We recover results by Ullmo-Yafaev and Peterzil-Starchenko on the closure of the image of an algebraic variety in a compact complex torus. Our approach uses directed closed currents and allows us to extend the result for dimension 1 flows to the setting of commutative complex Lie groups which are not necessarily compact. A version of the classical Ax-Lindemann-Weierstrass theorem for commutative complex Lie groups is also given.

中文翻译：

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可交换复李群上的代数流

我们恢复了 Ullmo-Yafaev 和 Peterzil-Starchenko 对紧复圆环中代数变体图像的闭合的结果。我们的方法使用定向闭合电流，并允许我们将维度 1 流的结果扩展到不一定紧凑的交换复李群的设置。还给出了可交换复李群的经典 Ax-Lindemann-Weierstrass 定理的一个版本。