Overview
This research identified a necessary condition for the existence of characteristic zero liftings for specific algebraic varieties. The varieties considered are smooth and proper, defined in positive characteristic, and possess finite etale fundamental groups. The developed condition utilizes etale homotopy theory and extends Wall's finiteness obstruction concept.
Research Context
The study addresses the problem of lifting algebraic varieties from positive characteristic to characteristic zero. This type of lifting problem is a fundamental question in algebraic geometry, investigating when geometric objects defined over fields of positive characteristic can be 'transported' to fields of characteristic zero while preserving certain properties.
Specifically, the work focuses on varieties with finite etale fundamental groups (denoted $\pi$). The etale fundamental group captures important topological information about a variety, and its finiteness is a key restriction in this context. The theoretical framework involves etale homotopy theory, which provides a way to apply topological methods to algebraic geometry.
Approach
The methodology introduced a concept of mod-l finite dominatedness, which is applicable to varieties possessing a finite etale fundamental group $\pi$. This concept is built upon the $F_l$-chain complex derived from the universal cover of the variety's l-profinite etale homotopy type.
The research extended Wall's discussions concerning finiteness obstructions. Specifically, these extensions were applied to l-profinite complete spaces that exhibit a finite fundamental group. This adaptation of Wall's work forms a crucial component in constructing the necessary condition for liftings.
Findings
The primary finding establishes a necessary condition for a variety $X$ (as described above) to be liftable to characteristic zero. This condition mandates that the $F_l$-chain complex of $X$ must be quasi-isomorphic to a bounded complex composed of finitely generated projective $F_l[\pi]$-modules.
This implies that if a smooth, proper variety in positive characteristic with a finite etale fundamental group fails to satisfy this quasi-isomorphism condition for its $F_l$-chain complex, then it cannot be lifted to characteristic zero. The condition provides a concrete algebraic criterion derived from homotopy-theoretic properties.
Why This Matters
The determination of necessary conditions for lifting varieties between characteristics contributes to the understanding of the relationship between algebraic geometry in positive and zero characteristics. Such conditions can constrain the types of varieties that allow for characteristic zero liftings.