Nowhere vanishing 1-forms on 4-folds

TL;DR

This work establishes Hao's conjecture for smooth projective 4-folds: a morphism f: X → A to a simple abelian variety A is smooth precisely when there exists a nowhere vanishing pullback of a 1-form from A. The authors combine MMP techniques, purity arguments for maps to abelian varieties, and a detailed analysis of conic bundles and del Pezzo fibrations to constrain the possible fibration structures, ultimately delivering a complete birational classification of X when Alb_X has no elliptic factor. The results show that in dimension four, the geometry of X is governed by either a smooth conic/del Pezzo fibration over a simple abelian base or by isogenies to abelian varieties, with explicit descriptions of blowups and their centers. These contributions link global differential forms to the birational and fibration structure of 4-folds, advancing understanding of how zeros of pullback forms reflect underlying morphisms to abelian varieties.

Abstract

In this note, we prove -- in dimension at most 4 -- a conjectue of Hao which says that a morphism $f : X \to A$ to a simple abelian variety $A$ is smooth if and only if there is a 1-form pulled back from A without any zeros. We also give a complete classification of 4-folds with a 1-form without zeros not admitting a map to an elliptic curve.

Theorems & Definitions (37)

• Conjecture 1
• Theorem 2: =Theorem \ref{['thm:structure_no_elliptic_factor']}
• Conjecture 3
• Theorem 4: =Theorem \ref{['thm:smooth_map_to_simpleAV']}
• Definition 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 2.4
• Proposition 2.5
• proof