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<p>We prove a comparison theorem between the étale cohomology of algebraic varieties over Stein compacta and the singular cohomology of their analytifications. We deduce that the field of meromorphic functions in a neighborhood of a connected Stein compact subset of a normal complex space of dimension&#xa0;<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1323_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> <EquationSource Format="TEX">$n$</EquationSource> </InlineEquation> has cohomological dimension&#xa0;<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1323_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> <EquationSource Format="TEX">$n$</EquationSource> </InlineEquation>. As an application of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1323_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Gal</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">/</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\mathrm {Gal}(\mathbb{C}/\mathbb{R})$</EquationSource> </InlineEquation>-equivariant variants of these results, we obtain a quantitative version of Hilbert’s 17th problem on compact subsets of real-analytic spaces.</p>

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Étale cohomology of algebraic varieties over Stein compacta

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Olivier Benoist

摘要

We prove a comparison theorem between the étale cohomology of algebraic varieties over Stein compacta and the singular cohomology of their analytifications. We deduce that the field of meromorphic functions in a neighborhood of a connected Stein compact subset of a normal complex space of dimension  n $n$ has cohomological dimension  n $n$ . As an application of Gal ( C / R ) $\mathrm {Gal}(\mathbb{C}/\mathbb{R})$ -equivariant variants of these results, we obtain a quantitative version of Hilbert’s 17th problem on compact subsets of real-analytic spaces.