Uniqueness of tangent currents for positive closed currents
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910820285284352 |
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| author | Nguyen, Viet-Anh Truong, Tuyen Trung |
| author_facet | Nguyen, Viet-Anh Truong, Tuyen Trung |
| contents | Let $X$ be a complex manifold $X$ of dimension $k,$ and let $V\subset X$ be a Kähler submanifold of dimension $l,$ and let $B\subset V$ be a piecewise $\mathcal{C}^2$-smooth domain. Let $T$ be a positive closed currents of bidegree $(p,p)$ in $X$ such that $T$ satisfies a mild reasonable assumption in a neighborhood of $\partial B$ in $X$ and that the $j$-th average mean $ν_j(T,B,r)$ for every $j$ with $\max(0,l-p)\leq j\leq\min(l,k-p)$ converges sufficiently fast to the $j$-th generalized Lelong number $ν_j(T,B)$ as $r$ tends to $0$ so that $r^{-1}(ν_j(T, B,r)-ν_j( T,B))$ is locally integrable near $r=0.$ Then we show that $T$ admits a unique tangent current along $B.$ A local version where we replace the condition of $T$ near $B$ by the conditions on a finite cover of $B$ by piecewise $\mathcal{C}^2$-smooth domains in $V$ is also given. When $T$ is a current of integration over a complex analytic set, we show that $ν_j(T,B,r)-ν_j(T,B)=O(r^ρ)$ for some $ρ>0,$ and hence this condition is satisfied. Our result may be viewed as a natural generalization of Blel-Demailly-Mouzali's criterion from the case $l=0$ to the case $l>0.$ The result has applications in the intersection theory of positive closed currents. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06532 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness of tangent currents for positive closed currents Nguyen, Viet-Anh Truong, Tuyen Trung Complex Variables Algebraic Geometry 32J25, 14C30 Let $X$ be a complex manifold $X$ of dimension $k,$ and let $V\subset X$ be a Kähler submanifold of dimension $l,$ and let $B\subset V$ be a piecewise $\mathcal{C}^2$-smooth domain. Let $T$ be a positive closed currents of bidegree $(p,p)$ in $X$ such that $T$ satisfies a mild reasonable assumption in a neighborhood of $\partial B$ in $X$ and that the $j$-th average mean $ν_j(T,B,r)$ for every $j$ with $\max(0,l-p)\leq j\leq\min(l,k-p)$ converges sufficiently fast to the $j$-th generalized Lelong number $ν_j(T,B)$ as $r$ tends to $0$ so that $r^{-1}(ν_j(T, B,r)-ν_j( T,B))$ is locally integrable near $r=0.$ Then we show that $T$ admits a unique tangent current along $B.$ A local version where we replace the condition of $T$ near $B$ by the conditions on a finite cover of $B$ by piecewise $\mathcal{C}^2$-smooth domains in $V$ is also given. When $T$ is a current of integration over a complex analytic set, we show that $ν_j(T,B,r)-ν_j(T,B)=O(r^ρ)$ for some $ρ>0,$ and hence this condition is satisfied. Our result may be viewed as a natural generalization of Blel-Demailly-Mouzali's criterion from the case $l=0$ to the case $l>0.$ The result has applications in the intersection theory of positive closed currents. |
| title | Uniqueness of tangent currents for positive closed currents |
| topic | Complex Variables Algebraic Geometry 32J25, 14C30 |
| url | https://arxiv.org/abs/2502.06532 |