Chern and Chern-Schwartz-MacPherson Classes

The total Chern class of a j-dimensional nonsingular variety V is defined as the Chern class of the tangent bundle TV, we write this as c(V) = c(TV)· [V] in the Chow ring ofV, A∗(V). See Fulton [10, §3.2] for a definition of the Chern class of

a vector bundle. In this chapter and in Chapter3we will abuse notation and write

c(V) for the pushforward to Pn of the total Chern class of V (as we also do with

cS M and Segre classes). As a consequence of the Gauss-Bonnet-Chern theorem (or the Grothendieck-Riemann-Roch theorem, see for example Sch¨urmann and Yokura [22]), we have that the degree of the zero dimensional component of the total Chern class of a projective variety is equal to the Euler characteristic, that is

Z

c(TV)·[V]= χ(V). (2.6) There are several known generalizations of the total Chern class to singular vari- eties. All of these notions agree with c(TV)· [V] for nonsingular V, however the Chern-Schwartz-Macpherson class is the only one of these that satisfies a property analogous to (2.6) for anyV, i.e.

Z

cS M(V)= χ(V). (2.7) We review here the construction of thecS M classes, given in the manner considered by MacPherson [21]. For a scheme V, letC(V) denote the abelian group of finite

linear combinations P

WmW1W, where W are (closed) subvarieties ofV, mW ∈ Z, and1W denotes the function that is 1 inW, and 0 outside ofW. Elements f ∈ C(V) are known as constructible functions and the groupC(V) is referred to as the group of constructible functions onV. To makeCinto a functor we letCmap a schemeV

to the group of constructible functions onV and a proper morphism f :V1 →V2is mapped byCto

C(f)(1W)(p)= χ(f−1(p)∩W), W ⊂V1, p∈V2 a closed point.

Another functor from algebraic varieties to albelian groups is the Chow group func- tor A∗. ThecS M class may be realized as a natural transformation between these two functors.

Definition 2.1.2. The Chern-Schwartz-MacPherson class is the unique natural trans- formation between the constructible function functor and the Chow group functor, that is cS M :C → A∗is the unique natural transformation satisfying:

• (Normalization) cS M(1V)= c(TV)·[V]for V non-singular and complete.

• (Naturality) f∗(cS M(φ)) =cS M(C(f)(φ)), for f : X →Y a proper transforma-

tion of projective varieties,φa constructible function on X.

For a scheme V let Vred denote the support of V, the notationcS M(V) is taken to meancS M(1V) and hence, since1V =1Vred, we denotecS M(V)=cS M(Vred).

To see how thecS M class satisfies the relation (2.7) consider the morphismf:V → point,applying the naturality property of thecS M class we have

f∗(cS M(V))=cS M(C(f)(1V))=cS M(χ(V)1point)=χ(V)cS M(point)=χ(V)[point]. This gives us (2.7). Note that the cS M classes (and constructible functions) also satisfy the same inclusion/exclusion relation as the Euler characteristic, i.e. for the Euler characteristic we have

χ(V1∪V2)=χ(V1)χ(V2)−χ(V1∩V2).

Constructible functions inherit this property from the Euler characteristic via the definition of the constructible function functor, specifically we have1V1∪V2 = 1V1 +

1V2 − 1V1∩V2. From this we see that the cS M classes will also possess an inclu-

sion/exclusion property, giving us the relation Recall that from the construction of thecS M class we see thatcS M classes will also possess an inclusion/exclusion prop- erty similar to that of the Euler characteristic, in particular forV1,V2subschemes of projective spacePnwe have that

cS M(V1∩V2)= cS M(V1)+cS M(V2)−cS M(V1∪V2). (2.8) Note that this relation forcS Mclasses will allow us to reduce all computation ofcS M classes to the case of hypersurfaces. From this property we obtain the following proposition, discussed informally by Aluffi [2]; Proposition 2.1.3follows directly from (2.8).

Proposition 2.1.3. Let V = X1∩ · · · ∩Xr = V(f1)∩ · · · ∩V(fr)be a subscheme of Pn = Proj(k[x0, . . . ,xn]). Write the polynomials defining V as F = (f1, . . . , fr)and let F{S} =Qi∈S fi for S ⊂ {1, . . . ,r}. Then,

cS M(V)=

X

S⊂{1,...,r}

(−1)|S|+1cS M V(F{S})

where|S|denotes the cardinality of the integer set S .

Remark 2.1.4. The following special case is from Suwa [25]. Let X be a smooth subvariety ofPnwhich is a global complete intersection, further suppose that X =

V(f0, . . . , fr)with di = degfi, then we have

cS M(X)=c(X)=(1+h)n+1· codimX Y i=0 dih 1+dih in A∗(Pn), (2.9)

recall that c(X)= c(TX)·[X]is the total Chern class of the smooth variety X. We note that using Remark 2.1.4 the computation of cS M classes could be made much more efficient in the particular case where the input scheme is a complete intersection which is known to be smooth.

As noted in Example1.2.1above, when working inPn, there is a very concrete rela- tionship between thecS M class and the Euler chacteristic of general linear sections, in particular it was shown by Aluffi[4] that there is an involution between these two objects, we state this result below.

Theorem 2.1.5(Theorem 1.1 Aluffi[4]). Let V be any locally closed set inPn. Let

Vr = V∩L1∩ · · · ∩Lr be the intersection of V with r general hyperplanes. Define

the polynomial having degree at most n specified by χV(t) :=

X

r≥0

χ(Vr)·(−t)r.

Define another polynomial of degree at most n given by γV(t) :=

X

r≥0

γr·(−t)r

hereγr = cS M(V)r is the coefficent of the dimension r componet of cS M(V), that is;

the polynomial γV(t) is obtained by replacing[Pr] hn−r with tr in cS M(V). Also

define the mapIspecifed by

p(t)7→ I(p) := t· p(−t−1)+p(0)

t+1 .

ThenIis an involution and we have:

χV(t)=I(γV(t)), γV(t)=I(χV(t)). (2.10)