On fibration stability after Dervan-Sektnan and singularities

Abstract

We introduce f-stability, a modification of fibration stability of Dervan-Sektnan [12], and show that f-semistable fibrations have only semi log canonical singularities. Moreover, f-stability puts restrictions on semi log canonical centers on Fano fibrations. Contents 1. Introduction 1 Outline of this paper 4 Acknowledgements 4 2. Notation 4 3. Review of Basic Results on K-stability and singularities 9 4. Fibration stability after Dervan-Sektnan 11 5. Singularities of semistable algebraic fiber spaces 18 5.1. Normal case 18 5.2. Non-normal case 22 References 26

On K-stability of Calabi-Yau fibrations

“…Then, A (X,B+T 1 ) (v) = A (X,B) (v) < 0 for general divisor T 1 ∼ Q T . Therefore, the theorem follows from the proof of [45,Theorem 5.1]. The rest also follows from the same argument of [45,Theorem 5.1] and is left to the reader.…”

“…In this section, we define K-stability of polarized log pairs as [11]. For deminormal schemes, we follow the definitions of test configuration, J-stability and K-stability from [45].…”

“…Here, we will denote H + f * L by H + L for simplicity. See the detail in Notation after [45,Remark 2.17].…”

“…By the generic smoothness [42, III Corollary 10.7], Y has reduced fibers and Y 0 + ρ * Y Θ A 1 is simple normal crossing over the generic points of the irreducible components of the central fiber C 0 over 0 ∈ A 1 . By [45,Proposition 5.7], there exists a G m -equivariant birational morphism ϕ : C ′ → C of test configurations such that ϕ is an isomorphism in codimension 1 and irreducible components of central fiber…”

Canonical metrics on holomorphic fibre bundles

“…It was observed by Dervan-Sektnan [DS21c] that in the non-isotrivial but smooth Fano case, the optimal symplectic connection equation still admits a simplification to a Hermite-Einsteintype equation. Recent work on the algebraic side by Hattori [Hat22] indicates that stable singular fibrations have mild singularities which may make the study of optimal symplectic connections with singular fibres tractable, and in the case where the smooth locus is isotrivial, one may hope to relate such singular fibrations to singular principal bundles, and singular Hermite-Einstein metrics.…”