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<p>In this paper, we introduce a necessary condition for the existence of characteristic zero liftings of certain smooth, proper varieties in positive characteristic, using étale homotopy theory and Wall’s finiteness obstruction. For a variety with finite étale fundamental group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>, we define a notion of mod-<i>l</i> finite dominatedness based on the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {F}_l\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>l</mi> </msub> </math></EquationSource> </InlineEquation>-chain complex of the universal cover of its <i>l</i>-profinite étale homotopy type. We prove that such a variety <i>X</i> can be lifted to characteristic zero only if the above chain complex of <i>X</i> is quasi-isomorphic to a bounded complex of finitely generated projective <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {F}_l [\pi ]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>l</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-modules. To prove this result, we extend Wall’s discussions of finiteness obstructions to <i>l</i>-profinite complete spaces with finite fundamental group.</p>

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A necessary condition for liftings of positive characteristic varieties with finite fundamental groups

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  • 发表日期：

    2026年6月17日

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  • Ruida Di,
  • Runjie Hu,
  • Siqing Zhang

摘要

In this paper, we introduce a necessary condition for the existence of characteristic zero liftings of certain smooth, proper varieties in positive characteristic, using étale homotopy theory and Wall’s finiteness obstruction. For a variety with finite étale fundamental group \(\pi \) π , we define a notion of mod-l finite dominatedness based on the \(\mathbb {F}_l\) F l -chain complex of the universal cover of its l-profinite étale homotopy type. We prove that such a variety X can be lifted to characteristic zero only if the above chain complex of X is quasi-isomorphic to a bounded complex of finitely generated projective \(\mathbb {F}_l [\pi ]\) F l [ π ] -modules. To prove this result, we extend Wall’s discussions of finiteness obstructions to l-profinite complete spaces with finite fundamental group.