Hirzebruch-Milnor classes of local complete intersections, minimal exponent, and applications to higher singularities

TL;DR

This work defines spectral Hirzebruch-Milnor classes for arbitrary local complete intersections by combining deformation to the normal cone with Verdier-Saito specialization, enabling a Hodge-theoretic, homological probe of singularities. The core contribution is a precise vanishing framework: the spectral Hirzebruch-Milnor classes vanish above certain thresholds governed by the minimal exponent $\tilde{\alpha}(X)$, generalizing hypersurface results to lci varieties. A central theme is the sharp link between $\tilde{\alpha}(X)$ and classical invariants like the log canonical threshold, yielding homological criteria to detect higher Du Bois and higher rational singularities, particularly when Sing$(X)$ is projective. The paper also develops the machinery of V-filtration, vanishing cycles, and spectral Hirzebruch classes for lci, plus duality relations with the Hodge spectrum, thereby enriching the toolkit for understanding and classifying singularities in algebraic geometry.

Abstract

In this paper we use the deformation to the normal cone and the corresponding Verdier-Saito specialization to define and study (spectral) Hirzebruch-Milnor type homology characteristic classes for local complete intersections. Our main results describe vanishing properties of these classes in relation to the minimal exponent. As applications, we show how Hirzebruch-Milnor classes of local complete intersections with a projective singular locus can be used to detect higher Du Bois and higher rational singularities.

Theorems & Definitions (69)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Definition 2.2
• Theorem 2.3