arXiv:2611.13267v1 Announce Type: new Abstract: The Cayley--Bedregal reduction replaces a complete intersection $\{f_1=\dots=f_r=0\}$ by the hypersurface $\sum_i\varphi_if_i=0$ in the projectivized dual bundle. For an isolated complete intersection singularity of corank $c$, we express the Milnor class of this hypersurface on its singular fibre $\mathbb P^{c-1}$ in terms of the Milnor numbers of generic systems obtained by taking linear combinations of the defining equations. Its degree is the original Milnor number, and the higher components recover the Milnor numbers of the systems with fewer equations. These numbers are invariants of the analytic germ. Our second main result concerns the Hodge-theoretic refinement. For a complete intersection defined by a regular section of a rank-$r$ algebraic vector bundle, the Hirzebruch--Milnor class is determined by that of the Cayley--Bedregal hypersurface: the push-forward of the normalized Hirzebruch--Milnor class of the hypersurface, in the sense of Cappell, Maxim, Sch\"urmann and Shaneson, is $(-y)^{r-1}$ times that of the complete intersection, with no positivity assumption on the bundle. The factor $(-y)^{r-1}$ comes from the motivic decomposition of the projective bundle, and at $y=-1$ the identity recovers Aluffi's comparison of Milnor classes. When the tautological bundle is very ample, the hypersurface formula of Maxim, Saito and Sch\"urmann then gives the Hirzebruch--Milnor class of the complete intersection through the vanishing cycles of a single function, avoiding their induction through intermediate complete intersections and their restriction to powers of one line bundle. In integral homology we prove the analytic form of Aluffi's identity, and obtain formulas for the Milnor class in terms of local Milnor fibres and for the Chern--Schwartz--MacPherson class of any projective scheme.
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