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<p>In this paper, we derive the explicit Bergman kernel functions for the classical domain of third class based on the Forelli-Rudin structure, using the orthonormal basis and a holomorphic invariant. We also prove <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\partial \log K_{\widetilde{\Omega }}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mo>log</mo> <msub> <mi>K</mi> <mover accent="true"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">~</mo> </mover> </msub> </mrow> </math></EquationSource> </InlineEquation> is <i>d</i>-bounded. As an application, we present the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-cohomology vanishing theorem for the domain.</p>

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The vanishing theorem for \(L^2\)-cohomology on certain Forelli-Rudin structures

  • 期刊论文

  • 发表日期：

    2026年1月19日

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  • Xiaoliang Cheng,
  • Yixuan He,
  • Mingyue Chi,
  • An Wang

摘要

In this paper, we derive the explicit Bergman kernel functions for the classical domain of third class based on the Forelli-Rudin structure, using the orthonormal basis and a holomorphic invariant. We also prove \(\partial \log K_{\widetilde{\Omega }}\) ∂ log K Ω ~ is d-bounded. As an application, we present the \(L^2\) L 2 -cohomology vanishing theorem for the domain.