arXiv subjects

Martin de Borbon

Publications and source records attributed to Martin de Borbon.

24 records · Page 2Linked to original sources

arXiv2018

ALE Calabi-Yau metrics with conical singularities along a compact divisor

We construct ALE Calabi-Yau metrics with cone singularities along the exceptional set of resolutions of $\mathbb{C}^n / Γ$ with non-positive discrepancies. In particular, this includes the case of the minimal resolution of two dimensional quotient singularities for any finite subgroup $Γ\subset U(2)$ acting freely on the three-sphere, hence generalizing Kronheimer's construction of smooth ALE gravitational instantons. Finally, we show how our results extend to the more general asymptotically conical setting.

math.DG↗

arXiv2018

Local models for conical Kähler-Einstein metrics

In this note we use the Calabi ansatz, in the context of metrics with conical singularities along a divisor, to produce regular Calabi-Yau cones and Kähler-Einstein metrics of negative Ricci with a cuspidal point. As an application, we describe singularities and cuspidal ends of the completions of the complex hyperbolic metrics on the moduli spaces of ordered configurations of points in the projective line introduced by Thurston and Deligne-Mostow.

math.DG↗

arXiv2017

Asymptotically conical Ricci flat Kähler metrics on $\mathbb{C}^2$ with cone singularities along a complex curve

We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.

math.DG↗

arXiv2017

The Gibbons-Hawking ansatz over a wedge

We discuss the Ricci-flat `model metrics' on $\mathbb{C}^2$ with cone singularities along the conic $\{zw=1\}$ constructed by Donaldson using the Gibbons-Hawking ansatz over wedges in $\mathbb{R}^3$. In particular we describe their asymptotic behavior at infinity and compute their energies.

math.DG↗

arXiv2016

Kähler metrics with cone singularities along a divisor of bounded Ricci curvature

Let $D$ be a smooth divisor in a compact complex manifold $X$ and let $β\in (0,1)$. We show that in any positive co-homology class on $X$ there is a Kähler metric with cone angle $2πβ$ along $D$ which has bounded Ricci curvature. We use this result together with the Aubin-Yau continuity method to give an alternative proof of a well-known existence theorem for Kahler-Einstein metrics with cone singularities.

math.DG↗

arXiv2016

Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities.

math.DG↗