On birational trivial families and Adjoint quadrics

TL;DR

This paper establishes criteria for when fibers in certain polarized families of Abelian varieties are birationally equivalent, utilizing liftability of specific differential forms and applying Nori's theorem for notable applications.

Contribution

It introduces general criteria for birational triviality in families of Abelian varieties based on form liftability, connecting to Nori's theorem for new applications.

Findings

Fibers are birationally equivalent if certain forms are liftable.

Provides criteria for birational triviality in polarized Abelian families.

Connects form liftability with Nori's theorem for applications.

Abstract

Let $\pi\colon \mathcal{X}\to B$ be a family whose general fiber $X_b$ gives a $(d_1,...,d_a)$ polarisation of a general Abelian variety where $1\leq d_i\leq 2$, $i=1,...,a$ and $a\geq 4$. We show that the fibers are in the same birational class if all the $(m,0)$ forms on $X_b$ are liftable to $(m,0)$ forms on $\mathcal{X}$ where $m=1$ and $m=a-1$. Actually we show general criteria to find families with fibers in the same birational class, which leads together with a famous theorem of Nori to some interesting applications.