Irregular fibrations of derived equivalent varieties

TL;DR

This work examines how irregular fibrations, defined as surjective morphisms onto bases of positive dimension with maximal Albanese-dimension desingularizations, behave under derived equivalence of smooth projective varieties. The authors introduce and exploit Rouquier-stable subvarieties in Pic$^0$, along with their associated base morphisms and Stein factorizations, to prove that for a pair ${\mathbf D}(X) \simeq {\mathbf D}(Y)$, irregular fibrations of $X$ over bases of general type correspond to fibrations on $Y$ with birationally equivalent bases and fiberwise derived-equivalent general fibers; moreover, if the general fiber has big (anti)canonical, the varieties are $K$-equivalent. The results generalize Kawamata’s birational reconstruction and extend prior work on irrational pencils to higher-dimensional irregular fibrations, providing a robust framework (via Rouquier stability) to transfer fibration data across Fourier–Mukai partners. They also establish a base-preserving bijection between equivalence classes of irregular fibrations of derived-equivalent varieties, highlighting deep structural links between the derived category and the birational geometry of fibrations.

Abstract

We study the behavior of irregular fibrations of a variety under derived equivalence of its bounded derived category. In particular we prove the derived invariance of the existence of an irregular fibration over a variety of general type, extending the case of irrational pencils onto curves of genus $g\geq 2$. We also prove that a derived equivalence of such fibrations induces a derived equivalence between their general fibers.

Theorems & Definitions (26)

• Theorem 1.0.1
• Proposition 1.0.2
• Theorem 1.0.3
• Corollary 1.0.4
• Theorem 1.0.5
• Theorem 1.0.6
• proof
• Definition 2.0.1
• Theorem 3.0.1
• proof