Chern Classes and Transversality for Singular Spaces

Schürmann, Jörg

doi:10.1007/978-3-319-39339-1_13

Cited by 21 publications

(15 citation statements)

References 27 publications

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“…The next result, relating characteristic cycles to (signed) CSM classes, has a long history. See [Sab85, Lemme 1.2.1], and more recently [PP01,(12)], [Sch05, §4.5], [Sch17, §3], especially diagram (3.1) in [Sch17].…”

Section: Characteristic Classes Via Characteristic Cycles; Proof Of T...mentioning

confidence: 99%

“…Then by definition, f is non-characteristic with respect to the support supp(CC(ϕ)) of the characteristic cycle of ϕ if and only if by [Sch17,Theorem 3.3], and this cycle is effective if CC(ϕ) is effective. Indeed the proof of [Sch17, Theorem 3.3] is done in two steps: first for a submersion, where our claim follows from the case (3) above; then the case of a closed embedding of a nonsingular subvariety is (locally) reduced by induction to the case of a hypersurface of codimension one (locally) given by an equation {f = 0}.…”

Section: Effective Charactersitic Cycles (Ii)mentioning

confidence: 99%

“…Further, if Y = Z is a nonsingular closed subvariety of X, then s SM (Y, X) ∈ A * Y equals the ordinary Segre class s(Y, X); in general, the two classes differ (as discussed in subsection 8.2 for Y a global complete intersection in a nonsingular projective variety X). See [Alu03] and (in the equivariant case) [Ohm06] for general properties of SM classes, and [Sch17] for their compability with transversal pullbacks.…”

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confidence: 99%

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Positivity of Segre-MacPherson classes

Aluffi¹,

Mihalcea²,

Schürmann³

et al. 2019

Preprint

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Let X be a complex nonsingular variety with globally generated tangent bundle. We prove that the signed Segre-MacPherson (SM) class of a constructible function on X with effective characteristic cycle is effective. This observation has a surprising number of applications to positivity questions in classical situations, unifying previous results in the literature and yielding several new results. We survey a selection of such results in this paper. For example, we prove general effectivity results for SM classes of subvarieties which admit proper (semi-)small resolutions and for regular or affine embeddings. Among these, we mention the effectivity of (signed) Segre-Milnor classes of complete intersections if X is projective and an alternation property for SM classes of Schubert cells in flag manifolds; the latter result proves and generalizes a variant of a conjecture of Fehér and Rimányi. Among other applications we prove the positivity of Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety and the signed-effectivity of the intersection homology Chern class of the theta divisor of a non-hyperelliptic curve; and we extend the (known) non-negativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety.

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Section: Characteristic Classes Via Characteristic Cycles; Proof Of T...mentioning

confidence: 99%

Section: Effective Charactersitic Cycles (Ii)mentioning

confidence: 99%

mentioning

confidence: 99%

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Positivity of Segre-MacPherson classes

Aluffi¹,

Mihalcea²,

Schürmann³

et al. 2019

Preprint

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“…Notice that for r = 1 this is just the definition of the class M(X). There is also [241] where the author gives a general transversality formula that throws light on the theory of Chern classes for complete intersections.…”

Section: Milnor Classesmentioning

confidence: 99%

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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“…The transversality condition in this theorem can be relaxed (see Section 2). Similar transversality conditions were used in [22] to prove a refined intersection formula for the Chern-Schwartz-MacPherson classes.…”

Section: Introductionmentioning

confidence: 99%

On the Chern classes of singular complete intersections

Callejas-Bedregal

Morgado

Seade

2020

Journal of Topology

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We consider two classical extensions for singular varieties of the usual Chern classes of complex manifolds, namely the total Schwartz–MacPherson and Fulton–Johnson classes, cSMfalse(Xfalse) and cFJfalse(Xfalse). Their difference (up to sign) is the total Milnor class M(X), a gener‐alization of the Milnor number for varieties with arbitrary singular set. We get first Verdier‐Riemann–Roch type formulae for the total classes cSMfalse(Xfalse) and cFJfalse(Xfalse), and use these to prove a surprisingly simple formula for the total Milnor class when X is defined by a finite number of local complete intersection X1,.….,Xr in a complex manifold, satisfying certain transversality conditions. As applications, we obtain a Parusiński–Pragacz type formula and an Aluffi type formula for the Milnor class, and a description of the Milnor classes of X in terms of the global Lê classes of the Xi.

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Chern Classes and Transversality for Singular Spaces

Cited by 21 publications

References 27 publications

Positivity of Segre-MacPherson classes

Positivity of Segre-MacPherson classes

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

On the Chern classes of singular complete intersections

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