This paper shows the existence of local wedge products of two positive closed currents of higher bi-degree on a complex manifold X. We prove that if T, S are positive closed currents of higher bi-degree (p, p) and (q, q), respectively, on X then in each patch <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$U_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> of X the wedge product <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T\hspace{1.111pt}{\wedge }\hspace{1.111pt}S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mspace width="1.111pt" /> <mo>∧</mo> <mspace width="1.111pt" /> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation> is well defined under the assumption that T or S has the representation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T=dd^c U_T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>=</mo> <mi>d</mi> <msup> <mi>d</mi> <mi>c</mi> </msup> <msub> <mi>U</mi> <mi>T</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$U_T$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mi>T</mi> </msub> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(p-1,p-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-current satisfying certain conditions. This result can be applied to currents T on complex manifolds in which local potentials exist.

Local wedge products of positive closed currents of higher bi-degree on complex manifolds

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发表日期：

2025年10月14日

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Le Mau Hai,
Trinh Tung

摘要

This paper shows the existence of local wedge products of two positive closed currents of higher bi-degree on a complex manifold X. We prove that if T, S are positive closed currents of higher bi-degree (p, p) and (q, q), respectively, on X then in each patch $U_0$ $U_{0}$ of X the wedge product $T\hspace{1.111pt}{\wedge }\hspace{1.111pt}S$ $T \land S$ is well defined under the assumption that T or S has the representation $T=dd^c U_T$ $T = d d^{c} U_{T}$ where $U_T$ $U_{T}$ is a $(p-1,p-1)$ $(p - 1, p - 1)$ -current satisfying certain conditions. This result can be applied to currents T on complex manifolds in which local potentials exist.