Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2208.13367

Kähler-Einstein metrics and obstruction flatness of circle bundles

Peter Ebenfelt·Ming Xiao·Hang Xu
View original source ↗Read source document ↗Export metadata ↓

Abstract

Obstruction flatness of a strongly pseudoconvex hypersurface $Σ$ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of $Σ$, complete up to $Σ$, has a potential $-\log u$ such that $u$ is $C^\infty$-smooth up to $Σ$. In general, $u$ has only a finite degree of smoothness up to $Σ$. In this paper, we study obstruction flatness of hypersurfaces $Σ$ that arise as unit circle bundles $S(L)$ of negative Hermitian line bundles $(L, h)$ over Kähler manifolds $(M, g).$ We prove that if $(M,g)$ has constant Ricci eigenvalues, then $S(L)$ is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and $(M,g)$ is complete, then we show that the corresponding disk bundle admits a complete Kähler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of $S(L)$ when $(M, g)$ is a Kähler surface $(\dim M=2$) with constant scalar curvature.

Explore related subjects

math.CVmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Peter Ebenfelt, Ming Xiao, Hang Xu. 2022-08-29. Kähler-Einstein metrics and obstruction flatness of circle bundles. https://arxiv.org/abs/2208.13367

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Explicit Estimates for the Bergman Kernel Form

Let $(L,e^{-ϕ})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and let $ω=i\partial\overline{\partial}ϕ$. We obtain explicit pointwise estimates for the Bergman form of the tensor power $mL$. If $\mathrm{Ric}\,ω\leqω$ and the shortest nonconstant closed geodesic has length at least $2π$, then \[ K_{mϕ}\geq \frac{2m-1}{4π}\,ω, \] with sharpness holding for $(\mathbb P^1,\mathcal O_{\mathbb P^1}(2))$. We also obtain a local version, depending on an upper curvature bound and the injectivity radius, which recovers the first two terms of the Bergman expansion when the curvature is constant. We also find a higher dimensional version. Under the two-sided bound $-ω\leq\mathrm{Ric}\,ω\leqω$ and the same closed-geodesic hypothesis, we also prove \[ K_{mϕ}\leq \frac{mω}{2π} \left(1+\frac{3}{2m}\right). \] The lower estimates use the deformation to the tangent space version of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound via Błocki--Zwonek and isoperimetric inequalities.

math.CV↗
arXiv2026

The Complete Crouzeix Conjecture in Dimension Three and the Clouâtre-Ostermann-Ransford conjecture

We settle the complete Crouzeix conjecture for matrices of order at most three and prove the Q-algebra case of the completely bounded Clouâtre--Ostermann--Ransford (COR) conjecture for homomorphisms into matrices of order at most three, via sharp abstract column and row estimates. Our approach also establishes the scalar COR conjecture in a stronger form, for homomorphisms with commutative range on Banach algebras with unity satisfying von Neumann's inequality. Under contractivity of the symmetrized map, this result holds on arbitrary Hilbert spaces without initial boundedness assumptions on the homomorphism or the antilinear map. For arbitrary operator algebras, we disprove the complete COR conjecture by an exact three-dimensional example with target matrix order two. We also prove the sharp complete bound for every matrix subalgebra containing the diagonal, in arbitrary matrix order and on arbitrary target Hilbert spaces. We also obtain column and row square-function inequalities with sharp norm bounds, strict scalar bounds for operators similar to normal operators, sharp complete spectral constants for scaled $q$-numerical ranges in dimensions two and three, and rigidity, stability, and representing-measure results.

math.CV↗
arXiv2026

Solving non-oscillatory solutions of the Hill equation via the Tumura--Clunie method

We consider the Hill equation $f''-(\sum_{i=-\mathbf{l}}^{\mathbf{k}}b_{i}e^{iz})f=0$ ($†$), where $\mathbf{k}\geq 1$ and $\mathbf{l}\geq 0$ are integers and $b_{-\mathbf{l}}$, $\cdots$, $b_{\mathbf{k}}$ are constants such that $b_{\mathbf{k}}\not=0$. We point out that there is a full correspondence between the class of non-oscillatory solutions such that $λ(f)<\infty$ of equation ($†$) and the class of Liouvillian solutions of equation $x^2u''-(\sum_{i=-\mathbf{l}}^{\mathbf{k}}b_{i}x^{i})u=0$ ($‡$). Then this paper has twofold purposes. First, parallel to Kovacic's algorithms to find the Liouvillian solutions of equation ($‡$), we develop the Tumura--Clunie method to find the non-oscillatory solutions of a higher order version of the Hill equation. Second, for the particular Hill equation $f''-(b_{\mathbf{k}}e^{\mathbf{k}z}+b_{\mathbf{s}}e^{\mathbf{s}z}+b_0)f=0$, where $\mathbf{k}>\mathbf{s}\geq 1$ are integers and $b_{\mathbf{k}}b_{\mathbf{s}}\not=0$, we use the Tumura--Clunie method to determine the non-oscillatory solution $f$ with an additional zero property.

math.CV↗