Let X be a reduced Stein space of pure dimension n, D an open set in X, and q an integer, such that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2024_706_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le q\le n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. Assume that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2024_706_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="208" /> </InlineMediaObject> <EquationSource Format="TEX">$H^{n-1}(D,\mathscr {O})\rightarrow H^{n-1}(D,\mathscr {M})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi mathvariant="script">O</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msup> <mi>H</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi mathvariant="script">M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is injective and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2024_706_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$H^k(D,\mathscr {O})=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>k</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi mathvariant="script">O</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2024_706_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">$k=q,\ldots ,n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mi>q</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Then, we prove that D is locally q-complete with corners at every point <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2024_706_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="131" /> </InlineMediaObject> <EquationSource Format="TEX">$x\in \partial D\setminus \textrm{Sing}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>∂</mi> <mi>D</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mtext>Sing</mtext> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. As a corollary, we obtain Eastwood–Vigna Suria’s theorem and a new characterization theorem of Steinness.

q-complete with corners open sets and vanishing cohomology groups

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发表日期：

2025年1月27日

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Shun Sugiyama

摘要

Let X be a reduced Stein space of pure dimension n, D an open set in X, and q an integer, such that $1\le q\le n$ $1 \leq q \leq n$ . Assume that $H^{n-1}(D,\mathscr {O})\rightarrow H^{n-1}(D,\mathscr {M})$ $H^{n - 1} (D, O) \to H^{n - 1} (D, M)$ is injective and $H^k(D,\mathscr {O})=0$ $H^{k} (D, O) = 0$ for every $k=q,\ldots ,n-2$ $k = q, \dots, n - 2$ . Then, we prove that D is locally q-complete with corners at every point $x\in \partial D\setminus \textrm{Sing}(X)$ $x \in \partial D \ Sing (X)$ . As a corollary, we obtain Eastwood–Vigna Suria’s theorem and a new characterization theorem of Steinness.