The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties). The generalization for singular varieties is due to Brasselet–Schürmann–Yokura. Following the work of Weber, we investigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed points of the torus action and a formula for the local class in terms of the defining fan are recalled. After this review part, we prove the positivity of local Hirzebruch classes for all toric varieties, thus proving false the alleged counterexample given by Weber.

中文翻译：

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局部等变Hirzebruch类对复曲面品种的正性

本文研究的中心对象是Hirzebruch类，它是Hirzebruch给出的Todd类的变形（对于光滑品种）。奇异变体的泛化归因于Brasselet–Schürmann–Yokura。在Weber的工作之后，我们研究了（可能是单数）复曲面品种的等变版本。回顾了Hirzebruch类在环面作用的固定点上的局部分解以及关于局部扇形的局部公式。在本部分的回顾之后，我们证明了本地Hirzebruch类对所有复曲面品种的积极性，从而证明了Weber所称的反例是错误的。