We prove that for a surjective holomorphic endomorphism f of a compact Kähler manifold X of dimension <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and for some integer p with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$1\le p\le k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>, there exists a proper invariant analytic subset E for f such that if a positive closed (p, p)-current S can be represented by a smooth form in a neighborhood of E, the sequence <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d_p^{-n}(f^n)^*(S-\alpha _S)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>d</mi> <mi>p</mi> <mrow> <mo>-</mo> <mi>n</mi> </mrow> </msubsup> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>-</mo> <msub> <mi>α</mi> <mi>S</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> converges to 0 exponentially fast in the sense of currents, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> is the dynamical degree of order p and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha _S$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>α</mi> <mi>S</mi> </msub> </math></EquationSource> </InlineEquation> is a smooth closed (p, p)-form in the de Rham cohomology class of S.

Inverse Images of Positive Closed Currents Under Holomorphic Endomorphisms of Compact Kähler Manifolds

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发表日期：

2026年7月27日

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Taeyong Ahn

摘要

We prove that for a surjective holomorphic endomorphism f of a compact Kähler manifold X of dimension $k\ge 2$ $k \geq 2$ and for some integer p with $1\le p\le k$ $1 \leq p \leq k$ , there exists a proper invariant analytic subset E for f such that if a positive closed (p, p)-current S can be represented by a smooth form in a neighborhood of E, the sequence $d_p^{-n}(f^n)^*(S-\alpha _S)$ $d_{p}^{- n} {(f^{n})}^{*} (S - α_{S})$ converges to 0 exponentially fast in the sense of currents, where $d_p$ $d_{p}$ is the dynamical degree of order p and $\alpha _S$ $α_{S}$ is a smooth closed (p, p)-form in the de Rham cohomology class of S.