Singular Locus and Differential Forms

In this section we characterize singular points of complex spaces and of mor- phisms of complex spaces. One of the aims is to show that these sets are analytically closed.

Recall thatxis aregular(orsmooth)point ofX, iff the local ringOX,xis regular (cf. Definition 1.40);xis a singular point iff it is not regular. The set of singular points of X is referred to as the singular locus of X, denoted by Sing(X).

IfX ispure dimensional, that is, if the dimension dim(X, x) is independent of x∈X, then we can easily give a local description of Sing(X). Since any isomorphismX →Y of complex spaces maps Sing(X) isomorphic to Sing(Y) (since dim(X, x) and edim(X, x) are preserved under isomorphisms), we may assume thatX is a complex model space. In this situation we have

Proposition 1.104.LetX be a puren-dimensional complex subspace ofC^m with ideal sheafI. Ifx∈X andIx=f1, . . . , fk · OC^m,x withf1, . . . , fkholo- morphic functions in a neighbourhood U ofxthen

Sing(X)∩U = y∈X∩Urank_∂fi

∂xj(y)

< m−n .

In particular, there is a canonical ideal sheaf JSing(X) such thatJSing(X)|U is generated by f₁, . . . , f_k and all(m−n)-minors of the Jacobian matrix _∂fi

∂x_j

with V(JSing(X)) = Sing(X).

Proof. By Lemma 1.22, rank_∂fi

∂xj(y)

= jrk(Iy) =m−edim(X,y). Hence, y∈Sing(X)∩U iff rank_∂fi

∂x_j(y)

< m−dim(X,y). The result follows since

X is purelyn-dimensional.

IfX has several irreducible components,X =X₁∪. . .∪X_rthen Sing(X) =

,r i=1

Sing(X_i)∪ ,

i<j

(X_i∩X_j),

as will be shown in the exercises. As X_i is pure dimensional, Sing(X_i) is analytic in X, by Proposition 1.104. The intersection of two irreducible com- ponents is analytic, too. Hence, Sing(X) is analytic. We can use locally a primary decomposition of 0 ⊂ OX,x to define an ideal for Sing(X) locally at x. But, since a primary decomposition is not unique it is not clear how to glue these locally defined sheaves to get a well-defined global ideal sheaf for Sing(X).

In the following, we shall give a different proof of the analyticity of Sing(X), which provides Sing(X) with a canonical structure, even if X is not pure dimensional. For this, we use differential forms.

Before we introduce differential forms, let us first recall the notion of derivations.

Definition 1.105.Let A be a B-algebra and M an A-module. Then a B- derivation with values inM is aB-linear mapδ:A→M satisfying the prod- uct rule, also called theLeibniz rule,

δ(f g) = δ(f)g+f δ(g), f, g∈A . The set

DerB(A, M) := δ:A→Mδis aB-derivation

⊂HomB(A, M) is via (a·δ)(f) :=a·δ(f) an A-module, the module of B-derivations of A with values inM.

We consider first the case B=C. It is easy to see that for A = C{x} = C{x1, . . . , xn} the partial derivatives _∂x^∂

i,i= 1, . . . , n, are a basis of the free C{x}-module Der_C(C{x},C{x}).

For each local ring (A,m) and each A-derivation δ:A→M we have δ(m^k)⊂m^k−1M for allk >0. Note also that, by the Leibniz rule and Krull’s intersection theorem, any derivationδis already uniquely determined by the valuesδ(xi) forx1, . . . , xn a set of generators form.

In particular, for A=C{x1, . . . , xn}, each δ∈Der_C(C{x}, M) has a unique expression

δ = n i=1

δ(xi)· ∂

∂xi

. (1.10.1)

Now, let us define differential forms.

Theorem 1.106.Let Abe an analyticC-algebra.

(1) There exists a pair (Ω¹_A, dA) consisting of a finitely generated A-module Ω¹_A and a derivation dA:A→Ω_A¹ such that for each finitely generated A-moduleM the A-linear morphism

θM : HomA(Ω_A¹, M)−→Der_C(A, M), ϕ −→ϕ◦dA, is an isomorphism ofA-modules.

(2) The pair(Ω¹_A, dA) is uniquely determined up to unique isomorphism.

(3) If A=C{x1, . . . , xn} then Ω¹_A is free of rank n with basis dx1, . . . , dxn

andd=dA:A→Ω¹_A is given by df =

n i=1

∂f

∂xi

dxi. (4) IfA=C{x₁, . . . , x_n}/I then

Ω_A¹ = Ω_C{¹ _x}

I·Ω_C{¹ _x}+C{x} ·dI

withdA:A→Ω_A¹ induced byd:C{x} →Ω_C{¹ _x}. In particular,Ω¹_Ais gen- erated, as A-module, by the classes ofdx1, . . . , dxn.

The pair (Ω_A¹, dA) is called the module of (K¨ahler) differentials. We usually write dinstead ofdA.

Proof. Once we have shown the defining property of the modules constructed in (3) and (4), (1) is obviously satisfied. Moreover, (2) follows from (1), by the usual abstract argument.

(3) Ω_A¹ =Adx1⊕. . .⊕Adxnis finitely generated andd:A→Ω_A¹ is a deriva- tion. We have to show thatθ=θM is bijective. Ifθ(ϕ) = 0 then

θ(ϕ)(x_i) = ϕ(dx_i) = 0, i= 1, . . . , n , hence ϕ= 0, andθ is injective.

Given a derivationδ∈Der_C(A, M) defineϕ∈HomA(Ω_A¹, M) byϕ(dxi) = δ(xi). Then

θ(ϕ)(f) = ϕ(df) = ϕ _n

i=1

∂f

∂x_idxi

= n i=1

∂f

∂x_iδ(xi) = δ(f), by (1.10.1). That is, θis surjective, too.

(4) One checks directly that d:C{x}/I→Ω_C{¹ _x}/(I·Ω¹_C{x}+C{x} ·dI) is well-defined (by the Leibniz rule), and a derivation. If M is a finite A- module then it is also a finite C{x}-module. We set N =C{x}dI+IΩ, Ω=Ω¹_C{x}. Induced by the exact sequences 0→N/IΩ→Ω/IΩ→Ω_A¹ →0 and 0→I→C{x} →A→0, we have a commutative diagram with exact rows

Hom_C{x}(N/IΩ, M) Hom_C{x}(Ω/IΩ, M)

∼=

Hom_C{x}(Ω_A¹, M) 0 Der_C(I, M) Der_C(C{x}, M) Der_C(A, M) 0 where the vertical arrows are given byϕ →ϕ◦d. The middle arrow is bijective by (3) and the left one is injective by a direct check. It follows that the right-

hand one is bijective, too.

Lemma 1.107.For each analytic C-algebra A there are canonical isomor- phisms

Ω_A¹

mAΩ_A¹ −→^∼= mA

m²_A, Der_C(A,C)−→^∼= Hom_C(mA/m²_A,C). In particular, edim(A) = mng(Ω_A¹) = dim_C

Der_C(A,C) . Proof. Consider thepoint derivation (puttingm=mA)

δ0:A−→m/m², f −→(f−f(0))(mod m²), which is an element of Der_C(A,m/m²).

Since HomA(Ω_A¹,m/m²)→Der_C(A,m/m²) is bijective, there is a unique homomorphismϕ0:Ω_A¹ →m/m² such thatϕ0◦d=δ0. Sinceδ0is surjective, so is ϕ₀. From ϕ₀(mΩ_A¹)⊂m(m/m²) = 0 we get that ϕ₀ induces a surjec- tive morphismϕ₀:Ω_A¹/mΩ_A¹ →m/m². On the other hand, Theorem 1.106 (4) implies mng(Ω_A¹) = dim_C(Ω_A¹/mΩ_A¹)≤dim_C(m/m²). Hence,ϕ₀is an isomor- phism.

Dualizing this isomorphism and using Hom_C(M/mM,C) = Hom_C(M,C) and the universal property ofΩ_A¹, we get the second isomorphism.

Proposition 1.108.For each morphismϕ:A→Bof analytic algebras there is a unique A-module homomorphism dϕ:Ω¹_A→Ω_B¹ making the following diagram commutative

A ^ϕ

dA

B

dB

Ω¹_A ^dϕ Ω_B¹

dϕis called thedifferential ofϕ. It satisfies thechain ruled(ψ◦ϕ) =dψ◦dϕ.

Proof. SinceΩ_A¹ =A·dAA,dϕmust satisfy

dϕ

i

gi·dA(fi)

=

i

gi·dϕ(dA(fi)) =

i

gi·dB(ϕ(fi)), and, hence, dϕ is uniquely defined if it exists. For the existence, let A = C{x}/I,B=C{y}/Jandϕ:C{x} →C{y}a lifting ofϕ(Lemma 1.14). We define

dϕ:Ω_C{¹ _x}→Ω_C{¹ _y}, dϕ(dx i) :=d_C{y}(ϕ(x i)),

which is well-defined, sinceΩ_C{¹ _x} is free and generated by thedxi. It is now straightforward to check thatdϕinduces, via the surjections of Theorem 1.106,

anA-linear mapΩ_A¹ →Ω_B¹.

Now, letX be a complex space andx∈X. Moreover, letU ⊂X be an open neighbourhood ofxwhich is isomorphic to a local model spaceY defined by a coherent ideal sheafI ⊂ OD. Here,Dis an open subset ofCⁿ, and we may assume thatI is generated f1, . . . , fk∈Γ(D,OD). The sheaf Ω¹_D is defined to be the free sheaf ODdx1⊕. . .⊕ ODdxn and the derivation d:OD→Ω_D¹ is defined bydf=n

i=1

∂f

∂x_idxi.

Definition 1.109.LetOY =OD/I andI=f1, . . . , fkOD. We define Ω_Y¹ := Ω_D¹

IΩ_D¹+ODdI

Y ,

whereODdI is the subsheaf ofΩ¹_Dgenerated bydf1, . . . , dfk, andIΩ¹_Dis the subsheaf of Ω¹_D generated by fjdxi, i= 1, . . . , n, j= 1, . . . , k. The induced derivation is denoted bydY :OY →Ω¹_Y.

Finally, if ϕ:U →Y is an isomorphism to the local model space Y, we de- fineΩ_U¹ :=ϕ^∗Ω_Y¹ whereϕ^∗Ω_Y¹ is the analytic preimage sheaf (A.6). Theorem 1.106 (2) implies that Ω_U¹ is, up to a unique isomorphism, independent of the choice of ϕ.

It follows that we can glue the locally defined sheavesΩ_U¹ to get a unique sheafΩ_X¹ onX, thesheaf of holomorphic (K¨ahler) differentialsorholomorphic 1-formsonX, and a unique derivationd_X:OX →Ω_X¹ (A.2).Ω_X¹ is a coherent OX-module (A.7), and it satisfies

Ω¹_X,x=Ω_O¹

X,x for eachx∈X.

It is now easy to prove the important regularity criterion for complex spaces.

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