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<p>Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((D,\pi )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a connected Riemann domain over an <i>n</i>-dimensional Stein manifold which satisfies <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(H_1(D,{\mathbb {Z}})=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(2\le k\le n-1.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Then <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\((D,\pi )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is Stein if and only if <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\((D,\pi )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfies the <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\partial \bar{\partial }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mover accent="true"> <mrow> <mi>∂</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </math></EquationSource> </InlineEquation>-lemma.</p>

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\(\pmb {\partial \bar{\partial }}\)-Lemma and Steinness for Riemann domains over Stein manifolds

  • 期刊论文

  • 发表日期：

    2026年3月9日

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  • Shun Sugiyama

摘要

Let \((D,\pi )\) ( D , π ) be a connected Riemann domain over an n-dimensional Stein manifold which satisfies \(H_1(D,{\mathbb {Z}})=0\) H 1 ( D , Z ) = 0 and \(H^k(D,{\mathscr {O}})=0\) H k ( D , O ) = 0 for \(2\le k\le n-1.\) 2 ≤ k ≤ n - 1 . Then \((D,\pi )\) ( D , π ) is Stein if and only if \((D,\pi )\) ( D , π ) satisfies the \(\partial \bar{\partial }\) ∂ ∂ ¯ -lemma.