# 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.

## Key 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.