Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.27800

Canonical metric on a mildly singular Kähler varieties with an intermediate log Kodaira dimension

Hassan Jolany
View original source ↗Read source document ↗Export metadata ↓

Abstract

Existence of canonical metric on a canonical model of projective singular variety was a long standing conjecture and the major part of this conjecture is about varieties which do not have definite first Chern class (most of the varieties do not have definite first Chern class). There is a program which is known as Song-Tian program for finding canonical metric on canonical model of a projective variety by using Minimal Model Program. In this paper, we apply Song-Tian program for mildly singular pair $(X;D)$ via Log Minimal Model Program where $D$ is a simple normal crossing divisor on $X$ with conic singularities. We show that there is a unique $C^\infty$-fiberwise conical Kähler-Einstein metric on $(X;D)$ with vanishing Lelong number which is twisted by logarithmic Weil-Petersson metric and an additional term of Fujino-Mori [72] as soon as we have fiberwise KE-stability or Kawamata's condition of Theorems 2.28, or 2.30(in $C^0$ case). In final we highlight that how the complete answer of this question can be reduced to CMA equation corresponding to fiberwise Calabi-Yau foliation(due to H.Tsuji) which is still open.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hassan Jolany. 2026-08-17. Canonical metric on a mildly singular Kähler varieties with an intermediate log Kodaira dimension. https://arxiv.org/abs/2609.27800

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

KEEP EXPLORING

Related papers

arXiv2026

Curvature equations coupling symmetric tensors with a metric

There are described hierarchies of equations coupling a metric with a trace-free tensor having prescribed symmetries and in the kernel of certain generalized gradients. These specialize, when the tensor vanishes identically, to the usual hierarchy of constant sectional curvature (projectively flat), Einstein, and constant scalar curvature. At the Ricci curvature level these equations are formal analogues of the Einstein-Maxwell and supergravity equations that couple differential forms with a metric. The particular cases coupling a metric with trace-free symmetric tensors satisfying the Codazzi or conformal Killing equations are studied in detail. Examples of solutions are obtained from mean curvature zero immersions, affine spheres, isoparametric hypersurfaces, and related algebraic constructions. The formalism yields a hierarchy of curvature equations for statistical structures. There are deduced constraints on the scalar curvature of the metric occurring in a solution that generalize classical results of Simons, for mean curvature zero hypersurfaces in spheres, and of Calabi, for hyperbolic affine spheres.

math.DG↗
arXiv2026

The signature of geometrically decomposable aspherical 4-manifolds

We construct examples of geometrically decomposable aspherical 4-manifolds with non-zero signature. We show that all such 4-manifolds satisfy the inequality (of Bogomolov--Miyaoka--Yau type) $χ\geq 3|σ|$. We also construct examples attaining the equality that are non-geometric and have non-zero signature. Finally, we prove that for higher graph 4-manifolds, with complex-hyperbolic vertices, the strict inequality always holds. Moreover, we construct infinitely many examples of higher graph 4-manifolds with non-zero signature and prove that the inequality is strict and sharp in this class.

math.DG↗
arXiv2026

Proper affine deformations of positive representations

We define for every positive Anosov representation of a nonabelian free group into $\mathrm{SO}(2n,2n-1)$ a family of $\mathbb{R}^{4n-1}$-valued cocycles which induce proper affine actions on $\mathbb{R}^{4n-1}$. We construct fundamental domains in $\mathbb{R}^{4n-1}$ bounded by generalized crooked planes for these affine actions, and deduce that the quotient manifolds are homeomorphic to handlebodies.

math.DG↗