On the Chern classes of singular complete intersections

Callejas-Bedregal, R.; Morgado, M. F. Z.; Seade, José

doi:10.1112/topo.12129

Cited by 4 publications

(4 citation statements)

References 25 publications

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“…In the case of local complete intersections, the hypersurface X was introduced in [CBMS]. In fact, in [CBMS,Theorem 6.4], Callejas-Bedregal, Morgado, and Seade obtain a different expression relating the Milnor classes of X and X in the local complete intersection case. Comparing (3.7) and the expression from [CBMS] may lead to nontrivial identities for Chern classes of bundles associated with local complete intersections.…”

Section: 2mentioning

confidence: 99%

The Chern–schwartz–macpherson Class of an Embeddable Scheme

Aluffi

2019

Forum of Mathematics, Sigma

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The Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety may be computed directly from the Segre class of the jacobian subscheme of the hypersurface; this has been known for a number of years. We generalize this fact to arbitrary embeddable schemes: for every subscheme X of a nonsingular variety V , we define an associated subscheme Y of a projective bundle over V and provide an explicit formula for the Chern-Schwartz-MacPherson class of X in terms of the Segre class of Y . If X is a local complete intersection, a version of the result yields a direct expression for the Milnor class of X.For V = P n , we also obtain expressions for the Chern-Schwartz-MacPherson class of X in terms of the 'Segre zeta function' of Y .

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Section: 2mentioning

confidence: 99%

The Chern–schwartz–macpherson Class of an Embeddable Scheme

Aluffi

2019

Forum of Mathematics, Sigma

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“…In [54] there is a surprisingly simple formula for the total Milnor class when X is defined by a finite number of hypersurfaces X 1 , · · · , X r in a complex manifold M , satisfying certain transversality conditions:…”

Section: Milnor Classesmentioning

confidence: 99%

“…Milnor classes spring from [9] and they are an active field of current research with significant applications to other related areas. There is a large literature on Milnor classes, for instance [10,11,25,43,42,38,52,54,57,178,179,212,225,240]. Milnor classes encode much information about the varieties in question, and this is being studied by various authors from several points of view.…”

Section: Milnor Classesmentioning

confidence: 99%

“…Most of the literature on Milnor classes is for hypersurfaces, though there are some recent works that throw light on the subject in the case of complete intersections. In [53] the authors study the total Milnor class of complete intersections Z(s) defined by a regular section s of a rank r holomorphic bundle E over a compact manifold M . It is noticed that s determines a hypersurface Z(s) in the total space of the projectivization P(E ∨ ) of the dual bundle E ∨ , and one has a formula expressing the total Milnor class of Z(s) in terms of the Milnor classes of the hypersurface Z(s).…”

Section: Milnor Classesmentioning

confidence: 99%

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On Milnor’s fibration theorem and its offspring after 50 years

Seade

2018

Bull. Amer. Math. Soc.

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To Jack, whose profoundness and clarity of vision seep into our appreciation of the beauty and depth of mathematics.Abstract. Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own, with a vast literature, plenty of different viewpoints, a large progeny and connections with many other branches of mathematics. In this work we revisit the classical theory in both the real and complex settings, and we glance at some areas of current research and connections with other important topics. The purpose of this article is two-fold. On the one hand, it should serve as an introduction to the topic for non-experts, and on the other hand, it gives a wide perspective of some of the work on the subject that has been and is being done. It includes a vast literature for further reading. IntroductionMilnor's fibration theorem in [189] is a milestone in singularity theory that has opened the way to a myriad of insights and new understandings. This is a beautiful piece of mathematics, where many different branches, aspects and ideas, come together. The theorem concerns the geometry and topology of analytic maps near their critical points.Consider the simplest case, a holomorphic map (C n+1 , 0) f → (C, 0) taking the origin into the origin, with an isolated critical point at 0. As an example one can have in mind the Pham-Brieskorn polynomials:(0.1) z → z a 0 0 + · · · + z an n , a i ≥ 2 for all i = 0, 1, · · · , n .Since f is analytic, there exists r > 0 sufficiently small so that 0 ∈ C is the only critical value of the restriction f | Br , where B r is the open ball of radius r and center at 0. Set V := f −1 (0) and V * := V \ {0} .

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