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<p>This paper is concerned with a class of the finite time noncollapsing of continuity method over a compact Kähler manifold. It is shown that the Gromov-Hausdorff limit <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((X_T,d_T)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi>T</mi> </msub> <mo>,</mo> <msub> <mi>d</mi> <mi>T</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is isometric to the completion of ample locus of limit class with respect to limit metric. Furthermore, we can prove the regular part of limit space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(X_T\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mi>T</mi> </msub> </math></EquationSource> </InlineEquation> is geodesically convex and every tangent cone of limit space is homeomorphic to a normal affine algebraic variety. In particular, we can obtain that the limit space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((X_T,d_T)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi>T</mi> </msub> <mo>,</mo> <msub> <mi>d</mi> <mi>T</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is homeomorphic to a manifold under some assumption of non-Kähler locus.</p>

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Geometry on the finite time noncollapsing along continuity method

  • 期刊论文

  • 发表日期：

    2025年11月10日

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  • Lei Zhang,
  • Zhenlei Zhang

摘要

This paper is concerned with a class of the finite time noncollapsing of continuity method over a compact Kähler manifold. It is shown that the Gromov-Hausdorff limit \((X_T,d_T)\) ( X T , d T ) is isometric to the completion of ample locus of limit class with respect to limit metric. Furthermore, we can prove the regular part of limit space \(X_T\) X T is geodesically convex and every tangent cone of limit space is homeomorphic to a normal affine algebraic variety. In particular, we can obtain that the limit space \((X_T,d_T)\) ( X T , d T ) is homeomorphic to a manifold under some assumption of non-Kähler locus.