arXiv subjects

Alexander Bednarek

Publications and source records attributed to Alexander Bednarek.

4 recordsLinked to original sources

arXiv2026

On Miyaoka-Yau Inequalities and Weil-Petersson Metrics

We use the Kähler-Ricci flow to give alternate proofs of several known Miyaoka-Yau inequalities and slope semi-stabilities, in particular, for the case of compact Kähler manifolds with semi-ample canonical line bundles and K-semistable Fano manifolds. Moreover, when the canonical line bundle is semi-ample, and the Kodaira dimension is $n-1$, we prove the Miyaoka-Yau quantity is equal to the intersection of the Weil-Petersson metric and the Fubini-Study metric on the canonical model. Consequently, equality holds in the Miyaoka-Yau inequality if and only if the pluricanonical map is a holomorphic fibre bundle.

math.DG↗

arXiv2025

Fano Fibrations and Twisted Kähler-Einstein Metrics I

This is the first of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. Given a Fano fibration which is generated by Kawamata's theorem from a compact Kähler manifold $X$ endowed with an ample, rational line bundle $L$ and non-nef canonical line bundle $K_X$, we construct a $(1,1)$-form on the regular part of the base analytic variety which is related to the Weil-Petersson metric. It is also proven that the singular Kähler metric constructed by Zhang, Zhang, on the base analytic variety satisfies a twisted Kähler-Einstein equation involving this $(1,1)$-form and, for a submersion, that the Chern classes of $X$ and the base manifold decompose in terms of this $(1,1)$-form.

math.DG↗

arXiv2025

Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow

This is the second of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. We assume that the Kähler-Ricci flow on a compact Kähler manifold has a rational initial metric and develops a singularity in finite time such that the manifold admits a Fano fibration structure. Moreover, it is assumed that the volume form of the flow collapses uniformly at the rate of $C^{-1}(T-t)^{n-m} Ω\leq ω(t)^n\leq C(T-t)^{n-m}Ω$. Under this setting, a diameter bound is obtained in any compact set away from singular fibres and the diameter of the fibres is proven to collapse at the optimal rate $\sqrt{T-t}$. Furthermore, several precise $C^0$-estimates are proven for the potential of the complex Monge-Ampere flow which involve the potentials of singular twisted Kähler-Einstein metrics on the base variety from part I. Finally, in the case of Kähler-Einstein Fano fibres, we deduce Type I scalar curvature in any compact set away from singular fibres and globally for a submersion.

math.DG↗

arXiv2024

Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities

We consider the Kähler-Ricci flow $(X, ω(t))_{t \in [0,T)}$ on a compact manifold where the time of singularity, $T$, is finite. We assume the existence of a holomorphic map from the Kähler manifold $X$ to some analytic variety $Y$ which admits a Kähler metric on a neighbourhood of the image of $X$ and that the pullback of this metric yields the limiting cohomology class along the flow. This is satisfied, for instance, by the assumption that the initial cohomology class is rational, i.e., $[ω_0] \in H^{1,1}(X,\mathbb{Q})$. Under these assumptions we prove an $L^4$-like estimate on the behaviour of the Ricci curvature and that the Riemannian curvature is Type $I$ in the $L^2$-sense.

math.DG↗