THE UNIQUENESS THEOREM FOR GYSIN COHERENT CHARACTERISTIC CLASSES OF SINGULAR SPACES MARKUS BANAGL AND DOMINIK J. WRAZIDLO

2025-05-06 0 0 471.61KB 39 页 10玖币

侵权投诉

THE UNIQUENESS THEOREM FOR GYSIN COHERENT CHARACTERISTIC

CLASSES OF SINGULAR SPACES

MARKUS BANAGL AND DOMINIK J. WRAZIDLO

ABSTRACT. We establish a general computational scheme designed for a systematic com-

putation of characteristic classes of singular complex algebraic varieties that satisfy a Gysin

axiom in a transverse setup. This scheme is explicitly geometric and of a recursive nature

terminating on genera of explicit characteristic subvarieties that we construct. It enables us

e.g. to apply intersection theory of Schubert varieties to obtain a uniqueness result for such

characteristic classes in the homology of an ambient Grassmannian. Our framework applies

in particular to the Goresky-MacPherson L-class by virtue of the Gysin restriction formula

obtained by the ﬁrst author in previous work. We illustrate our approach for a systematic

computation of the L-class in terms of normally nonsingular expansions in examples of sin-

gular Schubert varieties that do not satisfy Poincar´

e duality over the rationals.

CONTENTS

1. Introduction 2

2. Whitney Transversality 4

3. Generic Transversality 7

4. Preliminaries on Schubert Varieties 8

4.1. Partitions 8

4.2. Flags 9

4.3. Schubert Varieties 9

4.4. The Homology of Schubert Varieties 9

4.5. The Singular Set of a Schubert Variety 10

5. Intersection Theory of Schubert Cycles 11

5.1. Transversality of Flags 11

5.2. Segre Products in Ambient Grassmannians 13

6. Gysin Coherent Characteristic Classes 17

7. Normally Nonsingular Expansion 19

7.1. Normally Nonsingular Integration 19

7.2. Genera of Characteristic Subvarieties 28

7.3. Proof of Theorem 7.1 30

8. Proof of Theorem 6.4 30

9. The Goresky-MacPherson L-Class 32

Date: October 25, 2022.

2020 Mathematics Subject Classiﬁcation. 57R20, 55R12, 55N33, 57N80, 32S60, 32S20, 14M15, 14C17,

57R40, 32S50.

Key words and phrases. Gysin transfer, Characteristic Classes, Singularities, Stratiﬁed Spaces, Intersection Ho-

mology, Goresky-MacPherson L-class, Verdier-Riemann-Roch formulae, Schubert varieties, Intersection theory,

Transversality.

This work is funded in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)

through a research grant to the ﬁrst author (Projektnummer 495696766). The second author was supported in addi-

tion by a Feodor Lynen Return Fellowship of the Alexander von Humboldt Foundation.

arXiv:2210.13009v1 [math.AT] 24 Oct 2022

2 MARKUS BANAGL AND DOMINIK J. WRAZIDLO

10. An Example: The L-Class of X3,2,134

References 38

1. INTRODUCTION

We establish a general computational scheme that allows for a systematic computation of

characteristic classes of singular complex algebraic varieties that satisfy a Gysin axiom. This

scheme is explicitly geometric and of a recursive nature terminating on genera of explicit

characteristic subvarieties that we construct.

We introduce and investigate the notion of Gysin coherent characteristic classes cde-

ﬁned for inclusions of certain subvarieties in ambient smooth complex algebraic varieties

(here, varieties are understood to be pure-dimensional complex and quasiprojective). In Def-

inition 6.2, we deﬁne such a class to be a pair c= (c∗,c∗)consisting of a function c∗

that assigns to every inclusion f:M→Wof a smooth closed subvariety M⊂Win a smooth

variety Wa normalized element c∗(f)∈H∗(M;Q), and a function c∗that assigns to every

inclusion i:X→Wof a compact possibly singular subvariety X⊂Win a smooth variety W

an element c∗(i)∈H∗(W;Q)whose highest non-trivial homogeneous component is the am-

bient fundamental class of Xin Wsuch that the following axioms hold. Apart from the Gysin

restriction formula f!c∗(i) = c∗(f)∩c∗(MtX⊂M)in a transverse setup, we also require

that c∗is multiplicative under products, that c∗and c∗transform naturally under isomor-

phisms of ambient smooth varieties, and that c∗is natural with respect to inclusions in larger

ambient smooth varieties. The genus |c∗|associated to a Gysin coherent characteristic class

cis deﬁned as the composition of c∗with the homological augmentation, |c∗|=ε∗c∗∈Q.

Often, such classes carise from generalizations c∗to singular varieties of bundle theo-

retic cohomological classes c∗. In such a situation, the pair cis of the form c∗(f) = c∗(νf),

where νfis the normal bundle of the smooth embedding f, and c∗(i) = i∗c∗(X)for inclu-

sions iof compact possibly singular subvarieties Xin ambient smooth varieties. By virtue

of the Verdier-Riemann-Roch type formulae derived by the ﬁrst author in [6], our framework

applies in particular to the topological characteristic class c∗=L∗of Goresky and MacPher-

son [21] that generalizes the cohomological Hirzebruch class c∗=L∗[25] to singular spaces

(see Theorem 9.2). In this case, the associated genus is the signature σ(X)of the Goresky-

MacPherson-Siegel intersection form on middle-perversity intersection homology of the Witt

space X.

By its very deﬁnition, the Goresky-MacPherson L-class is uniquely determined by signa-

ture normalization in degree zero, and by compatibility with Gysin restriction associated to

normally nonsingular topological embeddings of topological singular spaces with trivial nor-

mal bundle (see Cappell-Shaneson [12]). Uniqueness can be shown by a Thom-Pontrjagin

type approach via transverse regular maps to spheres. On the other hand, by dropping the triv-

iality assumption for normal bundles, the Gysin axiom introduced in the present paper seems

especially well-suited for concrete computations in transverse situations within the realm of

complex algebraic geometry. However, the Thom-Pontrjagin method is usually not directly

applicable when ranging only over algebraic varieties rather than all topological Witt spaces.

(Regular level sets of PL representatives of homotopy classes of maps from a variety to a

sphere are not subvarieties in general.) Our main result is the following uniqueness theorem

for Gysin coherent characteristic classes of singular varieties embedded in Grassmannians.

THE UNIQUENESS THEOREM FOR GYSIN COHERENT CHARACTERISTIC CLASSES OF SINGULAR SPACES 3

Theorem 1.1 (Uniqueness Theorem).Let cand e

cbe Gysin coherent characteristic classes.

If c∗=e

c∗and |c∗|=|e

c∗|for the associated genera, then we have c∗(i) = e

c∗(i)for all

inclusions i:X→G of compact irreducible subvarieties in ambient Grassmannians.

Since the inclusion of Schubert subvarieties in Grassmannians induces an injective map

on homology with rational coefﬁcients, we obtain

Corollary 1.2. Let c∗and e

c∗be generalizations to singular varieties of a bundle theoretic

cohomological class c∗such that |c∗|=|e

c∗|for the associated genera. If c∗and e

c∗induce

Gysin coherent characteristic classes as explained above, then we have c∗(X) = e

c∗(X)for all

Schubert varieties X.

The reader can directly verify our Theorem 1.1 for the toy example of inclusions into

ambient projective spaces by applying the Gysin axiom inductively to intersections of sub-

varieties with generic hyperplanes. To provide enough ﬂexibility for applications, we state

Theorem 1.1 in a slightly more general form (see Theorem 6.4) that accounts for a ﬁxed

family Xof admissible inclusions i:X→Wwhich satisfy an analog of the Kleiman-Bertini

transversality theorem for an appropriate notion of transversality. In Section 8, we derive

Theorem 6.4 by induction on the dimension of the ambient Grassmannian from a more tech-

nical result (see Theorem 7.1) that exploits the intersection theory of Schubert cycles, and

speciﬁcally the Segre product of subvarieties of Grassmannians in an ambient Grassmannian.

The additional value of our Theorem 7.1 is that it yields a systematic method for the re-

cursive computation of Gysin coherent characteristic classes in ambient Grassmannians in

terms of the normal geometry of Schubert varieties. First examples of these normally non-

singular expansions appered in [6, Section 4]. There, the ﬁrst author computed the Goresky-

MacPherson L-class in (real) codimension 4 for the Schubert varieties X2,1and X3,2, which are

sufﬁciently singular so as not to be rational Poincar´

e complexes. Beyond these examples, it

seems to be an open problem to compute L-classes of singular Schubert varieties. The Chern-

Schwartz-MacPherson classes of Schubert varieties were computed by Alufﬁ and Mihalcea

[1]. Our recursive formula (10) reduces computations to concrete Kronecker products (inte-

grals) that capture the normal geometry of Schubert varieties with L-shaped Young diagrams

over triple intersections of Schubert varieties (see Remark 7.9), and to genera of explicitly

constructed characteristic subvarieties (see Remark 7.11). These characteristic subvarieties

are obtained by taking the product of the given embedded variety with a certain Schubert

variety, and then intersecting the Segre embedded product in the larger Grassmannian with

another appropriate Schubert variety. If the given variety is a Schubert variety, then our char-

acteristic varieties turn out to be triple intersections of Schubert varieties. In the literature,

intersections of more than two general translates of Schubert varieties were studied by Billey

and Coskun [8] as a generalization of Richardson varieties [31]. They employed Kleiman’s

transversality theorem [26] to determine the singular locus of such intersection varieties. The

explicit computation of the integrals and genera that appear in our recursive formula (10)

requires separate techniques and is hence not pursued in the present paper.

Despite the general applicability of our recursive technique asserted by Theorem 7.1, it can

be more convenient in practice to implement slightly modiﬁed algorithms for doing concrete

computations. In Section 10, we illustrate such a related recursive method to compute the

Goresky-MacPherson L-class for the example of the singular Schubert variety X3,2,1of real

dimension 12, which does not satisfy global Poincar´

e duality over the rationals. Note that the

computation of L4(X3,2,1)goes beyond the scope of the computations in [6, Section 4], which

are limited to the L-class in real codimension 4.

4 MARKUS BANAGL AND DOMINIK J. WRAZIDLO

In [6], the ﬁrst author derived Verdier-Riemann-Roch type formulae for the Gysin restric-

tion of both the topological characteristic classes L∗of Goresky and MacPherson [21] and the

Hodge-theoretic intersection Hirzebruch characteristic classes IT1∗of Brasselet, Sch¨

urmann

and Yokura [9]. For an introduction to characteristic classes of singular spaces via mixed

Hodge theory in the complex algebraic context see Sch¨

urmann’s expository paper [32]. The

formulae in [6] have the potential to yield new evidence for the equality of the characteristic

classes L∗and IT1∗for pure-dimensional compact complex algebraic varieties, as conjectured

by Brasselet, Sch¨

urmann and Yokura in [9, Remark 5.4]. Cappell, Maxim, Sch¨

urmann and

Shaneson proved the conjecture in [11, Cor. 1.2] for orbit spaces X=Y/G, with Ya projec-

tive G-manifold and Ga ﬁnite group of algebraic automorphisms. They also showed the con-

jecture for certain complex hypersurfaces with isolated singularities [10, Theorem 4.3]. The

conjecture holds for simplicial projective toric varieties as shown by Maxim and Sch¨

urmann

[29, Corollary 1.2(iii)]. Furthermore, the conjecture was established by the ﬁrst author in [5]

for normal connected complex projective 3-folds Xthat have at worst canonical singularities,

trivial canonical divisor, and dimH1(X;OX)>0. Generalizing the above cases, Fern´

andez de

Bobadilla and Pallar´

es [17] proved the conjecture for all compact complex algebraic varieties

that are rational homology manifolds. In ongoing work with J¨

org Sch¨

urmann, we apply the

methods developed in the present paper to prove the ambient version of the conjecture for

a certain class of subvarieties in Grassmannians. This class includes all Schubert varieties.

Since the homology of Schubert varieties injects into the homology of ambient Grassmanni-

ans, this would imply the conjecture for all Schubert varieties. Furthermore, we shall clarify

how other algebraic characteristic classes such as Chern classes ﬁt into the framework.

Acknowledgement. We thank J¨

org Sch¨

urmann and Laurentiu Maxim for insightful com-

ments on an earlier version of the paper.

Notation. (Co)homology groups will be with rational coefﬁcients unless otherwise stated.

Complex algebraic varieties are not assumed to be irreducible. If Xand Yare subvarieties of

a smooth variety W, then the symbol X∩Ydenotes the set theoretic intersection of Xand Y

in W. Given a set S, the Kronecker delta of two elements a,b∈Sis δab =1 for a=b, and

δab =0 else. The Kronecker product h−,−i is deﬁned by hξ,xi=ε∗(ξ∩x)∈Qfor classes

ξ∈H∗(A)and x∈H∗(A)on a topological space A, where ε∗:H∗(A)→H∗(pt) = Qdenotes

the augmentation map induced by the unique map A→pt.

2. WHITNEY TRANSVERSALITY

Let Wbe a smooth manifold. Recall that a Whitney stratiﬁcation of a closed subset Z⊂W

is a certain decomposition of Zinto locally closed smooth submanifolds of Wsuch that the

pieces of this decomposition ﬁt together via Whitney’s conditions A and B (for details, see

e.g. [23, Section 1.2, p. 37]). Two Whitney stratiﬁed subsets Z1,Z2⊂Ware called transverse

if every stratum of Z1is transverse to every stratum of Z2as smooth submanifolds of W. The

transverse intersection Z1∩Z2then has a canonical Whitney stratiﬁcation in Wwhose strata

are given by the intersections of the strata of Z1and Z2.

Pure-dimensional closed subvarieties of a nonsingular complex algebraic variety can al-

ways be Whitney stratiﬁed by strata of even codimension. In the context of this paper, we

call two such subvarieties Whitney transverse if they can be equipped with transverse Whit-

ney stratiﬁcations. The present section collects various results about Whitney stratiﬁed spaces

that will be applied to Whitney transverse subvarieties later on in the paper.

Lemma 2.1. Let Z1,Z2⊂W be Whitney stratiﬁed subspaces of a smooth manifold W that

are transverse to each other. If U ⊂W is a smooth submanifold that is open as a subset of

Z2, then every stratum of Z1is transverse to U in W .

THE UNIQUENESS THEOREM FOR GYSIN COHERENT CHARACTERISTIC CLASSES OF SINGULAR SPACES 5

Proof. Let z∈Z1∩U. Let S⊂Z1and S0⊂Z2denote the strata containing z. Then, Sand

S0are transverse at zin W. Note that U0:=W\(Z2\U)is an open subset of Wbecause Z2

is a closed subset of W. Since z∈U0, it follows that U0∩Sand U0∩S0are transverse at zin

U0. Since U⊂U0is a smooth submanifold that contains U0∩S0, it follows that U0∩Sand

Uare transverse at zin U0. Consequently, Sand Uare transverse at zin W, and the claim

follows. 

The following topological version of Kleiman’s transversality theorem [26] will be applied

for the canonical action of the Lie group G=GL(V)on the Grassmannian W=Gk(V)for a

complex vector space Vof ﬁnite dimension.

Theorem 2.2. (See e.g. [23, p. 39, Theorem 1.3.6 and Examples 1.3.7].) Let W be a smooth

manifold that is homogeneous under the action of a Lie Group G. Given Whitney stratiﬁed

subspaces Z1,Z2⊂W , the set U of all g ∈G such that g ·Z1and Z2are transverse in W is

dense in G. Moreover, if Z1is compact, then U is also open in G.

Recall that an inclusion g:Y→Xof topological spaces is called (oriented) normally non-

singular (of codimension r) if there is a (oriented) real vector bundle ν:E→Y(of rank r),

a neighborhood U⊂Eof the zero section of ν(which we also denote by Y), and a homeo-

morphism j:U→Xonto an open subset j(U)⊂Xsuch that gfactorizes as the composition

Yincl

−−→ Uj

−→ X(see e.g. [23, Section 1.11, p. 46f]). We also call νa normal bundle of the

normally nonsingular inclusion g. For example, transverse intersections give rise to normally

nonsingular inclusions as follows.

Theorem 2.3. (See Theorem 1.11 in [23, p. 47].) Let X ⊂W be a Whitney stratiﬁed subset

of a smooth manifold W . Suppose that M ⊂W is a smooth submanifold of codimension r

that is transverse to every stratum of X, and set Y =M∩X . Then, the inclusion g :Y→X is

normally nonsingular of codimension r with respect to the normal bundle ν=νM⊂W|Ygiven

by restriction of the normal bundle νM⊂Wof M in W .

An oriented normally nonsingular inclusion g:Y→Xof a closed subset Y⊂Xwith

normal bundle ν:E→Yof rank rinduces a Gysin homomorphism

g!:H∗(X;Q)→H∗−r(Y;Q)

given by the composition

H∗(X)incl∗

−−→ H∗(X,X\Y)e∗

←−

∼

=H∗(E,E0)u∩−

−−→

∼

=H∗−r(E)ν∗

−→

∼

=H∗−r(Y),

where u∈Hr(E,E0)is the Thom class with E0=E\Ythe complement of the zero section of

νin E, and e∗denotes the excision isomorphism induced by the open embedding j:U→X.

Gysin homomorphisms are compatible with pushforward under embeddings as follows.

Proposition 2.4 (Base Change).Consider a cartesian square

Lβ//



Kα

//X

of topological spaces and continuous maps, where f and g are oriented normally nonsingular

inclusions of closed subsets with normal bundles νfand νg, respectively, such that νg=β∗νf.

Then,

β∗g!=f!α∗.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

THEUNIQUENESSTHEOREMFORGYSINCOHERENTCHARACTERISTICCLASSESOFSINGULARSPACESMARKUSBANAGLANDDOMINIKJ.WRAZIDLOABSTRACT.Weestablishageneralcomputationalschemedesignedforasystematiccom-putationofcharacteristicclassesofsingularcomplexalgebraicvarietiesthatsatisfyaGysinaxiominatransversesetup.Thisschemeisexp...

展开>> 收起<<

THE UNIQUENESS THEOREM FOR GYSIN COHERENT CHARACTERISTIC CLASSES OF SINGULAR SPACES MARKUS BANAGL AND DOMINIK J. WRAZIDLO.pdf

共39页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

THE UNIQUENESS THEOREM FOR GYSIN COHERENT CHARACTERISTIC CLASSES OF SINGULAR SPACES MARKUS BANAGL AND DOMINIK J. WRAZIDLO

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: