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Geodesic-Einstein metrics and Nonlinear Stabilities
Abstract
In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional to show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relationships between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.
... [12][13][14]26,34]). On the other hand, in the paper [19], we introduced the notion of the geodesic-Einstein metric on a relative ample line bundle L over X . For any holomorphic vector bundle E, there is a canonical associated projective bundle pair (P(E), O P(E) (1)). ...
... In this case, a geodesic-Einstein metric is same as a Finsler-Einstein metric by the natural one-to-one correspondence between Finsler metrics on E and the metrics on the tautological line bundle O P(E) (−1) (cf. [23, p. 82], [19]). Combining with [19,Theorem 0.3], it shows that the existence of geodesic-Einstein metrics is equivalent to the existence of Hermitian-Einstein metrics, which is also equivalent to polystability of the holomorphic vector bundle. ...
... [23, p. 82], [19]). Combining with [19,Theorem 0.3], it shows that the existence of geodesic-Einstein metrics is equivalent to the existence of Hermitian-Einstein metrics, which is also equivalent to polystability of the holomorphic vector bundle. ...
In this paper, we introduce the associated geodesic-Einstein flow for a relative ample line bundle L over the total space \(\mathcal {X}\) of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair \((\mathcal {X}, L)\) is nonlinear semistable if the associated Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time existence property. We also define the associated S-classes and C-classes for \((\mathcal {X}, L)\) and obtain two inequalities between them when L admits a geodesic-Einstein metric. Finally, in the appendix of this paper, we prove that a relative ample line bundle is geodesic-Einstein if and only if an associated infinite rank bundle is Hermitian–Einstein.
... Other generalizations of Theorem 1.1 include generalizations to compact Hermitian manifolds ([Buc99; LY87]), and very recent work of Feng-Liu-Wan [FLW18], which expanded Theorem 1.1 to include the existence of Finsler-Einstein metrics. ...
We give new proofs of two implications in the Donaldson--Uhlenbeck--Yau theorem. Our proofs are based on geodesic rays of Hermitian metrics, inspired by recent work on the Yau--Tian--Donaldson conjecture.
... The Poisson-Kähler condition is in general stronger than the geodesic-Einstein condition in [21,43]. It is known that (see [4,40]) every relative Kähler fibration is Poisson-Kähler locally in the following sense: for every t ∈ B, there exists a small open neighborhood U of t such that the fibration from p −1 (U) to U possesses a Poisson-Kähler structure. ...
We start from a finite dimensional Higgs bundle description of a result of Burns on negative curvature property of the space of complex structures, then we apply the corresponding infinite dimensional Higgs bundle picture and obtain a precise curvature formula of a Weil--Petersson type metric for general relative K\"ahler fibrations. In particular, our curvature formula implies a Burns type negative curvature property of the base manifold for a special class of maximal variation K\"ahler fibrations (named Poisson--K\"ahler fibrations)
... Lemma 2.1 [17] The following decomposition holds, ...
Let $\pi:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relative ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle $\pi_*(L^k+K_{\mathcal{X}/M})$ up to the lower order terms than $k^{n-1}$, for large $k$. As an application we prove that the analytic torsion $\tau_k(\bar{\partial})$ satisfies $\partial\bar{\partial}\log(\tau_k(\bar{\partial}))^2=o(k^{n-1})$, where $n$ is the dimension of fibers.
We consider harmonic maps u(z):Xz→N in a fixed homotopy class from Riemann surfaces Xz of genus g≥2 varying in the Teichmüller space T to a Riemannian manifold N with non-positive Hermitian sectional curvature. The energy function E(z)=E(u(z)) can be viewed as a function on T and we study its first and the second variations. We prove that the reciprocal energy function E(z)−1 is plurisuperharmonic on Teichmüller space. We also obtain the (strict) plurisubharmonicity of logE(z) and E(z). As an application, we get the following relationship between the second variation of logarithmic energy function and the Weil-Petersson metric if the harmonic map u(z) is holomorphic or anti-holomorphic and totally geodesic, i.e.,(0.1)−1∂∂¯logE(z)=ωWP2π(g−1). We consider also the energy function E(z) associated to the harmonic maps from a fixed compact Kähler manifold M to Riemann surfaces {Xz}z∈T in a fixed homotopy class. If u(z) is holomorphic or anti-holomorphic, then (0.1) is also proved.
We consider harmonic maps$u(z): \mathcal{X}_z\to N$ in a fixed homotopy class from Riemann surfaces $\mathcal{X}_z$ of genus $g\geq 2$ varying in the Teichm\"u{}ller space $\mathcal T$ to a Riemannian manifold $N$ with non-positive Hermitian sectional curvature. The energy function $E(z)=E(u(z))$ can be viewed as a function on $\mathcal T$ and we study its first and the second variations. We prove that the reciprocal energy function $E(z)^{-1}$ is plurisuperharmonic on Teichm\"uller space. We also obtain the (strict) plurisubharmonicity of $\log E(z)$ and $E(z)$. As an application, we get the following relationship between the second variation of logarithmic energy function and the Weil-Petersson metric if the harmonic map $u(z)$ is holomorphic or anti-holomorphic and totally geodesic, i.e., $$ \sqrt{-1}\p\b{\p}\log E(z)=\frac{\omega_{WP}}{2\pi(g-1)}. $$ We consider also the energy function $E(z)$ associated to the harmonic maps from a fixed compact K\"ahler manifold $M$ to Riemann surfaces ${\mathcal{X}_z\}_{z\in\mathcal{T}}$ in a fixed homotopy class. If $u(z)$ is holomorphic or anti-holomorphic, then the above equation is also proved.
In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we show a Schwarz Lemma from a compete Riemann manifold to a complex Finsler manifold. We also prove that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identically zero holomorphic sectional curvature has negative Kodaira dimension under an extra condition.
In this paper, we solve a problem of Kobayashi posed in \cite{Ko4} by
introducing a Donaldson type functional on the space $F^+(E)$ of strongly
pseudo-convex complex Finsler metrics on $E$ -- a holomorphic vector bundle
over a closed K\"ahler manifold $M$. This Donaldson type functional is a
generalization in the complex Finsler geometry setting of the original
Donaldson functional and has Finsler-Einstein metrics on $E$ as its only
critical points, at which this functional attains the absolute minimum.
Let $p:\sXS$ be a proper K\"ahler fibration and $\sE\sX$ a Hermitian
holomorphic vector bundle.
As motivated by the work of Berndtsson(\cite{Berndtsson09a}), by using basic
Hodge theory, we derive several general curvature formulas for the direct image
$p_*(K_{\sX/S}\ts \sE)$ for general Hermitian holomorphic vector bundle $\sE$
in a simple way.
A straightforward application is that, if the family $\sXS$ is
infinitesimally trivial and Hermitian vector bundle $\sE$ is Nakano-negative
along the base $S$, then the direct image $p_*(K_{\sX/S}\ts \sE)$ is
Nakano-negative. We also use these curvature formulas to study the moduli space
of projectively flat vector bundles with positive first Chern classes and
obtain that, if the Chern curvature of direct image $p_*(K_{X}\ts E)$--of a
positive projectively flat family $(E,h(t))_{t\in \mathbb D}X$--vanishes, then
the curvature forms of this family are connected by holomorphic automorphisms
of the pair $(X,E)$.
Given a family $f:\mathcal X \to S$ of canonically polarized manifolds, the
unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the
relative canonical bundle $\mathcal K_{\mathcal X/S}$. We use a global elliptic
equation to show that this metric is strictly positive on $\mathcal X$, unless
the family is infinitesimally trivial.
For degenerating families we show that the curvature form on the total space
can be extended as a (semi-)positive closed current. By fiber integration it
follows that the generalized Weil-Petersson form on the base possesses an
extension as a positive current. We prove an extension theorem for hermitian
line bundles, whose curvature forms have this property. This theorem can be
applied to a determinant line bundle associated to the relative canonical
bundle on the total space. As an application the quasi-projectivity of the
moduli space $\mathcal M_{\text{can}}$ of canonically polarized varieties
follows.
The direct images $R^{n-p}f_*\Omega^p_{\mathcal X/S}(\mathcal K_{\mathcal
X/S}^{\otimes m})$, $m > 0$, carry natural hermitian metrics. We prove an
explicit formula for the curvature tensor of these direct images. We apply it
to the morphisms $S^p \mathcal T_S \to R^pf_*\Lambda^p\mathcal T_{\mathcal
X/S}$ that are induced by the Kodaira-Spencer map and obtain a differential
geometric proof for hyperbolicity properties of $\mathcal M_{\text{can}}$.
Inspired by Berndtsson's results \cite{Bern06} and \cite{Bern09}, we shall
define the curvature operator $\Theta_{j\bar k}$ for a family of weighted
Bergman spaces associated to a family of smoothly bounded strongly pseudoconvex
domains $\{D_t\}$. What's more, we shall introduce the notion of geodesic
curvature, say $\theta_{j\bar k}(\rho)$, for $\{D_t\}$ and use it to decode the
boundary term of the curvature formula. In particular, we get the
$\partial\dbar$-Bochner type variation formula for the Bergman projection. A
flatness criterion for $\Theta_{j\bar k}$ and its applications are also given
in this paper.
Let $X$ be a K\"ahler manifold which is fibered over a complex manifold $Y$
such that every fiber is a Calabi-Yau manifold. Let $\omega$ be a fixed
K\"ahler form on $X$. By the theorem due to Yau, there exists a unique
Ricci-flat K\"ahler form $\rho\vert_{X_y}$ for each fiber, which is
cohomologous to $\omega\vert_{X_y}$. This family of Ricci-flat K\"ahler forms
$\rho\vert_{X_y}$ induce a smooth $(1,1)$-form $\rho$ on $X$. In this paper, we
prove that $\rho$ is semi-positive on the total space $X$. We also discuss
several byproducts, among them the local triviality of families of Calabi-Yau
manifolds.
In this paper, we present two kinds of total Chern forms $c(E,G)$ and
$\mathcal{C}(E,G)$ as well as a total Segre form $s(E,G)$ of a holomorphic
Finsler vector bundle $\pi:(E,G)\to M$ expressed by the Finsler metric $G$,
which answers a question of J. Faran (\cite{Faran}) to some extent. As some
applications, we show that the signed Segre forms $(-1)^ks_k(E,G)$ are positive
$(k,k)$-forms on $M$ when $G$ is of positive Kobayashi curvature; we prove,
under an extra assumption, that a Finsler-Einstein vector bundle in the sense
of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler
metric, which is weaker than Aikou's one (\cite{Aikou}) and prove that a
holomorphic vector bundle is Finsler flat in our sense if and only if it is
Hermitian flat.
Donaldson conjectured [16] that the space of Kähler metrics is geodesically convex by smooth geodesics and that it is a metric space. Following Donaldson's program, we verify the second part of Donaldson's conjecture completely and verify his first part partially. We also prove that the constant scalar curvature metric is unique in each Kähler class if the first Chern class is either strictly negative or 0. Furthermore, if C1 ≤ 0, the constant scalar curvature metrics: realizes the global minimum of the Mabuchi K energy functional; thus it provides a new obstruction for the existence of constant curvature metrics: if the infimum of the K energy (taken over all metrics in a fixed Kähler class) is not bounded from below, then there does not exist a constant curvature metric. This extends the work of Mabuchi and Bando [3]: they showed that K energy bounded from below is a necessary condition for the existence of Kähler-Einstein metrics in the first Chern class.
We develop some results from [4] on the positivity of direct image bundles in the particular case of a trivial vibration over a one-dimensional base. We also apply the results to study variations tions of Kähler metrics.
This paper is a sequel to \cite{Berndtsson}. In that paper we studied the
vector bundle associated to the direct image of the relative canonical bundle
of a smooth K\"ahler morphism, twisted with a semipositive line bundle. We
proved that the curvature of a such vector bundles is always semipositive (in
the sense of Nakano). Here we adress the question if the curvature is strictly
positive when the Kodaira-Spencer class does not vanish. We prove that this is
so provided the twisting line bundle is stricty positive along fibers, but not
in general.
Let X be a complex manifold fibered over the base S and let L be a relatively
ample line bundle over X. We define relative Kahler-Ricci flows on the space of
all Hermitian metrics on L with relatively positive curvature. Mainly three
different settings are investigated: the case when the fibers are Calabi-Yau
manifolds and the case when L is the relative (anti-) canonical line bundle.
The main theme studied is whether posivity in families is preserved under the
flows and its relation to the variation of the moduli of the complex structures
of the fibres. The quantization of this setting is also studied, where the role
of the Kahler-Ricci flow is played by Donaldson's iteration on the space of all
Hermitian metrics on the finite rank vector bundle over S defined as the zeroth
direct image induced by the fibration. Applications to the construction of
canonical metrics on relative canonical bumdles and Weil-Petersson geometry are
given. Some of the main results are a parabolic analogue of a recent elliptic
equation of Schumacher and the convergence towards the K\"ahler-Ricci flow of
Donaldson's iteration in a certain double scaling limit.
A complex Finsler structure F on a complex manifold M is a function on the tangent bundle T(M) with the following properties. (We denote a point of T(M) symbolically by (z, ζ), where z represents the base coordinate and ζ the fibre coordinate.)
We define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a Kahler metric of constant scalar curvature. The main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces. We prove that, under the stability hypothesis, the Mabuchi functional is bounded below on invariant metrics, and that minimising sequences have a certain convergence property. In the reverse direction, we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature. The proofs use a general framework, developed by Guillemin and Abreu, in which invariant Kahler metrics correspond to convex functions on the moment polytope of a toric variety.
Let $L$ be a (semi)-positive line bundle over a Kahler manifold, $X$, fibered
over a complex manifold $Y$. Assuming the fibers are compact and non-singular
we prove that the hermitian vector bundle $E$ over $Y$ whose fibers over points
$y$ are the spaces of global sections over $X_y$ to $L\gr K_{X/Y}$ endowed with
the $L^2$-metric is (semi)-positive in the sense of Nakano. We also discuss
various applications, among them a partial result on a conjecture of Griffiths
on the positivity of ample bundles. This is a revised and much expanded version
of a previous preprint with the title `` Bergman kernels and the curvature of
vector bundles''.
Complex Finsler Vector Bundles
- S Kobayashi
S. Kobayashi, Complex Finsler Vector Bundles. Contemp. Math., vol. 196, pp. 133C144. American Mathematical Society, Providence, RI (1996).