# Relative Severi inequality for fibrations of maximal Albanese dimension over curves

## TL;DR

This paper establishes a new inequality relating the relative canonical divisor and the Euler characteristic for fibrations of maximal Albanese dimension over curves, confirming a conjecture and providing new insights into Severi inequalities.

## Contribution

It proves the conjectured inequality for fibrations of maximal Albanese dimension and derives a new proof of the Severi inequality, also characterizing cases of equality.

## Key findings

- Proves $K_{X/B}^n \,\ge\, 2n! \chi_f$ for fibrations of maximal Albanese dimension.
- Shows that equality implies the general fiber satisfies the Severi equality.
- Provides sharper results under additional assumptions.

## Abstract

Let $f: X \to B$ be a relatively minimal fibration of maximal Albanese dimension from a variety $X$ of dimension $n \ge 2$ to a curve $B$ defined over an algebraically closed field of characteristic zero. We prove that $K_{X/B}^n \ge 2n! \chi_f$, which was conjectured by Barja in [2]. Via the strategy outlined in [5], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and $\chi_f > 0$, we prove that the general fiber $F$ of $f$ has to satisfy the Severi equality that $K_F^{n-1} = 2(n-1)! \chi(F, \omega_F)$. We also prove some sharper results of the same type under extra assumptions.