Differential forms and quadrics of the canonical image

TL;DR

This paper investigates the properties of families of algebraic varieties with high irregularity, focusing on the canonical map's image and Torelli-type problems, establishing conditions for birationality and generic Torelli, and providing counterexamples.

Contribution

It proves that under certain conditions, the fibers of a family are birational and generic Torelli holds, extending understanding of canonical images and infinitesimal Torelli for high irregularity varieties.

Findings

Fibers are birational if Albanese map degree is 1 and forms lift holomorphically.

Generic Torelli holds for minimal fibers with unique minimal models under specified conditions.

Counterexamples exist when the canonical image lies in low-rank quadrics.

Abstract

Let $\pi\colon\mathcal{X}\to B$ be a family over a smooth connected analytic variety $B$, not necessarily compact, whose general fiber $X$ is smooth of dimension $n$, with irregularity $\geq n+1$ and such that the image of the canonical map of $X$ is not contained in any quadric of rank $\leq 2n+3$. We prove that if the Albanese map of $X$ is of degree $1$ onto its image then the fibers of $\pi\colon\mathcal{X}\to B$ are birational under the assumption that all the $1$-forms and all the $n$-forms of a fiber are holomorphically liftable to $\mathcal{X}$. Moreover we show that generic Torelli holds for such a family $\pi\colon \mathcal{X}\to B$ if, in addition to the above hypothesis, we assume that the fibers are minimal and their minimal model is unique. There are counterexamples to the above statements if the canonical image is contained inside quadrics of rank $\leq 2n+3$. We also solve the infinitesimal Torelli problem for an $n$-dimensional variety $X$ of general type with irregularity $\geq n+1$ and such that its cotangent sheaf is generated and the canonical map is a rational map whose image is not contained in a quadric of rank less or equal to $2n+3$.