Overview
This research investigates the reduction properties of Albanese varieties associated with projective geometrically normal varieties. Specifically, it establishes conditions under which the Albanese variety of such a variety over the fraction field of a discrete valuation ring (DVR) achieves semi-abelian reduction over the same base. An application of this finding is presented, extending a known nonexistence theorem concerning abelian schemes, originally established by Fontaine and Abrashkin, to include singular contexts.
Research Context
The study operates within the domain of algebraic geometry, focusing on varieties defined over the fraction field of a DVR. A key concept is the Albanese variety, Alb$_{X_K/K}$, associated with a projective geometrically normal variety $X_K$. The reduction behavior of such varieties is a central theme, particularly the concept of semi-abelian reduction. The work also references a classical nonexistence theorem by Fontaine and Abrashkin concerning abelian schemes over the integers (Z).
Approach
The core of the research involves demonstrating a specific property for Albanese varieties. The approach posits that if a projective geometrically normal variety $X_K$ over the fraction field $K$ of a DVR admits a projective model, and the special fiber of this model possesses at worst nodes in codimension one, then its Albanese variety, Alb$_{X_K/K}$, will exhibit semi-abelian reduction over the base field. The established property of semi-abelian reduction then serves as a basis for its application.
Findings
- If a variety $X_K$, defined as a projective geometrically normal variety over the fraction field $K$ of a DVR, satisfies the condition that it admits a projective model whose special fiber has at worst nodes in codimension one, then its Albanese variety, Alb$_{X_K/K}$, possesses semi-abelian reduction over the same base.
- As an application, the classical nonexistence theorem of Fontaine and Abrashkin regarding abelian schemes over Z has been extended to encompass the singular setting.
- A specific instance of this extension states that if a flat projective Z-scheme $X$ has normal and geometrically connected fibers, then the first cohomology group $H^1(X_Q, O_{X_Q})$ is zero. This result pertains to the singular context.
Why This Matters
The findings presented here contribute to the understanding of the reduction properties of Albanese varieties and their implications for the existence of abelian schemes. By extending a classical nonexistence theorem to singular settings, the research broadens the applicability and scope of theoretical results in algebraic geometry.