements are bounded,

• •

for any $2\leq i,j,k\leq n$,

[TABLE]

• •

the Christoffel symbols of the connection of $g_{\varphi}$ for any $2\leq i,j,k\leq n$,

[TABLE]

Remark 2.2*.*

In the previous corollary, the same results hold if we replace $\nabla^{cone}$ by $\nabla^{D}$.

2.3. Second order elliptic equations with conical singularities

We first quote a proposition of the general linear elliptic equation which essentially uses Donaldson’s estimates [MR2975584] (see also Brendle [MR3144178], Calamai-Zheng [MR3405866]). Consider the boundary value problem

[TABLE]

Here $g^{i\bar{j}}$ is the inverse matrix of a Kähler cone metric $\omega$ in $C^{,\alpha,\beta}$. We also denote the vector field $h^{i}\partial_{i}$ to be $\mathbf{h}$ and $b^{i}\partial_{i}$ to be $\mathbf{b}$. Moreover, we are given the following data.

[TABLE]

Proposition 2.3** ([MR3405866, MR2975584, MR3144178]).**

Fix $\alpha$ with $0<\alpha<\frac{1}{b}-1$. Then there is a constant $C$ depending on $\beta$, $n$, $\alpha$, $|\mathbf{b}|_{C^{,\alpha,\beta}}$, $|c|_{C^{,\alpha,\beta}}$ such that for all the functions $f\in C^{,\alpha,\beta}$ and $\mathbf{h}\in C^{1,\alpha,\beta}$, we have Schauder estimate of the weak solution of equation (2.11),

[TABLE]

2.4. Some properties of cscK cone metrics

In this section we review some recent progress on the theory of cscK cone metrics and show some extra properties.

Recall the definition of the cscK cone metrics in [zhengcscKcone].

Definition 2.4**.**

We say that $\omega_{cscK}$ is a cscK metric with conical singularities if

• •

$\omega_{cscK}$ is a cscK metric on the regular part $M$;

• •

$\omega_{cscK}$ is quasi isometric to the model metric $\omega_{cone}$;

• •

the potential of $\omega_{cscK}$ lies in $C^{2,\alpha,\beta}$.

From the definition, the cscK cone metric satisfies the equation on the regular part $M$,

[TABLE]

Remark 2.3*.*

We only require the second order behavior of the cscK cone metric in this definition. There are different ways to define cscK metrics with conical singularities and different notions are compared in [LiL]. However, a crucial issue is the question of higher regularities of such metrics.

We write the cscK cone metric $\omega_{cscK}$ using $\omega_{D}$-potentials i.e

[TABLE]

Because $\varphi_{cscK}$ is $C^{2,\alpha,\beta}$, the 4th order equation (2.13) could be re-written as the couple system of two second order elliptic equations

[TABLE]

The following higher regularity theorem is proved in [LZ2].

Theorem 2.4.1** ([LZ2]).**

Assume that $\varphi_{cscK}$ is the potential of a cscK cone metric satisfying (2.14) with $\varphi_{cscK}$ is $C^{2,\alpha,\beta}(X)$. Assume that the angle $2\pi\beta$ and the Hölder exponent $\alpha$ satisfy Condition (C). Then $\varphi_{cscK}$ is actually in $C^{4,\alpha,\beta}(X;\omega_{D})$ and the Ricci curvature of $\omega_{cscK}$ is $C^{,\alpha,\beta}(X)$.

Next lemma appeared in [MR3313212] in the case of Kähler-Einstein cone metrics.

Proposition 2.4**.**

Suppose that $\omega_{cscK}\in[\omega]$ is a cscK cone metric with $C^{2,\alpha,\beta}$ potential under Condition (C). The average of the scalar average of the cscK cone metric is

[TABLE]

Proof.

Since $\omega_{cscK}$ has $C^{2,\alpha,\beta}$ potential, $\omega_{cscK}^{n}=|s|_{h}^{2\beta-2}\omega_{0}^{n}e^{\psi}$ for some $\psi$ in $C^{,\alpha,\beta}$ and $h$ a smooth hermitian metric on $\mathcal{O}(D)$. Then we have

[TABLE]

as an equality of closed currents. By definition the scalar curvature is trace of Ricci curvature, and so

[TABLE]

The first two terms are what we need, i.e equal to

[TABLE]

We claim that the third term vanishes. From regularity Theorem 2.4.1, the potential of our cscK cone metric is actually $C^{4,\alpha,\beta}$. Hence, we have $Ric(\omega_{cscK})\leq C_{0}\cdot\omega_{cscK}$ for some constant $C_{0}>1$. Moreover, there is a constant $C_{1}>1$ such that $Ric(\omega_{0})-(1-\beta)\Theta(h)\geq-C_{1}\cdot\omega_{cscK}$ on $X$. We set $C_{2}=C_{0}+C_{1}$. Using now (2.16), we obtain

[TABLE]

We notice that $\varphi_{cscK}$ and $\varphi_{cscK}+\frac{1}{C_{2}}\psi$ are both $\omega_{0}$-psh functions which are globally bounded on $X$. We only need to check the vanishing along the direction $z^{1}$. According to the integration by part formula in [MR2746347, Theorem 1.14], we conclude the claim. ∎

Proposition 2.5**.**

Consider $X$ a smooth Fano manifold. Suppose that $D$ is a smooth divisor which is $\mathbb{Q}$-linearly equivalent to $-\lambda K_{X}$, with $\lambda\in\mathbb{Q}^{*}_{+}$. Consider $\omega=\omega_{cscK}$ is a cscK cone metric in the class $2\pi c_{1}(X)$ along $D$ with angle $2\pi\beta$ and Hölder exponent $\alpha$. Assume that $(\alpha,\beta)$ satisfy Condition (C). Then $\omega$ is actually a Kähler-Einstein cone metric satisfying the equation

[TABLE]

with $\nu=1-(1-\beta)\lambda$. Conversely, such a Kähler-Einstein cone metric is also a cscK cone metric.

Proof.

Using same notations as above, the function $z^{1}$ is the local defining function of the divisor $D$. The Poincaré-Lelong equation tells us that

[TABLE]

so the trace reduces to $g^{1\bar{1}}\delta_{\{z^{1}=0\}}$. Note that $\omega$ is $C^{,\alpha,\beta}$ and quasi-isometric to the standard cone metric

[TABLE]

Since $\delta$ function is a generalized function of order 0 (i.e. its action can be continuously extended to $C^{0}$ functions), this implies $g^{1\bar{1}}\delta_{\{z^{1}=0\}}=0$. Consequently, we have

[TABLE]

From Equation (2.16), $Ric(\omega)$ is actually a representative of $2\pi c_{1}(X)-2\pi(1-\beta)c_{1}(D)=\nu c_{1}(X)$. By considering cohomology classes, we can find a smooth real valued function $f$ such that

[TABLE]

Taking trace with respect to $\omega$ and using (2.17), we have

[TABLE]

Now, using the cscK condition, we obtain $\triangle_{\omega}f=0$. Thus $f$ is constant and we can conclude using again (2.18). ∎

Consider $V_{f}$ a holomorphic vector field on $X$ with holomorphy potential $f\in C^{\infty}(X,\mathbb{C})$ i.e $\iota_{V_{f}^{1,0}}\omega=-\bar{\partial}f$. Given the Kähler class $[\omega]$ and the vector field $V_{f}$, one can define the Futaki invariant as

[TABLE]

where $\underline{S}$ is the average of the scalar curvature of any Kähler form in the class $[\omega]$ and the log-Futaki invariant for vector fields $V_{f}$ that are furthermore tangent to $D$ as

[TABLE]

Both Futaki invariants depend only on the class $[\omega]$. As pointed out in [Hasfutaki], the log-Futaki invariant is the differential-geometric interpretation of the algebraic log Donaldson-Futaki invariant that can be defined using test configurations, see [MR2975584] and [MR3403730].

Next corollary is known for Kähler-Einstein cone metrics on Fano manifolds, see for instance [MR3470713], or in the smooth case for the classical Futaki invariant. Consider $\omega_{cscK}\in[\omega]$ a cscK cone metric. It satisfies globally in the sense of distributions

[TABLE]

Applying Proposition (2.4), we obtain the following result.

Corollary 2.3**.**

Under assumption (C), the log-Futaki invariant $Fut_{D,\beta,[\omega]}$ vanishes on Kähler classes $[\omega]$ which contain cscK cone metric with cone singularities along $D$ with cone angle $2\pi\beta$.

Remark 2.4* (Expansion close to the divisor).*

For any cone angle $0<\beta<1$, general expansion formulas for Kähler-Einstein cone metrics appear in [YZ]. They come from the study of a singular Monge-Ampère equation. We expect that similar expansion formulas hold for the cscK cone metrics.

Remark 2.5* (Uniqueness of cscK cone metrics).*

The study of uniqueness of cscK cone metrics has been initiated in [zhengcscKcone, LZ, LZ2, LZ3].

3. Lichnerowicz equations with conical singularities

The Lichnerowicz operator at a cscK metric $\omega_{cscK}$ is defined on functions $u$,

[TABLE]

We remark that when a Kähler metric has constant scalar curvature, the first variation of the scalar curvature is given by the Lichnerowicz operator.

We say that a Kähler cone metric $\omega$ has bounded Christoffel symbols of the connection, if for any $2\leq i,j,k\leq n$, the following items are bounded,

[TABLE]

We say that a Kähler potential $\varphi$ is a $C^{4,\alpha,\beta}$* (or $C^{3,\beta}$) cscK potential* if $\omega=\omega_{D}+i\partial\bar{\partial}\varphi$ is a cscK metric and also $\varphi\in C^{4,\alpha,\beta}$ (or $C^{3,\beta}$ respectively).

We also consider the following operator on functions $u$,

[TABLE]

under the assumption that the coefficient metric $\omega=\omega_{D}+i\partial\bar{\partial}\varphi$ has $C^{4,\alpha,\beta}$ cscK potential $\varphi$.

We are going to solve the equation for $f\in C^{,\alpha,\beta}$

[TABLE]

with solution $u\in C^{4,\alpha,\beta}$ and prove the Fredholm alternative of the Lichnerowicz operator Theorem 1.

3.1. Sobolev spaces for cone metrics

Since the volume element of the reference cone metric $\omega$ is an $L^{p}$ function (for some $p\geq 1$) with respect to the Euclidean metric, it gives rise to a measure $\omega^{n}$ on $M$. Thus, we can introduce the following Sobolev spaces with respect to $\omega^{n}$. We shall use the following complete Banach spaces on the whole manifold $X$.

Definition 3.1** (Sobolev spaces $W^{1,p,\beta}(\omega)$).**

For a Kähler cone metric $\omega$, the Sobolev spaces $W^{1,p,\beta}(\omega)$ for $p\geq 1$ are defined with respect to the reference Kähler cone metric $\omega$. The $W^{1,p,\beta}(\omega)$ norm is

[TABLE]

Definition 3.2** (Sobolev spaces $W^{2,p,\beta}(\omega)$).**

For a Kähler cone metric $\omega$ with bounded Christoffel symbols of the connection, the Sobolev spaces $W^{2,p,\beta}(\omega)$ for $p\geq 1$ are defined to be the completion of the space of smooth functions with finite $W^{2,p,\beta}(\omega)$ norm which is the combination of $W^{1,p,\beta}(\omega)$ norm and $W^{2,p,\beta}(\omega)$ semi-norm. The $W^{2,p,\beta}(\omega)$ semi-norm is defined with respect to the reference Kähler cone metric $\omega$ with cone angle $\beta$,

[TABLE]

Definition 3.3** (Strong Sobolev spaces $W_{\bf s}^{2,p,\beta}(\omega)$).**

For a Kähler cone metric $\omega$ with bounded Christoffel symbols of the connection, the Sobolev spaces $W_{\bf s}^{2,p,\beta}(\omega)$ for $p\geq 1$ are defined to be the completion of the space of smooth functions with finite $W^{1,p,\beta}(\omega)$ norm and $W_{\bf s}^{2,p,\beta}(\omega)$ semi-norm. The $W_{\bf s}^{2,p,\beta}(\omega)$ semi-norm is defined with respect to the reference Kähler cone metric $\omega$,

[TABLE]

In both (3.4) and (3.5), the second order pure covariant derivatives mean, for any $1\leq a,b\leq n$,

[TABLE]

The Christoffel symbols $\Gamma_{ab}^{c}(\omega)$ of the connection satisfy the properties of Corollary 2.1.

Remark 3.1*.*

From (3.6), we could see clearly that why the bounded Christoffel symbols of the connection of the background metric $\omega$ are required in the global definitions of the higher order Sobolev spaces.

Definition 3.4**.**

We define the Sobolev space $H^{2,\beta}:=W^{2,2,\beta}(\omega)$ and $H_{0}^{2,\beta}=\{u\in H^{2,\beta}|\int_{M}u\omega^{n}=0\}$. The Sobolev norm remains the same. The strong spaces $H_{\bf s}^{2,\beta}:=W_{\bf s}^{2,2,\beta}(\omega)$ and $H_{{\bf s},0}^{2,\beta}$ are defined in a similar way.

Lemma 3.1**.**

Assume that $u\in W^{1,p,\beta}(\omega)$. If $p<2n$, then there exists a constant $C$ such that

[TABLE]

for any $q\leq\frac{2np}{2n-p}$.

Proof.

We consider the function which is supported in a cone chart, we use the map $W$ defined in (2.2) and the Sobolev inequality in Euclidean space to obtain the desired inequality. The general case follows from a partition of unity. ∎

Lemma 3.2** (Sobolev embedding theorem).**

Assume that $u\in W_{\bf s}^{2,p,\beta}(\omega)$, $p<2n$ and $q\leq\frac{2np}{2n-p}$. There exists a constant $C$ independent of $u$ such that

[TABLE]

Proof.

From the lemma above,

[TABLE]

So the conclusion follows from applying classical Kato inequality that gives $\|\nabla|\nabla u|\|_{L^{p}(\omega)}\leq\|\nabla\nabla u\|_{L^{p}(\omega)}.$ The R.H.S is controlled by the term $\|u\|_{W_{\bf s}^{2,p,\beta}(\omega)}$. ∎

Lemma 3.3** (Kondrakov compactness theorem).**

Assume that $u\in W_{\bf s}^{2,p,\beta}(\omega)$, $p<2n$ and $q<\frac{2np}{2n-p}$. The Sobolev embedding

[TABLE]

is compact.

Proof.

We cover the manifold $X$ by a finite number of coordinates charts $\{U_{i},\psi;1\leq i\leq N\}$ and let $\rho_{i}$ be the smooth partition of unity subordinate to $\{U_{i}\}$. Let $f_{m}$ be a bounded sequence in $W_{\bf s}^{2,p,\beta}(\omega)$. In the charts which do not intersect with the divisor, we let $\tilde{f}_{m}=(\rho_{i}f_{m})\circ\psi_{i}^{-1}$. While, in the cone chart $U$ among $\{U_{i}\}$ we let $\tilde{f}_{m}=(\rho_{i}f_{m})\circ\psi_{i}^{-1}\circ W^{-1}$. Then we are able to pick Cauchy subsequence of $f_{m}$ in each charts for $i=1,2,...,N$ because of precompactness of $\tilde{f}_{m}$ in each $U_{i}$. ∎

Proposition 3.1** (Interpolation inequality).**

Suppose that $\epsilon>0$ and $1<p<\infty$. There exists a constant $C$ such that for all $u\in W_{\bf s}^{2,p,\beta}$, we have

[TABLE]

Proof.

It follows from Lemma 3.3 by using standard contradiction argument. We assume that the conclusion fails and for each $C_{i}=i$, there exists a $u_{i}\in W_{\bf s}^{2,p,\beta}$ with $\|u_{i}\|_{W_{\bf s}^{2,p,\beta}(\omega)}=1$ such that

[TABLE]

Thus $\|u_{i}\|_{W^{1,p,\beta}(\omega)}\geq\epsilon>0$ and $\|u_{i}\|_{L^{p}(\omega)}\rightarrow 0$, as $i\rightarrow+\infty$. On one hand, from Kondrakov compactness (Lemma 3.3), after taking a subsequence, $u_{i}$ converges to $u_{\infty}$ in $W^{1,p,\beta}(\omega)$ norm. Also $\|u_{\infty}\|_{W^{1,p,\beta}(\omega)}\geq\epsilon>0$. On the other hand, from Sobolev embedding (Lemma 3.2), $\|u_{i}\|_{W^{1,p,\beta}(\omega)}$ is uniformly bounded and then $u_{i}$ converges to zero in $L^{p}(\omega)$ as $i\rightarrow\infty$. Thus $u_{\infty}=0$, contradiction!

∎

Remark 3.2*.*

One may wonder whether we could replace the $W_{\bf s}^{2,p,\beta}$ with the partial norm $W^{2,p,\beta}$ in all Lemma 3.2, Lemma 3.3 and Proposition 3.1. But, after examining the proof of Lemma 3.2, it is obvious that a different Kato inequality would be needed.

Lemma 3.4** (Poincaré inequality).**

There is a constant $C_{P}$ such that for any $u\in H_{0}^{1,\beta}=\{u\in H^{1,\beta}|\int_{M}u\omega^{n}=0\}$,

[TABLE]

Proof.

The Poincaré inequality follows from the compactness theorem i.e. the inclusion $W^{1,p,\beta}(\omega)\subset L^{q}(\omega)$ with $q<\frac{2np}{2n-p}$ is compact. Actually, any minimizing sequence $u_{i}$ of $\|\nabla u\|_{L^{2}(\omega)}$ over $H=\{u\in H_{0}^{1,\beta}\text{ s.t }\|u\|_{L^{2}(\omega)}=1\}$, converges strongly in $L^{2}$ and weakly in $H^{1,\beta}_{0}$, to a limit $v$. So $\inf_{u\in H}\|\nabla u\|_{L^{2}(\omega)}$ is realized by $v$ and has to be positive. ∎

3.2. A partial $L^{p}$ estimate

In [chenwang] (Definition 2.1), it is defined a local Sobolev space $W_{loc}^{2,p,\beta}(\mathtt{B};\omega_{cone})$ over a ball $\mathtt{B}$ contained in a cone chart $U$. It contains the functions $u\in W^{1,p,\beta}(\mathtt{B};\omega_{cone})$ such that for all $2\leq i,j\leq n$,

• •

$|z^{1}|^{2(1-\beta)}\frac{\partial^{2}u}{\partial z^{1}\partial z^{\bar{1}}}\in L^{p}(\mathtt{B};\omega_{cone})$;

• •

$|z^{1}|^{1-\beta}\frac{\partial^{2}u}{\partial z^{1}\partial x^{j}}\in L^{p}(\mathtt{B};\omega_{cone})$, for all $2\leq j\leq 2n$, with $z^{i}=x^{i}+\sqrt{-1}x^{n+i}$, for all $2\leq i\leq n$;

• •

$\frac{\partial^{2}u}{\partial x^{j}\partial x^{k}}\in L^{p}(\mathtt{B};\omega_{cone})$, for all $2\leq j,k\leq 2n$;

• •

$u\in W^{2,2}(\mathtt{B}\setminus\mathtt{N}_{\epsilon};\omega)$ for any $\mathtt{N}_{\epsilon}\subset\mathtt{B}$, $\epsilon$-tubular neighbourhood of the divisor $D\cap\mathtt{B}$.

The semi-norm is defined to be

[TABLE]

It is also proved in [chenwang, Theorem 4.1] a $L^{p}$ estimate over $U$ with respect to the flat metric $\omega_{cone}$. Note that we define $\mathtt{B}_{r}$ the balls of radius $r$ times a small radii $r_{0}$ with respect to the cone metric $\omega_{cone}$.

Lemma 3.5** ([chenwang], Theorem 4.1).**

Assume that $u\in W_{loc}^{2,p,\beta}(\mathtt{B}_{1};\omega_{cone})$ for $2\leq p<\infty$, and $\triangle_{\omega_{cone}}u\in L^{p}(\mathtt{B}_{2};\omega_{cone})$. Then there exists a constant $C$ depending on $n,p,\beta$ such that

[TABLE]

In order to extend the local definition to the global manifold, we need the following lemmas. We could see that if we restrict the semi-norm $W^{2,p,\beta}$ defined by (3.4) over $U$, it is controlled by patching up these local $W_{loc}^{2,p,\beta}$ semi-norms on coverings.

Lemma 3.6**.**

Let $\omega=\omega_{D}+i\partial\bar{\partial}\varphi$ be a Kähler cone metric with bounded Christoffel symbols of the associated connection. There exists a constant $C>0$ such that for all function $u\in W_{loc}^{2,p,\beta}(\mathtt{B}_{1};\omega_{cone})$, it holds

[TABLE]

Proof.

Recall the definition

[TABLE]

In the cone charts, $\omega$ is equivalent to $\omega_{cone}$. We then examine term by term. $\partial_{1}\partial_{1}u$ is already in (3.9). Then

[TABLE]

$\partial_{k}\partial_{\bar{1}}$ and $\partial_{k}\partial_{\bar{j}}$ are also $L^{p}(\omega_{cone})$. Meanwhile, the second term

[TABLE]

From assumption, the Christoffel symbols $\Gamma_{1j}^{1}(\omega)$ and $|z^{1}|^{1-\beta}\Gamma_{1j}^{c}(\omega)$ for $2\leq c\leq n$ are all bounded. Therefore $\partial_{1}\partial_{j}u$ is $L^{p}(\omega_{cone})$ and so is the third term $\partial_{j}\partial_{k}u$. ∎

Then we consider the $W^{2,p,\beta}(\omega)$ solution of the linear equation on $M$,

[TABLE]

where $f\in L^{p}(\omega)$.

Proposition 3.2**.**

Let $\omega$ be a Kähler cone metric with bounded Christoffel symbols of the associated connection. Suppose that $u\in W^{2,p,\beta}(\omega)$ for $2\leq p<\infty$ is a classical solution of Equation (3.10) for $f\in L^{p}(\omega)$. Then there exists a constant $C$ depending on $n,p,\beta,M,\omega$ such that

[TABLE]

Proof.

We let the manifold $B$ be covered by a finite number of coordinates charts $\{U_{i},\psi;1\leq i\leq N\}$. We let $\rho_{i}$ be the smooth partition of unity, subordinate to $\{U_{i}\}$ and supported on $\mathtt{B}_{1}\subset\mathtt{B}_{3}\subset U_{i}$ for each $i$. We compute that

[TABLE]

Thus we apply Lemma 3.5 (over $\mathtt{B}_{1}$ and $\mathtt{B}_{2}$) to

[TABLE]

to obtain,

[TABLE]

and put together all estimates over each $U_{i}$ with Lemma 3.6,

[TABLE]

Then the conclusion follows.

∎

3.3. Weak solutions to bi-Laplacian equations: existence

For a constant $K>0$, we are looking for weak solutions to the following $K$-bi-Laplacian equation in $H_{0}^{2,\beta}(\omega)$,

[TABLE]

The positive constant $K$ will be determined later in this section.

We recall the definition of the semi-norm

[TABLE]

which does not involve the term $\|\partial_{1}\partial_{1}u\|_{L^{2}(\omega)}$. We introduce the following bilinear form.

Definition 3.5**.**

Define the bilinear form on $H_{0}^{2,\beta}(\omega)$ given by

[TABLE]

for all $u,\eta\in H_{0}^{2,\beta}(\omega)$.

Then we introduce the weak solution to (3.11) , whose leading coefficients are conical.

Definition 3.6** (Weak solution).**

We say that $u$ is the $H_{0}^{2,\beta}(\omega)$-weak solution of the $K$-bi-Laplacian equation (3.11), if it satisfies the following identity for all $\eta\in H_{0}^{2,\beta}(\omega)$,

[TABLE]

Lemma 3.7**.**

The bilinear form $\mathcal{B}^{K}$ is bounded and coercive for $K>C_{P}+1$. Here $C_{P}$ is the Poincaré constant of $\omega$.

Proof.

For all $u,\eta\in H_{0}^{2,\beta}$, the boundedness follows from using Cauchy-Schwarz inequality,

[TABLE]

Then we prove coercivity of the bilinear form. We first use the partial $L^{2}$ estimate of the cone metrics for the standard linear operator $\triangle_{\omega}$ (Proposition 3.2), i.e. that there exists a constant $C>0$ such that,

[TABLE]

Then, using the definition of the bilinear form, the R.H.S above is

[TABLE]

We use Poincaré inequality (Lemma 3.4) and denoting the Poincaré constant by $C_{P}$,

[TABLE]

Thus choosing $K>C_{P}+1$, we have

[TABLE]

∎

The next proposition proves the existence of weak solution.

Proposition 3.3**.**

Let $\omega$ be a Kähler cone metric with bounded Christoffel symbols of the connection. Suppose that $K>C_{P}+1$ and $f$ is in the dual space $(H_{0}^{2,\beta}(\omega))^{\ast}$. Then the $K$-bi-Laplacian equation (3.11) has a unique weak solution $u\in H_{0}^{2,\beta}(\omega)$.

Proof.

According to Lemma 3.7, the bilinear form $\mathcal{B}^{K}$ is bounded and coercive. Then the Lax-Milgram theorem tells us that there is a unique weak solution $v\in H_{0}^{2,\beta}$ to equation (3.11).

∎

We could define another 2nd Sobolev space $H^{2,\beta}_{\mathbf{w}}$ with semi-norm

[TABLE]

and norm

[TABLE]

Then following the same argument as above, we get a “very weak” solution, that lies in $H^{2,\beta}_{\mathbf{w},0}$, the space of functions of $H^{2,\beta}_{\mathbf{w}}$ with vanishing integral assuming that $\omega$ is merely a Kähler cone metric.

Proposition 3.4**.**

Assume that $\omega$ is a Kähler cone metric. Suppose that $K>C_{P}+1$ and $f$ is in the dual space $(H_{\mathbf{w},0}^{2,\beta}(\omega))^{\ast}$. Then the $K$-bi-Laplacian equation (3.11) has a unique weak solution $u\in H_{\mathbf{w},0}^{2,\beta}(\omega)$.

3.4. Weak solutions to bi-Laplacian equations: regularity

According to the existence theorem (Proposition 3.3), we now already have a weak solution

[TABLE]

We shall see we can obtain that the weak solution is actually $C^{4,\alpha,\beta}$ if additionally we impose more regularity on $f$.

Proposition 3.5** (Schauder estimate).**

With previous notations with $(\alpha,\beta)$ satisfying Condition (C) and $f\in C^{,\alpha,\beta}$, the weak solution $u\in H^{2,\beta}_{0}$ to equation (3.11) is actually $C^{4,\alpha,\beta}$. Moreover, there exists a constant $C$ such that

[TABLE]

Proof.

We rewrite (3.11) as

[TABLE]

According to the Schauder estimate for second order equation (see Section 2.3), we have proved that $\triangle_{\omega}u$ is in $C^{2,\alpha,\beta}$ from (3.12). Then we could use the angle restriction to conclude that $u\in C^{4,\alpha,\beta}$, according to Proposition 4.3 in [LZ2].

∎

Furthermore, we actually could weaken the condition on $\omega$ using Proposition 3.4, however we do not use it in this paper.

Proposition 3.6** (Schauder estimate).**

Let $\omega=\omega_{D}+i\partial\bar{\partial}\varphi$ be a Kähler cone metric with $\varphi\in C^{2,\alpha,\beta}$, $f\in C^{,\alpha,\beta}$ and the Hölder exponent satisfy $\alpha\beta<1-\beta$, the weak solution $u\in H^{2,\beta}_{\mathbf{w},0}$ to equation (3.11) is actually $C^{2,\alpha,\beta}$. Moreover, there exists a constant $C$ such that

[TABLE]

Proof.

We use (3.12) again, we only need $\omega$ to be Kähler cone metric to conclude $\triangle_{\omega}u$ is in $C^{\alpha,\beta}$. Then, 2nd order linear elliptic theory [MR2975584] tells us that $u\in C^{2,\alpha,\beta}$. ∎

3.5. Fredholm alternative for Lichnerowicz operator

We now use continuity method in the Hölder spaces. We define the operator

[TABLE]

and then define the continuity path $L_{t}^{K}:C^{4,\alpha,\beta}\rightarrow C^{,\alpha,\beta}$ with $0\leq t\leq 1$,

[TABLE]

Multiplying the equation with $u$ and integrating over the manifold $M$, we obtain the bilinear form

[TABLE]

We will need the following lemma.

Lemma 3.8**.**

Assume that $u\in C^{2,\alpha,\beta}$ and $Ric(\omega)$ is bounded. Then it holds

[TABLE]

Proof.

We apply the cutoff function $\chi_{\epsilon}$ which has been fully discussed in [LZ]. Then the argument is essentially Lemma 4.10 in [LZ2]. By dominated convergence theorem, we have

[TABLE]

On the other hand, using $\nabla Ric=0$ on $M$,

[TABLE]

The first term converges under the assumption on $u$ and $Ric(\omega)$. The second term also converges, since for $2\leq i,j\leq n$,

[TABLE]

∎

When $t=0$, $L_{0}u=\triangle^{2}_{\omega}u-K\triangle_{\omega}u$. We could solve $L_{0}u=f$ for any $f\in C^{,\alpha,\beta}$ and obtain an solution $u\in C^{4,\alpha,\beta}$ thanks to Propositions 3.3 and 3.5.

In order to apply the continuity method to our linear PDE (see e.g. Theorem 5.2 in [MR1814364] for Banach space setting), we need to prove the following key estimate.

Theorem 3**.**

Assume $\omega$ is a cscK cone metric with $C^{4,\alpha,\beta}$ potential, $(\alpha,\beta)$ satisfy the condition (C). Assume that $K>0$ is large enough, i.e. $K>1+2\|Ric(\omega)\|_{L^{\infty}}+3C_{P}$. There is constant $C_{1}$ such that for any $u\in C^{4,\alpha,\beta}$ along the continuity path (3.5) with $0\leq t\leq 1$, we have

[TABLE]

Proof.

Applying Proposition 4.3 in [LZ2] to the equation

[TABLE]

we have

[TABLE]

Note that $Ric(\omega)$ is $C^{,\alpha,\beta}$ and we apply the $\epsilon$-interpolation inequality of the Hölder spaces to the 2nd and 3rd terms on the right hand side, and so

[TABLE]

We use the $\epsilon$-interpolation inequality of the $L^{p}$ spaces to the 2nd term on the right hand side (Proposition 3.1, choosing $\epsilon$ small enough),

[TABLE]

Since $u\in C^{4,\alpha,\beta}$, we are able to apply the integration by parts to obtain a Gärding inequality as following. We first use the $L^{2}$ estimate of the cone metrics to the standard linear operator $\triangle_{\omega}u$ (Proposition 3.2), i.e. there exists a constant $C_{5}>0$ such that,

[TABLE]

Here the integration by parts works since $u\in C^{4,\alpha,\beta}$. Using the bilinear form $\mathcal{B}^{K}_{t}(u,u)$, the R.H.S of previous inequality is

[TABLE]

Then we use the Cauchy-Schwarz inequality to $\mathcal{B}^{K}_{t}(u,u)=\int_{M}uL_{t}^{K}u\omega^{n}$,

[TABLE]

Integrating by parts the term containing $Ric(\omega)$ using Lemma 3.8, the R.H.S of last inequality becomes

[TABLE]

Thus,

[TABLE]

Then we apply the Poincaré inequality (Lemma 3.4) to the 3rd term again and set $K_{0}=1-K+\|Ric(\omega)\|_{L^{\infty}}+2C_{P}$. We obtain

[TABLE]

Now we use the special form of the Lichnerowicz operator to estimate the term $\partial\partial u$ (since $u\in C^{4,\alpha,\beta}$), i.e.

[TABLE]

Thus we use (3.5), integration by parts (Lemma 3.8) and Cauchy-Schwarz inequality as before,

[TABLE]

We apply the Poincaré inequality (Lemma 3.4) to the 2nd term again and set $K_{1}=C_{P}+\|Ric(\omega)\|_{L^{\infty}}-K$,

[TABLE]

Thus we add (3.16) and $C_{5}$ times (3.17) together and have that

[TABLE]

We further choose $K_{0}+K_{1}<0$ i.e. $2K>1+2\|Ric(\omega)\|_{L^{\infty}}+3C_{P}$, then we have

[TABLE]

Together with (3.15), this allows us to conclude that

[TABLE]

∎

Proof of Theorem 1.

We have just solved

[TABLE]

and seen that the inverse map $({{\mathbb{L}\mathrm{ic}}}^{K}_{\omega})^{-1}:C^{4,\alpha,\beta}\rightarrow C^{4,\alpha,\beta}$ is compact. Now, we can solve (1.1), i.e

[TABLE]

Actually, this is equivalent, after taking $({{\mathbb{L}\mathrm{ic}}}^{K}_{\omega})^{-1}$, to

[TABLE]

Since $\mathfrak{T}:=-K({{\mathbb{L}\mathrm{ic}}}^{K}_{\omega})^{-1}:C^{4,\alpha,\beta}\rightarrow C^{4,\alpha,\beta}$ is compact, we can apply classical results of functional analysis and Riesz-Schauder theory (see [MR1814364, Theorem 5.3]) to the Lichnerowicz operator which is self-adjoint. Furthermore the reductivity of the automorphisms group of a Kähler manifold admitting cscK cone metric with $C^{4,\alpha,\beta}$ potential are proved in [LZ2], showing the one-one correspondence between the kernel of the Lichnerowicz operator and the holomorphic vector fields tangential to the divisor. ∎

4. Hermitian-Einstein metrics with conical singularities

4.1. Stable parabolic structures

From now, we consider $E\to B$ a holomorphic vector bundle over a base $B$, compact Kähler manifold endowed with a smooth Kähler metric $\omega_{0}$. Let $D=\sum^{m}_{i=1}D_{i}$ be a simple normal crossings divisor of $B$.

Definition 4.1**.**

A parabolic structure on $E$ with respect to $D$ consists of:

• •

a filtration of $E_{|D_{i}}$ for $1\leq i\leq m$ such that

[TABLE]

with $\mathcal{F}_{i}^{p+1}$ proper subbundle of $\mathcal{F}_{i}^{p}$ and the flags satisfy a natural compatibility condition: for every $I=(i_{1},...,i_{q})$, the restrictions $\{{\mathcal{F}_{i_{l}}^{p}}_{|D_{i_{1}}...D_{i_{q}}},1\leq l\leq q,1\leq p\leq l_{i_{l}}\}$ to $D_{i_{1}}...D_{i_{q}}$ yield to a flag of $E_{|D_{i_{1}}...D_{i_{q}}}$ which is a refined flag of $\{{\mathcal{F}_{i_{l}}^{p}}_{|D_{i_{1}}...D_{i_{q}}},1\leq p\leq l_{i_{l}}\}$ for every $1\leq l\leq q$.

• •

some real weights $\alpha_{i}^{1},...,\alpha_{i}^{l_{i}}$ attached to $\mathcal{F}_{i}^{p}$, $1\leq p\leq l_{i}$ satisfying the inequalities $0\leq\alpha_{i}^{1}<...<\alpha_{i}^{l_{i}}<1$.

We recall a classical definition.

Definition 4.2**.**

Given $E$ a parabolic structure, one can define its parabolic degree with respect to $\omega_{0}$ as

[TABLE]

and its parabolic slope as $\mathrm{par}\mu(E)=\mathrm{par}\deg(E)/\mathrm{rk}(E)$. Here the degree $\deg(E)$ is computed in the usual sense using the Kähler cone metric $\omega_{0}$ and depends only on the Kähler class. This definition extends to coherent subsheaves endowed with parabolic structures.

There exists a notion of stability for parabolic structures modeled on the notion of Mumford-Takemoto stability.

Definition 4.3**.**

Given a proper coherent subsheaf $F$ of a parabolic vector bundle $E$ along $D$, one can consider the induced parabolic structure for $F$. The only difficulty is to choose correctly the weights $\alpha_{i}^{p}(F)$. This is done by taking the maximum weights among the $\alpha_{i}^{p^{\prime}}(E)$ that respect the flag structure, i.e $\mathcal{F}_{i}^{p}(F)\subset\mathcal{F}_{i}^{p^{\prime}}(E)$. We say that $E$ is parabolic stable if for all proper coherent subsheaf $F$ of $E$, we have

[TABLE]

4.2. Hölder spaces for bundle endomorphisms

Let $V$ be a holomorphic vector bundle over the base manifold $B$. Let us fix a holomorphic frame $F_{V}=\{e_{\varsigma};1\leq\varsigma\leq\mathrm{rk}(V)\}$ and a finite covering $(U_{i})_{i=0,..,N}$ of $B$ composed of local cone charts around $D$. Consider a partition of unity $\{\psi_{i}\}_{i=0,..,N}$ where $\psi_{i}\in C^{\infty}(U_{i})$ have compact support, associated to the finite covering $(U_{i})_{i=0,..,N}$. Using the notations of subsection 2.1, we define the space $C^{,\alpha,\beta}(V)$ to be the space of sections $s$ of $V$ such that in $F_{V}$ the decomposition of $s$ is given by $\mathrm{rk}(V)$ functions $s_{1},..,s_{\mathrm{rk}(V)}$ that lie in the space $C^{,\alpha,\beta}$. More precisely, $s$ is a section in the Hölder space $C^{,\alpha}(V)$ such that if the local frame is defined close to the divisor $D$, over a cone chart $U_{i}$, each $\mathrm{rk}(V)$ complex valued functions defining $s$ lie in the space $C^{,\alpha,\beta}(U_{i})$. The advantage of fixing a frame and a partition of unity is that we can define now a norm $\|.\|_{C^{,\alpha,\beta}}$ by

[TABLE]

Note that the space of sections with bounded $\|.\|_{C^{,\alpha,\beta}}$ norm is independent of the covering, the partition of unity and the holomorphic frame.

Similarly to subsection 2.1, we can define the vector spaces $C^{2,\alpha,\beta}(V)$, $C^{3,\alpha,\beta}(V)$, $C^{4,\alpha,\beta}(V)$ and the associated norms by considering the analogue conditions on the decomposition of $s$. Eventually, all the spaces equipped with their natural norms are Banach spaces. With the previous reasoning, we can define this way the Hölder spaces of $End(E)$ that are denoted $C^{k,\alpha,\beta}(End(E))$.

The definition also applies to the space $\mathcal{H}^{+}(E)$ of hermitian metrics on the bundle $E$ seen as sections of the frame bundle and for $Herm(E,h)$ the space of hermitian endomorphisms of $E$ (over $M\setminus D$) with respect to the hermitian metric $h$.

4.3. Existence of Hermitian-Einstein cone metrics

Given $\omega_{0}$ and the divisor $D$, we can consider a model metric $\omega_{D}$ as in Section 2.2. Moreover, we will denote $\omega_{B}$ a Kähler cone metric in the same class and $C^{2,\alpha,\beta}$ potential, which is quasi-isometric to $\omega_{D}$.

We introduce now the Hermitian-Einstein equation and express it in coordinates. Let $H=\{H_{\varsigma\bar{\tau}}\}$ be the hermitian matrix induced by $h$ in a local holomorphic frame $\{e_{\varsigma};1\leq\varsigma\leq r\}$ for $E$ of rank $r$.

Definition 4.4**.**

We say a hermitian metric $h$ is $C^{k,\alpha,\beta}$ for $k\in\mathbb{Z}^{+}=\{0,1,2,3,\cdots\}$, if the associated hermitian matrix $H=\{H_{\varsigma\bar{\tau}}\}$ is $C^{k,\alpha,\beta}$, i.e. all its components $H_{\varsigma\bar{\tau}}$, $1\leq\varsigma,\tau\leq r$ are all $C^{k,\alpha,\beta}$.

The curvature is given as a 2 form

[TABLE]

with explicitly

[TABLE]

Lowing down the index, we have

[TABLE]

Fix $\omega_{B}=\frac{\sqrt{-1}}{2}(g_{B})_{k\bar{j}}dz^{k}\wedge d\bar{z}^{j}$ a Kähler metric on the base $B$. The Hermitian-Einstein equation

[TABLE]

reads in coordinates

[TABLE]

Definition 4.5** (Compatible metric with respect to parabolic structure).**

Let $h$ be a metric on the parabolic bundle $E$. We say that $h$ is compatible with the parabolic structure if the following holds. Given the parabolic structure, it is constructed by Li in [LiJ] a model metric $h_{0}$ on $E$ over $B\setminus D$, such that

[TABLE]

This model metric on the bundle is natural. In a nutshell, the norm of a local section of $\mathcal{F}_{i}$ with respect to $h_{0}$ restricted to $D$ has growth controlled by the weights of the filtration. To be compatible for $h$ metric on $E$ means that the 2 following conditions hold:

• •

$h,h_{0}$ are mutually bounded;

• •

$|\bar{\partial}(h_{0}^{-1}h)|_{h_{0}}\in L^{2}(B,\omega_{D})$.

Given $\omega_{D}$ the model metric and $h$ a metric on $E$, compatible with respect to the parabolic structure, it is possible to compute the analytic degree of $E$. It is given by the differential geometry as

[TABLE]

and a similar formula applies for the proper coherent subsheaves of $E$. In [LiJ] it is checked that $\widetilde{\deg}(E)$ is actually proportional to the parabolic degree of $\mathrm{par}\deg(E)$ and is an invariant of the space of hermitian metrics compatible with the parabolic structure. In other words, the bundle $E$ is parabolic stable if and only if it is stable with respect to the notion of slope induced by the analytic degree.

The same property holds if we replace $\omega_{D}$ by $\omega_{B}$ as we assumed it has $C^{2,\alpha,\beta}$ potential, and we have $\tilde{\mu}(E)=c\times\mathrm{par}\deg(E)$ for a certain constant $c>0$. The property of compatibility can also be defined using the Kähler cone metric $\omega_{B}$. In conclusion, we can speak of parabolic stability of the parabolic bundle $E$ with respect to $\omega_{B}$ by using the analytic degree.

We are ready to present a theorem of C. Simpson improved by J. Li.

Theorem 4.3.1** ( [LiJ, Theorem 6.3], [Sim]).**

Let $B$ be a base compact Kähler manifold endowed with a Kähler metric $\omega_{B}$ with conical singularities along $D\subset B$, smooth divisor. Let $E$ a parabolic stable vector bundle over $B$ with respect to $\omega_{B}$. There exists $\delta_{0}>0$ such that if the angle $2\pi\beta$ of $\omega_{B}$ satisfies $0<\beta_{i}\leq\delta_{0}$, then there exists a Hermitian-Einstein metric $h_{E}$ on $E$ compatible with the parabolic structure over $D$. It satisfies outside $D$ the Hermitian-Einstein equation,

[TABLE]

Here $Id_{E}$ is the identity endomorphism of $E_{|B\setminus D}$ and $\mathrm{Vol}$ the total volume of $B$ with respect to $\omega_{B}$.

We introduce the following definition of Hermitian-Einstein cone metric.

Definition 4.6**.**

As above, let $B$ be a base compact Kähler manifold endowed with a Kähler metric $\omega_{B}$ with conical singularities along $D$, smooth divisor. Let $E$ a parabolic vector bundle with respect to $D$ and $h_{E}$ a hermitian metric on $E_{|B\setminus D}$. We say that $h_{E}$ is a Hermitian-Einstein cone metric, if $h_{E}$ satisfies the Hermitian-Einstein equation (4.2) pointwisely over $B\setminus D$, $h_{E}$ is compatible with the parabolic structure and $h_{E}$ lies in $C^{2,\alpha,\beta}(\mathcal{H}^{+}(E))$.

Theorem 4**.**

*Under same assumptions as in Theorem 4.3.1, the Hermitian-Einstein metric $h_{E}$ is actually a Hermitian-Einstein cone metric in the sense of Definition 4.6.

Moreover, if $\alpha$ and $\beta$ satisfy the Condition (C), then $h_{E}\in C^{4,\alpha,\beta}(\mathcal{H}^{+}(E))$.*

Remark 4.1*.*

Note that the converse is true and constitutes the easy sense of the correspondence: an indecomposable parabolic vector bundle equipped with a Hermitian-Einstein cone metric compatible with its parabolic structure is actually parabolic stable. We refer [LiJ, Theorem 6.3], [Sim, Proposition 3.3].

Proof.

We start the proof by noticing that we could take a partition of unity $\{\theta_{p}\}$ of the base manifold $B$ subordinate to an open cover $\{U_{p}\}$ and construct the Hermitian-Einstein metric on each trivialization of the holomorphic vector bundle $E$ over each cover. It suffices to consider the cone chart $U$, which intersects with the divisor $D$, since far from the divisor all the arguments are the same to [MR1165874, MR765366, MR885784, MR861491, MR1215276].

The proof is divided in several steps. We first use Dirichlet problem for Donaldson’s flow to produce the weak solution $H$ to the Hermitian-Einstein equation away from the divisor and then prove the regularity of the Hermitian-Einstein limit metric. Of course, the new improvement with our theorem is the regularity of the weak Hermitian-Einstein metric. Note that we do not try to improve the regularity of the flow itself, which is a parabolic system and the Schauder estimate is not yet known in this case. But instead, we use the limit equation and improve the regularity by observing that the nonlinear term itself is Hölder. As a result, each equation in the system is independent, and we are able to apply the elliptic regularity theorem for second order equations with conical singularities to each single equation of the system.

Using $h_{0}$ the model metric fixed by Li and which satisfies (4.1), we write the endomorphism

[TABLE]

We let $s$ be a section of $L_{D}$, $h_{L}$ be a smooth positively curved hermitian metric on the line bundle $L_{D}$. We denote by $D_{\delta}$ a $\delta$ tubular neighbourhood of the divisor $D$ for small $0<\delta\leq 1$. We also use

[TABLE]

Eventually, we omit the factor $\frac{\sqrt{-1}}{2\pi}$ in from of the contraction operator $\Lambda_{\omega}$ to ease notations.

According to Donaldson [MR1165874], the Dirichlet problem for the following flow $h=h_{t}$,

[TABLE]

has a unique global solution for $0\leq t<+\infty$. We keep in mind that, we now obtain a sequence of solutions to Donaldson’s flow on $\delta$, and denote the solutions by $h_{\delta}$. As further shown in [MR1165874], the convergence of (4.3) is irrelevant of the delicate conditions of stability. We will need that the approximation flow $h_{\delta}$ converges to a limit flow as $\delta\rightarrow 0$, the limit flow converges to a Hermitian-Einstein metric $h_{\infty}$ with conical singularities as $t\rightarrow+\infty$ and this limit metric $h_{\infty}$ has higher order regularity across the divisor $D$. In order to achieves these goals, we need the following a priori estimates.

Step: Uniform bound of $A(x,t):=|\Lambda_{\omega_{B}}F_{h}|_{h}$. We have along the flow (4.3),

[TABLE]

Let $A(t)=\sup_{U_{\delta}}A(x,t)$, then we apply the maximum principle,

[TABLE]

So we prove that $\sup_{U_{\delta}}|\Lambda_{\omega_{B}}F_{h}|_{h}$ is non-increasing along the flow, and also it is uniformly bounded by the initial given data $|\Lambda_{\omega_{B}}F_{h_{0}}|_{h_{0}}$ and independent of $\delta$ and $t$.

Step: Zero order estimate. We are now aiming to prove that $h(t)$ converges to $h(T)$ in $C^{0}$ norm, for any finite time $T<\infty$. Donaldson’s distance function between two Hermitian metrics is used,

[TABLE]

It is known that for any two flows $h(t)$ and $k(t)$,

[TABLE]

For any $\epsilon>0$, we choose $\kappa>0$ such that in the $\kappa$-neighbourhood of $t=0$, i.e. for all $0\leq s,\tau<\kappa$,

[TABLE]

We now let $k_{\delta}(t)=h_{\delta}(t+s)$ in the inequality above, we see that $\sigma(h_{\delta}(t),k_{\delta}(t))$ is always zero on the boundary of the domain $U_{\delta}$ along the flow. From maximum principle, we see that in the $\kappa$-neighbourhood of $T$, i.e. when $T-\kappa<s^{\prime},\tau^{\prime}<T$,

[TABLE]

Thus $h_{t}$ is a uniform Cauchy sequence and converges in $C^{0}$ norm to $h_{T}$.

Step: Gradient estimate. It follows from the contradiction method, see [MR1040197, Lemma 6.4], that $h_{t}$ are bounded in $C^{1}$.

Step: $W^{2,p}(\omega_{B})$ estimate. From the gradient estimates above, $|H_{\varsigma\bar{\tau}}|_{C^{1}}$ is bounded and also is $\Lambda_{\omega_{B}}F_{h}$. So $-\triangle_{\omega_{B}}H_{\varsigma\bar{\tau}}$ is bounded by using the equation

[TABLE]

After applying the interior $L^{p}$ theory of the linear equation in $U_{\delta}$, we have for any $\varsigma,\tau$, $H_{\varsigma\bar{\tau}}\in W_{\bf s}^{2,p}(\omega_{B})(K)$ for any $K\subset\subset U_{\delta}$. And Proposition 3.2 for weaker Sobolev spaces tells us $H_{\varsigma\bar{\tau}}\in W^{2,p}(\omega_{B})(M)$. Thus $F_{h}$ is bounded in $L^{p}(M)$ norm for any $1\leq p<\infty$. Note that $W_{\bf s}^{2,p}$ is the strong Sobolev spaces and $W^{2,p}$ is the weaker one, see Definitions 3.3 and 3.2.

Step: Long time existence. In each $U_{\delta}$, we can see deduce from the estimates above that the solution to the approximation equation (4.3) has long time existence. For any compact subset $K\subset\subset B\setminus D$, we could choose a small enough $\delta_{K}$ such that for any $\delta<\delta_{K}$, $K\subset\subset U_{\delta}$. Since the metrics $h(x,t)$ have uniform $W^{2,p}(\omega_{B})(M)$ estimates for any $p$, and independently of $\delta$, $h_{\delta}$ converges to a limit flow $h=\lim_{\delta\rightarrow 0}h_{\delta}$ in $W^{2,p}(\omega_{B})(M)$-norm for any $p\geq 1$ and furthermore $h$ solves

[TABLE]

Step: Convergence. We need a subspace of the space of Hermitian metrics,

[TABLE]

Obviously, $h_{0}\in{\mathcal{H}}_{bounded}(E)$. Furthermore, for any $|h-h_{0}|_{C^{2,\alpha,\beta}}\leq\epsilon$, we still have $\sup_{B\setminus D}|\Lambda_{\omega_{B}}F_{h}|_{h}<2\sup_{B\setminus D}|\Lambda_{\omega_{B}}F_{h_{0}}|_{h_{0}}$, provided that $\epsilon$ is small enough. Under the topology induced by the $C^{2,\alpha,\beta}$-topology, we consider the path-connected branch of $h_{0}$, denoted by ${\mathcal{H}}_{bounded,h_{0}}(E)$, i.e. the set of metrics that be connected to $h_{0}$ by a path

[TABLE]

Donaldson’s functional for the cone version is well-defined on the space of Hermitian metrics ${\mathcal{H}}_{bounded,h_{0}}(E)$ with suitable asymptotic behavior near the divisor,

[TABLE]

where $h_{s}\in{\mathcal{H}}_{bounded,h_{0}}(E)$ is path connecting $h_{0}$ and $h$. The definition is independent of the choice of the path. Actually, one can adapt to our setting the proof of the classical smooth case. The proof consists in showing that the variation of $M_{D}(h_{0},.)$ is a closed 1-form. It requires to study the term $\phi_{h}:=\mathrm{tr}(h^{-1}\tilde{d}h\cdot F_{h})$ where

[TABLE]

is a smooth map and $\tilde{d}=(\partial_{s})ds+(\partial_{t})dt$ is the exterior differentiation on the domain $\Delta$. But the 1-form $\phi_{h}$ is well defined from our assumption on ${\mathcal{H}}_{bounded}(E)$ and one can apply Stokes theorem $\int_{\partial\Delta}\phi_{h}=\int_{\Delta}\tilde{d}\phi_{h}$. Then one can follow word by word the proof of [Kob, Lemma 3.6]. Alternatively, one can show that the curvature of $h$ is a moment map for the action of the Gauge group on the space of Chern connections associated to ${\mathcal{H}}_{bounded}(E)$, see for instance [MR2039989] and also [Sim, Lemma 7.2]. Then, as a classical result of the moment map theory, $M_{D}$ is the associated integral to this moment map and is consequently independent of the choice of the path.

We have by the arguments of [Sim, Proposition 5.3], that there are two constants $C_{1}$ and $C_{2}$ such that for any $h\in{\mathcal{H}}_{bounded}(E)$

[TABLE]

Since $\Lambda_{\omega_{B}}F_{h}$ and $h_{0}h^{-1}$ are both bounded, the following functional is well-defined along the flow $h_{t}$,

[TABLE]

We need its first variation formula along the flow,

[TABLE]

Thus the functional $M_{D}$ is non-increasing along the flow. This leads to a uniform upper bound to $\|\log\mathrm{tr}H\|_{L^{1}(\omega_{B})}$ from (4.10).

Now, we wish to apply De Giorgi-Nash-Moser iteration method for Kähler cone metrics of [LZ, Section 4] to the bounded $\log\mathrm{tr}H$ in the following functional inequality

[TABLE]

In order to do so, we need to examine the conditions of Proposition 4.8 in [LZ]. Firstly, the following Sobolev inequality with respect to $\omega_{B}$ holds, i.e. for any $w\in W^{1,2}(\omega_{B})$, there is a Sobolev constant $C_{S}(\omega_{B})<+\infty$ such that

[TABLE]

where $2^{\ast}=\frac{2n}{n+1}$. Secondly, we need to rewrite (4.12) to the following form by using integration by parts, i.e. $v=\log\mathrm{tr}H$ is a $W^{1,2}$ sub-solution of the linear equation in the weak sense, i.e. for any $\eta\in C^{2,\alpha,\beta}$,

[TABLE]

This is achieved by using the approximation sequence and $v$ vanishes on the exhaustion domains. So, outside the $\delta$ neighbourhood of the divisor $D$,

[TABLE]

The boundary $\int_{\partial D_{\delta}}v\cdot\partial\eta d\nu\rightarrow 0$, as $\delta\rightarrow 0$.

Let

[TABLE]

then there exists from [LZ, Proposition 4.8] a constant $C$ depending on the Sobolev constant $C_{S}$ with respect to $\omega_{B}$ such that for $p^{\ast}=\frac{2np}{2n+p}$,

[TABLE]

We thus obtain the $L^{\infty}$ bound of $H$,

[TABLE]

From the monotonicity of the energy along the flow (4.3), the right hand side is uniformly bounded. Letting $\eta=v$ in (4.13), we have

[TABLE]

Thus we are able to prove that $H_{t}$ converges in $C^{0}$ norm to some $H_{\infty}$ and then get $C^{1}$ norm of $H_{t}$ which is independent of $t$, as the gradient estimate above in Step: Gradient estimate. After applying the $L^{p}$ theory of the linear equation with respect to Kähler cone metrics developed in [chenwang] to the curvature equation (4.8), we have $H_{\varsigma\bar{\tau}}\in W^{2,p}(\omega_{B})$. Furthermore, the $W^{2,p}$ norm of $H(t)$ is independent of $t$, as the proof in Step: $W^{2,p}(\omega_{B})$ estimate.

Step: $W^{1,2}(\omega_{B})$ weak solution. Now we have a Hermitian-Einstein metric on the regular part, but we still need to verify that the limit metric satisfies the Hermitian-Einstein equation in $W^{1,2}(\omega_{B})$ sense,

[TABLE]

We fix $H=H_{\varsigma\bar{\tau}}$ and denote the nonlinear term

[TABLE]

We need

[TABLE]

It suffices to use the approximation sequence again and prove the boundary term $\int_{\partial D_{\delta}}\partial H\cdot\eta\rightarrow 0,$ as $\delta\rightarrow 0$. Thus is true, since $H$ has uniform $C^{1}$ norm.

Step: $C^{4,\alpha,\beta}$ estimate. Once we have $H_{\varsigma\bar{\tau}}$ is a $W^{1,2}(\omega_{B})$ weak solution and lies in $W^{2,p}(\omega_{B})$, we can apply the Sobolev embedding theorem [chenwang] to obtain $H_{\varsigma\bar{\tau}}\in C^{1,\alpha,\beta}$, thus returning to (4.18), the nonlinear term $\mathsf{N}$ is $C^{,\alpha,\beta}$. Then we have $H_{\varsigma\bar{\tau}}\in C^{2,\alpha,\beta}$ by Donaldson’s Schauder estimate and bootstrap to $H_{\varsigma\bar{\tau}}\in C^{4,\alpha,\beta}$ similar to the proof of Proposition 3.5 in Section 3.4.

∎

Remark 4.2*.*

From [LiJ], one gets that $\delta_{0}$ depends on the (difference of the) weights of the parabolic structure of $E$. So a priori, $\delta_{0}$ is fixed and we don’t know its size, it can be $>$ or $<1/2$. But we consider here the theorem only for angles $\beta<\min(1/2,\delta_{0})$. In general, we believe the asymptotic behaviour of Hermitian-Einstein metric $h_{E}$ could be well-understood with the method in [YZ] and our angle restriction could be removed.

4.4. Parabolic stability and holomorphic vector fields

It is well-known that Mumford stable vector bundles are simple. In the parabolic setting, we have the following result.

Lemma 4.1**.**

Assume $E$ is parabolic stable with parabolic structure along a simple normal crossings divisor $D=\sum^{m}_{i=1}D_{i}$. Then the holomorphic endomorphism are the homotheties, i.e

[TABLE]

Proof.

The proof is similar to the non parabolic case. Let $f$ be a holomorphic endomorphism which is not zero or an isomorphism. Then by holomorphicity of $f$, $\ker(f),\mathrm{Im}(f)$ have a coherent subsheaf of $E$ with quotient torsion free. One can obtain a parabolic structure for $F=\ker(f),\mathrm{Im}(f)$ by intersection $F_{|D_{i}}$ with the elements of the flag of $E_{|D_{i}}$, discarding the subspaces of $F_{|D_{i}}$ that coincide with another one, and considering the associated largest parabolic weights. Thanks to the parabolic stability of $E$, we have now the inequalities

[TABLE]

But, the parabolic weights of $\ker(f)$ and $\mathrm{Im}(f)$ satisfy also

[TABLE]

Using inequalities (4.20), (4.21) and Equation (4.22), one gets a contradiction: $\mathrm{par}\mu(\ker(f))<\mathrm{par}\mu(\mathrm{Im}(f))$ and $\mathrm{par}\mu(\mathrm{Im}(f))<\mathrm{par}\mu(\ker(f))$. Thus, $f$ is an isomorphism or trivial. In the first case, fix $x\in B$ and consider any eigenvalue of $f:E_{x}\rightarrow E_{x}$. Let’s call $\lambda_{0}$ this eigenvalue. Then by the reasoning as above, $f-\lambda_{0}Id_{E}$ is zero and consequently $f$ is an homothety. ∎

We need the following classical definition of logarithmic tangent bundle.

Definition 4.7**.**

Consider $B$ a complex manifold of complex dimension $n$ and $D$ a divisor with simple normal crossings singularities. In local coordinates $D=\{z\text{ s.t }\prod_{i=1}^{d}z^{i}=0\}$. Then the logarithmic tangent bundle $TB(-\log D)$ is the locally free sheaf generated by the vector fields $z^{i}\frac{\partial}{\partial z^{i}}$ where $1\leq i\leq d$ and the vector fields $\frac{\partial}{\partial z^{i}}$ for $d<i\leq n$.

Let us consider the stratification of $B$ given by $B_{0}=B\setminus D$, $B_{1}=D\setminus Sing(D)$ and recursively $B_{k}$ is the non-singular part of $Sing(B_{k-1})$. $T_{B}(-\log D)$ can be seen as the sheaf of holomorphic vector fields $v$ on $B$ such that for every $k\geq 0$, every $x\in B_{k}$, $v_{x}$ is tangent to $B_{k}$.

We denote the $X$ the projectivised bundle and $\pi$ the associated projection map to $B$,

[TABLE]

Next, we derive some information on the holomorphic automorphisms of $X$ when $E$ is parabolic stable and $B$ has no nontrivial holomorphic vector field.

Corollary 4.1**.**

*Assume $E$ is a parabolic stable vector bundle with respect to the Kähler cone metric $\omega_{B}$ and the base $B$ has no nontrivial holomorphic vector field.

Then the Lie algebra $Lie(Aut_{\mathcal{D}}(X,[k\pi^{*}\omega_{B}+\hat{\omega}_{E}]))$ is actually trivial.*

Proof.

Consider $TFibre(-\log\mathcal{D})$ the sheaf of logarithmic tangent vectors to the fibre of $\pi$ with respect to $\mathcal{D}$. There is an exact sequence

[TABLE]

which provides a long exact sequence

[TABLE]

Now, $H^{0}(X,\pi^{*}(TB(-\log D)))=0$ by assumption, and thus we obtain that the space $H^{0}(X,TFibre(-\log\mathcal{D}))$ is isomorphic to $H^{0}(X,TX(-\log\mathcal{D}))$.

On another hand, $H^{0}(B,\pi_{*}TFibre(-\log\mathcal{D}))$ can be identified with the holomorphic parabolic endomorphisms along $D$ that are trace free. By parabolic endomorphisms, we mean endomorphisms of $E$ that preserve the parabolic structure of $E$. Hence,

[TABLE]

Eventually, using Lemma 4.1, we obtain $H^{0}(X,TFibre(-\log D))=0$ and thus, $H^{0}(X,TX(-\log\mathcal{D}))=0$. This means that there is no nontrivial holomorphic vector fields tangent to $D$. ∎

5. Construction of cscK cone metrics over projective bundles

Given a (hermitian) vector space $\Xi$, there is an isomorphism between $\Xi$ and $H^{0}(\mathbb{P}\Xi^{*},\mathcal{O}_{\mathbb{P}\Xi^{*}}(1))$. This leads to define a metric on $\mathcal{O}_{\mathbb{P}\Xi^{*}}(1)$ by the following construction. For $v\in\Xi$, the element $\hat{v}\in H^{0}(\mathbb{P}\Xi^{*},\mathcal{O}_{\mathbb{P}\Xi^{*}}(1))$ is such that $\hat{v}(\xi)=\xi(v)$ for $\xi\in\Xi^{*}$. Then from any metric $h$ on $\Xi$, we get a metric $h^{*}$ on $\Xi^{*}$ and a Fubini-Study metric $\hat{h}$ on the line bundle $\mathcal{O}_{\mathbb{P}\Xi^{*}}(1)$ by the formula

[TABLE]

for $v,w\in\Xi$, and $\xi\in\Xi^{*}$.

Consequently, from the Hermitian metric $h_{E}$ on the holomorphic vector bundle $E$, we get a Hermitian metric $\hat{h}_{E}$ on the line bundle $\mathcal{O}_{\mathbb{P}E^{*}}(1)$.

5.1. Construction of background metrics

Over $X$, we consider the Kähler metric $\hat{\omega}_{E}\in c_{1}(\mathcal{O}_{\mathbb{P}E^{*}}(1))$ outside $\mathcal{D}$ given by the formula

[TABLE]

Proposition 5.1**.**

Let $h_{E}$ the Hermitian-Einstein cone metric obtained in Theorem 4. Then $\hat{\omega}_{E}$ is a (1,1)-form on $X={\mathbb{P}E^{*}}$ with conical singularity along $\mathcal{D}$ in $C^{2,\alpha,\beta}$ topology.

Proof.

The local computation of $\hat{\omega}_{E}$ using (5.1) involves only terms of the form $\partial_{k}H_{\varsigma\bar{\tau}}$, $\partial_{j}\partial_{\bar{k}}H_{\varsigma\bar{\tau}}$ where $H$ is the matrix representing $h$ in a local frame close to $D$. Thus, in order to have a cone metric we only need to have the entries of $H$ to be $C^{4,\alpha,\beta}$, and $h_{E}\in C^{4,\alpha,\beta}(\mathcal{H}^{+}(E))$, which is the case from Theorem 4. ∎

Thus, from the metric on the base $B$, we obtain a Kähler cone metric in $[k\pi^{*}\omega_{B}+\hat{\omega}_{E}]$, that we denote by

[TABLE]

Lemma 5.1**.**

Let $h_{E}$ the Hermitian-Einstein cone metric obtained in Theorem 4 and assume that condition (C) holds. Then we have $\omega_{k}\in C^{2,\alpha,\beta}$ and $S(\omega_{k})\in C^{,\alpha,\beta}$.

Proof.

This is an application of Theorem 2.4.1. The cscK cone metric $\omega_{B}$ on the base is $C^{2,\alpha,\beta}$, while $\hat{\omega}_{E}$ is also $C^{2,\alpha,\beta}$ from the previous result. ∎

5.2. Expansion of scalar curvature

Let us remember that we know from Hong’s techniques.

Lemma 5.2**.**

On the regular part of $\mathbb{P}E^{*}$,

[TABLE]

where $r=\mathrm{rk}(E)$, $[v]\in\mathbb{P}E^{*}$ and $[.]^{0}$ denotes the trace free part.

5.3. Approximate cscK cone metrics

In view of Theorem 2, we shall deform the metric $\omega_{k}$ to obtain Kähler metrics

[TABLE]

as in the following proposition, and then apply the contraction mapping theorem, following the main idea of [Hong2, Hong3].

Proposition 5.2**.**

Assume $\omega_{B}$ is cscK with conical singularities along $D$ with angles $\beta$ and that $Lie(Aut_{D}(B,[\omega_{B}]))$ is trivial. Assume the holomorphic vector bundle $E$ is parabolic stable and equipped with $h_{E}$ Hermitian-Einstein cone metric obtained via Theorem 4. Fix $p>0$. There exist deformations of

• •

the form $\omega_{B}$ as $\omega_{B}+\sqrt{-1}\partial\bar{\partial}\eta_{k,p}$, with

[TABLE]

• •

the Hermitian-Einstein metric $h_{E}$ as $h_{E}(Id_{E}+\Phi_{k,p})$ with

[TABLE]

such that the induced Kähler form $\tilde{\omega}_{k}\in[k\pi^{*}\omega_{B}+\hat{\omega}_{E}]$ on $X$ can be deformed to obtain an almost cscK cone metric outside $\mathcal{D}$ of order $p+1$, i.e there exists real valued functions on $X\setminus\mathcal{D}$,

[TABLE]

such that over $X\setminus\mathcal{D}$, we have

[TABLE]

where $\underline{S}_{\beta}$ is the topological constant.

We construct the approximation solution by the implicit function theorem inductively, using Lemma 5.3, Lemma 5.4 and Proposition 5.3. The proof is given at the end of this section, page 5.3, and requires several preliminary results. The deformation of the scalar curvature is divided into three parts, the function on $B$, the section $\Gamma(B,W)$ and the function on $\mathbb{P}E^{*}$. Actually, we need to deform all the metrics $\omega_{B},h_{E}$ and $\omega_{k}$.

Firstly, we need to understand the deformation of the cscK equation on $X$ with respect to $\omega_{B}$ and $h_{E}$, where $\omega_{B}\in\mathcal{H}_{\beta}(B,D)$ is a Kähler cone metric on $B$ and $h_{E}$ a hermitian metric on $E$ compatible to the parabolic structure with respect to $\omega$. In order to do so, we are going to study the maps

[TABLE]

When $\omega_{B}$ has constant scalar curvature and $h_{E}$ is Hermitian-Einstein, then the linearization of $A_{1}$ at $(\omega_{B},h_{E})$ is given by

[TABLE]

where $\eta\in C^{4,\alpha,\beta}$ and $\Phi$ is a hermitian endomorphism with respect to $h_{E}$.

Lemma 5.3**.**

Suppose that $Lie(Aut_{{D}}(B,[\omega_{B}]))$ is trivial and $E$ is parabolic stable with respect to $\omega_{B}$. Set $C^{\infty}(End(E))^{0}$ the space of trace-free endomorphisms of $E$ and ${C^{4,\alpha,\beta}_{0}}(B)$ the space of $C^{4,\alpha,\beta}$ real functions on $B$ with vanishing integral with respect to $\omega_{B}$. Then the map

[TABLE]

defined by (5.5), is an isomorphism.

Proof.

If $DA_{1}(\eta,\Phi)=0$, then we get the system of decoupled equations on $B\setminus D$ by considering the trace part,

[TABLE]

Solutions of this system live on $B$. Using Theorem 1, the first equation has for solution only the constants by assumption on $Lie(Aut_{{D}}(B,[\omega_{B}]))$. But then $\eta=0$ since $\eta$ has vanishing integral. From this fact, this leads from the second equation of the system, to

[TABLE]

and since $\Phi$ is trace-free and $C^{2,\alpha,\beta}(B,End(E))^{0}$,

[TABLE]

and then we get $\int_{B}|\bar{\partial}\Phi|^{2}\frac{\omega_{B}^{n}}{n!}=0$ and so $\Phi\in H^{0}(B,End(E))$, and thus constant from Lemma 4.1. Now, this constant vanishes since $\Phi$ is trace-free. Eventually we apply Fredholm alternative.

From the first equation, we are able to solve $\eta$ via the theory of the Lichnerowicz equation (Theorem 1). Putting it into the second equation, we could solve each component of $\Phi$, i.e. (5.7), by the theory of second order elliptic equations with conical singularities (see Section 2.3). ∎

Secondly, we deform the cscK equation on $X$ with respect to $\omega_{k}$.

5.4. Decomposition of the holomorphic tangent bundle

In the sequel of this section, we assume that we are under the setting of Proposition 5.2, and $h_{E}$ is a Hermitian-Einstein cone metric. Given $\hat{\omega}_{E}$ a form induced by the $h_{E}$ on $E$, one can define the following operator $\Delta_{V}$ on $C^{2,\alpha,\beta}$ functions on the ruled manifold $X\setminus\mathcal{D}$,

[TABLE]

where $n$ is the dimension of the base $B$ and $\pi:X\to B$ is the projection onto $B$.

Remark 5.1*.*

It is not a Laplacian. Nevertheless, once restricted to the fibers, it is a Laplacian with respect to the Fubini-Study metric.

Now, we know that for a general Kähler form $\omega$, the linearization of the scalar curvature is given by

[TABLE]

Thus for a smooth function $\phi$ the linearization of the operator the scalar curvature operator at the metric $\omega_{k}$ is given by

[TABLE]

when $k$ tends to $+\infty$ using (5.4). We deform $\omega_{k}$ and obtain

[TABLE]

Consider the metric $\hat{\omega}_{E}$ and its Laplacian acting on functions on $\mathbb{P}E^{*}_{x}$. We denote $W_{x}$ as the space of all eigenfunctions associated to the first nonzero eigenvalue of this Laplacian, which is $r=\mathrm{rk}(E)$.

This defines a vector bundle $W$ over $B$. We have the following result.

Lemma 5.4**.**

To any trace-free hermitian endomorphism

[TABLE]

one can associate

[TABLE]

such that its restriction over $B\setminus D$ belongs to $C^{,\alpha,\beta}(B\setminus D,W)$ (i.e is a eigenfunction). The converse is also true, i.e given a eigenfunction that belongs to $C^{,\alpha,\beta}(B\setminus D,W)\cap C^{,\alpha,\beta}(B)$, we obtain a trace free hermitian endomorphism in $C^{,\alpha,\beta}(B,End(E))^{0}.$

Proof.

It is known the eigenfunctions of the Laplacian on $\mathbb{CP}^{r-1}$ with respect to the Fubini-Study metric that are associated to the first non zero eigenvalue of the Laplacian. These eigenfunctions are given by harmonic polynomials on $\mathbb{C}^{r}$ that correspond to certain hermitian endomorphisms, see [MR704609], [MR3107685, Proposition 2.4]. We apply this correspondence over $B\setminus D$. Consequently, we obtain that given $\Phi$ as above, $\mathrm{tr}(\Phi\frac{v\otimes v^{*_{h_{E}}}}{\|v\|^{2}})\in C^{,\alpha,\beta}(B)$ is an eigenfunction over $B\setminus D$ and the regularity is clear. Now, for the converse, starting with an eigenfunction, we get an endomorphism $\Phi_{1}$ over $B\setminus D$ as before. We extend the subbundle $Im(\Phi_{1})$ as a subbundle of $C^{,\alpha,\beta}(B,End(E))$. For doing that, we apply [Sim, Lemma 10.6]. We just need to see that the curvature of $End(E)$ is $L^{1}$, but this is the case as $F_{End(E)}=F_{E,h_{E}}\otimes Id_{E^{*}}+Id_{E}\otimes F_{E^{*},h_{E}^{*}}$ and $h_{E}$ is Hermitian-Einstein. ∎

Remark 5.2*.*

With the lemma in hands, we see why it is not sufficient to deform the metrics $\omega_{B},h_{E}$ and that we need to also deform the metric $\omega_{k}$.

Now, for $p=2$ or $p=4$, one can consider the orthogonal space to $C^{p,\alpha,\beta}(B\setminus D,W)$ of functions in $C^{p,\alpha,\beta}(X\setminus\mathcal{D})$ with vanishing integral, i.e

[TABLE]

We consider the map

[TABLE]

Lemma 5.5**.**

$L_{V}$* is an injective.*

Proof.

For the injectivity, assume that $\Delta_{V}(\Delta_{V}-r)\phi=0$. Then over each fiber above $x\in B\setminus D$, $\Delta_{V}(\Delta_{V}-r)\phi_{x}=0$. By compactness of the fiber, it implies that $(\Delta_{V}-r)\phi$ is constant and since $\phi$ has vanishing integral, we get $(\Delta_{V}-r)\phi=0$ over $B\setminus D$. But then $\phi$ is an eigenfunction associated to the $r$-th eigenvalue, and thus by orthogonality, $\phi=0$.

∎

Next we prove that the map $L_{V}$ is also surjective. We adapt an idea from the theory of differential families of strongly elliptic differential operators, see Section 7 in Kodaira [MR2109686] for instance.

We first cut out a small open neighbourhood $D_{\delta}$ containing $D$ with radius $\delta$. On each fibre at $x\in B\setminus D_{\delta}$, the ellipticity of the operator $\Delta_{V}(\Delta_{V}-r)(\phi)$ tells us that there is a complete orthonormal set of eigenfunctions $\{e_{i}(x)\}$. Each corresponding eigenvalues $\lambda_{i}(x)$ is a continuous function of $x\in B\setminus D_{\delta}$.

Given $\psi\in\mathcal{Z}^{,\alpha,\beta}_{0}$ we construct a solution $\phi_{x}$ for all $x\in B\setminus D$ of

[TABLE]

over $\mathbb{P}E^{*}_{x}$. Along each fibre, we are able to solve the equation $L_{V,x}\phi_{x}=\psi_{x}$, since the operator is strongly elliptic and self-adjoint with respect to the inner product $\int_{\mathbb{P}E^{*}_{x}}(\cdot,\cdot)\hat{\omega}_{E}^{r-1}$.

We define the spaces $C_{V}^{k,\alpha}$, $k\geq 0$ as the Hölder spaces defined on the fibre $\mathbb{P}E^{*}_{x}$. Along each fibre, we then prove a priori estimates, with the help of the injectivity of the operator $L_{V}$.

Lemma 5.6**.**

Suppose that $\phi\in\mathcal{Z}^{4,\alpha,\beta}_{0}$ in $\mathbb{P}E^{*}$. There exists a constant $C>0$ depending on $\mathbb{P}E^{*}_{x}$ and the coefficients of $L_{V,x}$ such that

[TABLE]

Proof.

Since $L_{V,x}$ is strongly elliptic, we have

[TABLE]

We now assume the conclusion is not true. Then there exists a sequence of $\phi_{x}(k)\in\mathcal{Z}^{4,\alpha,\beta}_{0}$, $k\geq 1$ such that

[TABLE]

Thus from compactness of $C^{4,\alpha}_{V}$, $|\phi_{x}(\infty)|_{C^{,\alpha}_{V}}=1$ and $L_{V,x}\phi_{x}(\infty)=0$. But $\mathcal{Z}^{4,\alpha,\beta}_{0}$ is closed and then we use Lemma 5.5, there is no nontrivial kernel in $\mathcal{Z}^{4,\alpha,\beta}_{0}$. Thus $\phi_{x}(\infty)=0$, which is a contradiction. ∎

Let $x,y$ be two points in $B\setminus D_{\delta}$. We let $\phi_{x},\phi_{y}$ solve the equations $L_{V,x}\phi_{x}=\psi_{x}$ and $L_{V,y}\phi_{y}=\psi_{y}$ respectively. Now we are ready to prove in the next two lemma, the asymptotic behaviors of the solution $\phi_{x}$ with respect to the point $x$ in the base manifold $B\setminus D$.

Lemma 5.7**.**

The family of solution $\phi_{x}$ is continuous in $x$.

Proof.

We have

[TABLE]

From Lemma 5.6, we have

[TABLE]

Here $C$ depends on the geometry of the fibre $\mathbb{P}E^{*}_{x}$ and the coefficients of $L_{V}$ at $x$. Since the coefficients of $L_{V}$ and $\psi$ are both $C^{,\alpha,\beta}$, we see that $\phi_{y}\rightarrow\phi_{x}$ as $y\rightarrow x$. ∎

Lemma 5.8**.**

The family of solution $\phi_{x}$ is $C^{1,\beta}$ in $x$.

Proof.

We denote $\nabla_{x}$ be the covariant derivative with respect to the metric $\pi^{\ast}\omega_{B}$. We let $\eta_{x}$ solves the equation

[TABLE]

We also denote distance $\delta$ between $x,y$ are measured under the metric $\pi^{\ast}\omega_{B}$. It is sufficient to prove that as $y\rightarrow x$,

[TABLE]

From Lemma 5.7, it suffices to prove that as $y\rightarrow x$,

[TABLE]

Using the formulas above, it becomes

[TABLE]

The first two terms on the right hand side converge to zero by definition and the third term converges to zero, since the coefficients are in $C^{,\alpha,\beta}$.

∎

Proposition 5.3**.**

$L_{V}$* is bijective.*

Proof.

Thanks to Lemma 5.5, we just need to prove that $L_{V}$ is surjective. The problem now is how to glue the family of pointwise solutions together to produce a solution on the whole ruled manifold. Combining Lemma 5.7 and Lemma 5.8, we obtain $\phi\in C^{4,\alpha,\beta}$ by using the induction argument and repeating the strategy of Lemma 5.8. ∎

Remark 5.3*.*

Note that the proofs of Lemmas 5.5, 5.6, 5.7, 5.8 and Proposition 5.3 do not require $h_{E}$ to be Hermitian-Einstein but only to have the regularity obtained in Theorem 4.

Proof of Proposition 5.2.

The first two terms of the expansion of the scalar curvature $S(\tilde{\omega}_{k}+\sqrt{-1}\partial\bar{\partial}\phi_{k,p})$ are constant since $S_{1}(\omega_{B},h_{E})$ is constant by definition of $\omega_{B}$ and $h_{E}$. Now, we wish to make constant the $k^{-2}$ term of the expansion of $S(\tilde{\omega}_{k}+\sqrt{-1}\partial\bar{\partial}\phi_{k,p})$ constant. For that, writing the topological constant $\underline{S}_{\beta}=\underline{S}_{\beta}^{0}+k^{-1}\underline{S}_{\beta}^{1}+k^{-2}\underline{S}_{\beta}^{2}+...$ and considering (5.9), (5.10), we need to find $(\eta_{0},\Phi_{0},\phi_{0})$ solving the equation over $X\setminus\mathcal{D}$,

[TABLE]

Since $\underline{S}_{\beta}^{2}$ is a constant, we are lead to solve both equations $\Delta_{V}(\Delta_{V}-r)\phi_{0}=0$ and $DS_{1}(\eta_{0},\Phi_{0})=\underline{S}_{\beta}^{2}$. Note that first equation has an obvious solution. Hence, we can apply Lemma 5.3 and Lemma 5.4 to obtain $(\eta_{0},\Phi_{0})$.

Next, we need to make constant the $k^{-3}$ term. As there is a contribution coming from expansion of $S(\tilde{\omega}_{k})$, we are lead this time to solve

[TABLE]

for a certain function $\gamma_{3}\in C^{,\alpha,\beta}(X\setminus\mathcal{D},\mathbb{R})$. Here we use the fact that we found $\eta_{0}$ and $\phi_{0}$ in $C^{4,\alpha,\beta}$ and $\Phi_{0}\in C^{2,\alpha,\beta}$. The term $\gamma_{3}$ appears as a combination of 4th order derivatives of $\eta_{0},\phi_{0}$ and 2nd order derivatives in $\Phi_{0}$. Consequently, from the fact that $C^{,\alpha,\beta}$ is an algebra, $\gamma_{3}$ is an element of $C^{,\alpha,\beta}(X,\mathbb{R})$. Again, Lemma 5.3, Lemma 5.4 and Proposition 5.3 ensure that a solution $(\eta_{1},\Phi_{1},\phi_{1})$ can be found with the right regularity.

Then Proposition 5.2 is obtained by induction using the same reasoning for higher order terms, thanks to the fact that we have surjectivity from $C^{4,\alpha,\beta}_{0}(B)\times C^{2,\alpha,\beta}(B,End(E))^{0}\times\mathcal{Z}^{p,\alpha,\beta}_{0}$ onto $C^{,\alpha,\beta}(X\setminus\mathcal{D},\mathbb{R})$ and applying Lemmas 5.3 and 5.4 and Proposition 5.3.

∎

5.5. Proof of Theorem 2

Proof of Theorem 2.

Note that the proof is technically different to [LS], as in the smooth case it is used Sobolev spaces while we use Hölder spaces. Since $E$ is parabolic stable, we obtain a Hermitian-Einstein cone metric from Theorem 4. We can apply Proposition 5.2 to obtain an approximate cscK cone metric. To obtain a genuine cscK cone metric, we need to apply the implicit function theorem using Theorem 1 on $X$ the projectivisation of the bundle. This requires to solve the Lichnerowicz equation as the kernel of the associated operator is trivial, thanks to Corollary 4.1. ∎

6. Kähler-Einstein cone metrics and Tangent bundle

It is well-known that if $X$ is a compact Kähler manifold endowed with a smooth Kähler-Einstein metric (with positive, negative or zero curvature) then its tangent bundle $TX$ admits a Hermitian-Einstein metric and thus is Mumford polystable (with respect to the anticanonical polarization, canonical polarization, or any polarization respectively). In this section, we study the case when $X$ is a compact Kähler manifold and admits a Kähler-Einstein metric with conical singularities along a divisor. Let $n$ be the complex dimension of $X$.

Let us consider $\omega_{KE}$ a Kähler-Einstein metric with conical singularities along $D$ smooth divisor for which the Hölder exponent $\alpha$ and the angle $2\pi\beta$ satisfy Condition (C). Define $\nabla_{KE}$ the Chern connection associated to the induced hermitian metric $h_{KE}$ on the tangent bundle $TX$ and $F_{KE}=F_{\nabla_{KE}}\in\Omega^{1,1}(End(T^{1,0}X))$ its curvature. We obtain an operator

[TABLE]

by identifying $T^{1,0}X$ to $\Lambda^{0,1}X$ using $\omega_{KE}$. Consequently, $\widehat{\omega}_{KE}(\Lambda_{\omega_{KE}}F_{{KE}})$ can be seen as an element in $\Omega^{1,1}(X)$. A local computation that remains valid outside of the divisor $D$, shows that

[TABLE]

Using the Kähler-Einstein property, the previous equation provides a metric on $TX$ which is Hermitian-Einstein metric outside $D$ and has $C^{2,\alpha,\beta}$ regularity as $Ric(\omega_{KE})$ is $C^{,\alpha,\beta}$ from Theorem 2.4.1. We now check that we obtain furthermore a parabolic structure for which this metric is compatible. We consider the canonical section $\sigma$ of $D$ that vanishes precisely on $D$. Now, as the Kähler form $\omega_{KE}$ has conical singularities along $D$, it is quasi-isometric to

[TABLE]

using local cone chart coordinates. Here the $z^{1}$ is the local defining function of the hypersurface $D=\{\sigma=0\}$ where $p$ locates, $a_{1}$ is a smooth function, $0<\beta<1/2$, and $\tilde{\omega}$ is a smooth form. Consequently, from (6.1), the curvature $|F_{KE}|_{h_{KE}}$ of the metric $h_{KE}$ lies in $L^{p}(X)$ for $p<\frac{2}{1-\beta}$ as

[TABLE]

for a uniform constant $C>0$, see for instance the proof of [MR1701135, Lemma 5.2]. This bound implies the following statement.

Lemma 6.1**.**

With above assumptions, there exists $C>0$ such that

[TABLE]

with $p>2$.

Theorem 5**.**

Assume $(X,\omega_{KE})$ is a compact Kähler manifold endowed with a Kähler-Einstein cone metric along a smooth divisor $D$, with Hölder exponent $\alpha$ and angle $2\pi\beta$ satisfying Condition (C). Then its tangent bundle $TX$ is parabolic polystable with respect to $\omega_{KE}$.

Proof.

Firstly, note that in the case of a curve or a surface, we can use a strong result of Biquard that shows an equivalence between the category of hermitian bundles on $X\setminus D$ with $L^{p}$ curvature and the category of holomorphic bundles on $X$ with parabolic structure over $D$, see [MR1168354, MR1373062]. Applying Lemma 6.1, we obtain that the tangent bundle can be extended over $D$ together with a parabolic structure along the divisor and such that the metric $h_{KE}$ is compatible with this parabolic structure. Moreover this extension is essentially unique. Now, using the Hermitian-Einstein condition and [LiJ, Theorem 6.3] or [Sim], we obtain the parabolic stability of each component of the tangent bundle if it is not indecomposable, i.e its polystability.

In general, we can adapt the construction of [LiJ, Section 3] as the bundle we are interested in is already defined over the divisor. We choose local holomorphic coordinates in a neighbourhood $U=\{|z^{i}|<1,i=1,..,n\}$ of the point $p=(0,...,0)$ such that the intersection with the divisor can be written $D\cap U=\{z^{1}=0\}$. We may choose a holomorphic basis $\{e_{i}\}_{i=1,..,r}$ of $E_{|U}$ such that the matrix of the metric $h_{KE}$ in this basis is diagonal. We write $D_{h_{KE}}$ this matrix. The matrix $D_{h_{KE}}$ necessarily vanishes on $U\cap D$ as the curvature $F_{KE}$ is singular and thus, fixing $\|.\|_{D}$ a norm on $\mathcal{O}(D)$, we can write $D_{h_{KE}}$ as

[TABLE]

Here $\zeta_{j}$ are positive smooth functions, the $\gamma_{j}$ are non-negative real numbers and (6.2) ensures that $\gamma_{j}<1$ for all $j$. Without loss of generality we can assume that the $\gamma_{j}$ is an increasing sequence, by doing a permutation of $\{e_{i}\}$. We denote $r_{j}$ the integers which count the numbers of equal $\gamma_{j}$, i.e that $r_{1}=\mathrm{rk}(TM)=n$ and $r_{j+1}$ is defined inductively by $\gamma_{l}=\gamma_{n-r_{j}+1}$ for $n-r_{j}+1\leq l\leq n-r_{j+1}$. Let $l_{E}$ be the number of different integers $r_{i}$. We define $\mathcal{F}_{|D\cap U}^{i}=Vect(e_{n-r_{i}+1},...,e_{n})$ for $i=2,..,l_{E}$ and $\alpha_{i}=\gamma_{n-r_{i+1}}$. Clearly, the data $(\mathcal{F}^{i},\alpha_{i})$ defines a parabolic structure for $E_{|D\cap U}$. If we consider another neighbourhood $U^{\prime}$ that intersects $U\cap D$, then one can find as above a basis $\{e^{\prime}_{i}\}_{i=1,..,r}$ of $E_{|U^{\prime}}$ such that the matrix of $h_{KE}$ is diagonal with diagonal entries $\zeta^{\prime}_{i}\|\sigma\|^{2\gamma^{\prime}_{i}}_{D}$ for $i=1,..,r$ with $\gamma^{\prime}_{i}$ increasing sequence and $\sigma$ the canonical section of $D$ which vanishes precisely on $D$. The vanishing order must be the same on $U\cap U^{\prime}\cap D\neq\emptyset$ which forces $\gamma_{i}=\gamma_{i}^{\prime}$ and consequently $\mathcal{F}_{|D\cap U\cap U^{\prime}}$ extends to $\mathcal{F}_{|D\cap U^{\prime}}$. Consequently we have defined a filtration by subbundles of $E_{|D}$ and thus a parabolic structure for $E$ along $D$. Then, following [LiJ, Section 3], the metric $h_{KE}$ is compatible with the structure and $TM$ is endowed with a Hermitian-Einstein cone metric. We conclude as above using Remark (4.1). ∎

7. Further Applications and Remarks

7.1. Simple normal crossings divisors

We expect that Theorems 1, 2, 3, 4, 5 and Corollary 1.1 could be generalized to the case of simple normal crossings divisors $D=\sum_{i=1}^{m}D_{i}$ where $D_{i}$ are irreducible. It is possible to define Hölder and Sobolev spaces for Kähler cone metrics $\omega$ with angles $2\pi\beta_{i}$ along $D_{i}$. The condition (C) would be replaced by the condition (C’)

[TABLE]

The statements of our results would remain identical under above changes. The only missing step is a Schauder estimate that extends Proposition 3.5. This has been announced recently in the preliminary work of Guo-Song [GS].

7.2. Twisted conical path for cscK metric

Donaldson introduced a continuity method for conical Kähler-Einstein metrics [MR2975584, Equation (27)] on Fano manifolds. A natural extension of this path for general polarizations is given by the following equation, that we call scalar curvature twisted conical path.

[TABLE]

where $c_{t}$ is a constant that depends on the time given in terms of topological invariants, $[\omega_{\varphi(t)}]=2\pi c_{1}(L)$ for $L$ ample line bundle on the projective manifold $X$. Note that this is a variant of the continuity method introduced by Chen [chen, Equation (2.16)]. Scalar curvature twisted conical path is expected to be helpful for the construction of smooth cscK metric as $t\to 1$. It is natural to ask if the set of times $t$ for which (7.1) admits a solution is open and non empty. As far as we know, the existence of a solution at an initial time $t_{0}>0$ is not known. Nevertheless, [MR3403730, Corollary 5.10] shows the existence of an invariant $\beta(X,D)$ such for $0<t_{0}<\beta(X,D)$, $(X,D)$ is log K-stable for angle $t_{0}$. If an initial solution does exist at time $t_{0}<\min(1/2,\beta(X,D))$, Theorem 1 applies and provides the openness property if $Lie(Aut_{D}(X,L))$ is trivial. This is similar to the smooth case, see [chen, Theorems 1.5 and 1.8].

7.3. Other applications of Theorem 1

The linear theory proved in this paper could be used to construct a large family of cscK cone metrics over blow-ups, extending the previous work of Arezzo-Pacard [MR2275832]. We also expect a generalization of the work of Fine (see [Fine] and subsequent works) for construction at the adiabatic limit of cscK cone metrics over a holomorphic submersion between compact Kähler manifolds $\pi:X\to B$ where the fibers and the base do not admit non trivial holomorphic vector fields and each fiber admits a cscK cone metric.

7.4. Generalizations of Theorem 2

In view of [Hong3, LS, Bro, MR2807093, LiH] it is natural to ask whether Theorem 2 admits a generalization in the case the base manifold $B$ admits nontrivial holomorphic vector fields. As in the smooth case, this will not be automatically happen, and an extra condition on $(B,D)$ will be required. A related question is about the existence of extremal Kähler cone metric when the bundle $E$ splits as sum of parabolic stable bundles of different slopes. These problems will be investigated in a forthcoming paper.

7.5. Log K-stability

We expect that a purely algebraic version of Theorem 2 holds, i.e under the same assumptions, one gets also log-K-stability of $(X,\mathcal{D},[\omega_{k}]=[k\pi^{*}\omega_{B}+\hat{\omega}_{E}])$ for large $k$. We refer to [MR2975584] and [MR3313212] for the notion of log-K-stability and the logarithmic version of Yau-Tian-Donaldson conjecture. In the particular case of vector bundles over a curve, by analogy to the smooth case, we expect the following to be true.

Conjecture 1**.**

Let $C$ be a complex curve endowed with a cscK cone metric $\omega_{cscK}$ along a divisor $D$. Let $E$ be a parabolic vector bundle over $C$, with parabolic structure along $D$. Let $X=\mathbb{P}(E^{*})\to C$ be its projectivisation and $\mathcal{D}=\pi^{-1}(D)$. The following three conditions are equivalent:

1. (i)

$X$* admits a cscK cone metric along $\mathcal{D}$ in any ample class $2\pi c_{1}({\mathcal{L}})$ on $X$;* 2. (ii)

$(X,\mathcal{D})$* is log-K-polystable for any polarization ${\mathcal{L}}$ on $X$;* 3. (iii)

$E$* is parabolic polystable with respect to $\omega_{cscK}$, i.e it decomposes as the sum of stable parabolic bundles of same parabolic slopes.*

Before we provide some information on this conjecture, let us mention that from the work of Troyanov we have an algebraic characterization of complex curves that can be endowed with a cscK cone metric, see [MR1005085] and [MR3313212] for the relation with log K-stability.

(iii)$\Rightarrow$(i). At the boundary of the Kähler cone and when $E$ is irreducible, on can invoke Theorem 2 which provides a more precise result than [Ke1] in terms of regularity of the cscK cone metric. Nevertheless the implication (iii)$\Rightarrow$(i) is true in general as soon as condition (C) holds. Actually, starting from $\omega_{B}$ cscK cone metric and $h_{E}$ Hermitian-Einstein metric on $E$, the Kähler cone metric $\omega_{k}$ given by (5.3) has constant scalar curvature outside $\mathcal{D}$, like in the smooth case (to check this fact one can also refine Lemma 5.2 by obtaining a complete expansion when the base has dimension 1). Moreover $\omega_{k}$ has $C^{2,\alpha,\beta}$ potential from Lemma 5.1 and thus is a genuine cscK cone metric. Note that for this reasoning we don’t need to assume $k>>0$ and that when the base is a curve, any Kähler class on $X$ is of the form $[\omega_{k}]$.

(i)$\Leftrightarrow$(ii). Hashimoto [Hasfutaki] provides essentially the equivalence for a very particular bundle $E$ and $C=\mathbb{P}^{1}$. Remark that the considered bundle $E$ can be made parabolic polystable by using the techniques [Ke1]. Moreover, Corollary 2.3 gives evidence that the implication (i)$\Rightarrow$(ii) holds as product test configurations correspond to holomorphic vector fields with a holomorphy potential.

(ii)$\Rightarrow$(iii). A weaker version is shown in terms of asymptotic Chow polystability with angle in [Ke1].

Acknowledgments. The work of J. Keller has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). J. Keller is also partially supported by supported by the ANR project EMARKS, decision No ANR-14-CE25-0010. J. Keller thanks O. Biquard and P. Eyssidieux.

The work of K. Zheng has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 703949, and was also partially supported by the Engineering and Physical Sciences Research Council (EPSRC) on a Program Grant entitled Singularities of Geometric Partial Differential Equations reference number EP/K00865X/1.