Derived equivalence and fibrations over curves and surfaces

TL;DR

This paper demonstrates that the derived category of a smooth projective variety uniquely determines certain fibrations onto curves and surfaces, revealing deep links between derived categories and geometric structures.

Contribution

It establishes that the derived category reconstructs fibrations onto curves of genus ≥2 and, in low dimensions, onto certain surfaces, extending the understanding of derived invariance in algebraic geometry.

Findings

Derived category reconstructs fibrations onto curves of genus g ≥ 2.

In dimension ≤4, it reconstructs fibrations onto specific surfaces with positive Euler characteristic.

Shows derived invariance of certain fibrations under Hodge number conditions.

Abstract

We prove that the bounded derived category of coherent sheaves on a smooth projective complex variety reconstructs the isomorphism classes of fibrations onto smooth projective curves of genus $g\geq 2$. Moreover, in dimension at most four, we prove that the same category reconstructs the isomorphism classes of fibrations onto normal projective surfaces with positive holomorphic Euler characteristic and admitting a finite morphism to an abelian variety. Finally, we study the derived invariance of a class of fibrations with minimal base-dimension under the condition that all the Hodge numbers of type $h^{0,p}(X)$ are derived invariant.