We introduce spectral Hirzebruch–Milnor classes for singular hypersurfaces. These can be identified with Steenbrink spectra in the isolated singularity case, and may be viewed as their global analogues in general. Their definition uses vanishing cycles of mixed Hodge modules and the Todd class transformation. These are compatible with the pushforward by proper morphisms, and the classes can be calculated by using resolutions of singularities. Formulas for Hirzebruch–Milnor classes of projective hypersurfaces in terms of these classes are given in the case where the multiplicity of a generic hyperplane section is not 1. These formulas using hyperplane sections instead of hypersurface ones are easier to calculate in certain cases. Here we use the Thom–Sebastiani theorem for the underlying filtered D -modules of vanishing cycles, from which we can deduce the Thom–Sebastiani type theorem for spectral Hirzebruch–Milnor classes. For the Chern classes after specializing to $$y=-\,1$$ y = - 1 , we can give a relatively simple formula for the localized Milnor classes, which implies a new formula for the Euler numbers of projective hypersurfaces, using iterated hyperplane sections. Applications to log canonical thresholds and Du Bois singularities are also explained; for instance, the latter can be detected by using Hirzebruch–Milnor classes in the projective hypersurface case.

中文翻译：

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奇异超曲面的光谱 Hirzebruch-Milnor 类

我们为奇异超曲面引入了光谱 Hirzebruch-Milnor 类。在孤立的奇点情况下，这些可以用 Steenbrink 光谱识别，并且通常可以被视为它们的全局类似物。他们的定义使用混合 Hodge 模块的消失循环和 Todd 类转换。这些与适当态射的推进兼容，并且可以通过使用奇点的分辨率来计算类。在通用超平面截面的重数不为 1 的情况下，给出了根据这些类的 Hirzebruch-Milnor 类投影超曲面的公式。这些公式使用超平面截面而不是超曲面截面在某些情况下更容易计算。在这里，我们将 Thom-Sebastiani 定理用于消失循环的底层滤波 D 模，从中我们可以推导出谱 Hirzebruch-Milnor 类的 Thom-Sebastiani 类型定理。对于特化为 $$y=-\,1$$ y = - 1 后的 Chern 类，我们可以为局部化的 Milnor 类给出一个相对简单的公式，这意味着投影超曲面的欧拉数的一个新公式，使用迭代超平面截面。还解释了对数规范阈值和杜波依斯奇点的应用；例如，后者可以通过在投影超曲面情况下使用 Hirzebruch-Milnor 类来检测。使用迭代超平面部分。还解释了对数规范阈值和杜波依斯奇点的应用；例如，后者可以通过在投影超曲面情况下使用 Hirzebruch-Milnor 类来检测。使用迭代超平面部分。还解释了对数规范阈值和杜波依斯奇点的应用；例如，后者可以通过在投影超曲面情况下使用 Hirzebruch-Milnor 类来检测。