A note on smooth forms on analytic spaces
aa r X i v : . [ m a t h . C V ] J u l A NOTE ON SMOOTH FORMS ON ANALYTIC SPACES
MATS ANDERSSON & H˚AKAN SAMUELSSON KALM
Abstract.
We prove that any smooth mapping between reduced analytic spacesinduces a natural pullback operation on smooth differential forms. Introduction
There is a natural notion of smooth differential forms on any reduced analyticspace. The dual objects are the currents. Such forms and currents have turned outto be useful tools, e.g., in [4, 5, 7], in the analytic approach to intersection theory[2, 3], and in the context of the ¯ ∂ -equation on analytic spaces [1, 8].It is desirable to be able to take the direct image of a current under a propermap f : X → Z between reduced analytic spaces. By duality this amounts to takepullbacks of smooth forms. In some works, e.g., [2, 3], it is implicitly assumed thatthis is possible. There is an obvious tentative definition of f ∗ φ for a smooth form φ on Z . It is however not clear that it gives a well-defined pullback operation, noteven if f and φ are holomorphic and φ has positive degree; this case is settled in[5, Corollary 1.0.2]. The main problem is when f is the inclusion of an analyticsubvariety contained in Z sing . It was proved in [4, III Corollary 2.4.11] that if f is holomorphic, then the suggested definition indeed gives a functorial operation onsmooth forms. In this short note we give a new proof of this fact. Moreover, weextend it to the case when f is merely smooth, see Theorem 2.1 below. Our resultis implicitly claimed in [6], see Remark 2.2 below. Acknowledgment:
We are grateful to the referee for careful reading and impor-tant comments. 2.
Results
Let X be a reduced analytic space. Recall that, by definition, there is a neighbor-hood U of any point in X and an embedding i : U → D in an open set D ⊂ C N such that U can be identified with its image. For notational convenience we willsuppress U and say that i is a local embedding of X . A smooth ( p, q )-form φ on X reg is smooth on X , φ ∈ E p,q ( X ), if there is a smooth form ϕ in D such that i | ∗ X reg ϕ = φ. If j : X → D ′ is another local embedding, then the identity on X induces a biholo-morphism i ( X ) ∼ −→ j ( X ). Thus, again by definition, locally in D and D ′ , there areholomorphic maps g : D → D ′ and h : D ′ → D such that i = h ◦ j and j = g ◦ i . Since h ∗ ϕ is smooth in D ′ and j | ∗ X reg h ∗ ϕ = φ, Date : July 8, 2020. By an embedding we always mean a closed holomorphic embedding so that the image is ananalytic subvariety.
MATS ANDERSSON & H˚AKAN SAMUELSSON KALM it follows that the notion of smooth forms on X is independent of embedding.We will write i ∗ ϕ for the image of ϕ ∈ E ( D ) in E ( X ). Let [ i ( X )] be the Lelongcurrent of integration over i ( X ) reg . The kernel of i ∗ is closed since i ∗ ϕ = 0 ⇐⇒ ϕ ∧ [ i ( X )] = 0 . Thus, with the quotient topology E ( X ) = E ( D ) / Ker i ∗ is a Fr´echet space. To seethat this topology is independent of the embedding, notice that it is defined by thesemi-norms | φ | X,i := inf {| ϕ | D ; i ∗ ϕ = φ } , where | · | D are the semi-norms defining the topology on E ( D ). Let j , g , and h beas above and let | · | X,j be the analogously defined semi-norms induced by j . Since j ∗ ψ = φ implies that i ∗ g ∗ ψ = φ and since g ∗ : E ( D ′ ) → E ( D ) is continuous we get | φ | X,i = inf i ∗ ϕ = φ | ϕ | D ≤ inf j ∗ ψ = φ | g ∗ ψ | D ≤ C inf j ∗ ψ = φ | ψ | D ′ = C | φ | X,j . In the same way, | φ | X,j ≤ C ′ | φ | X,i and it follows that the semi-norms | φ | X,j give thesame topology as | φ | X,i .Let E p,qX be the sheaf of smooth ( p, q )-forms on X and let E rX = ⊕ p + q = r E p,qX . Wesay that a continuous map f : X → Z between reduced analytic spaces is smooth if f ∗ φ ∈ E X for any φ ∈ E Z . Notice that if i : X → D X and j : Z → D Z are localembeddings, then f is the restriction to i ( X ) of a smooth map D X → D Z . Theorem 2.1.
Let f : X → Z be a smooth map between reduced analytic spaces.There is a well-defined map f ∗ : E r ( Z ) → E r ( X ) with the following property: If φ is a smooth form on Z , i : X → D X and ι : Z → D Z are local embeddings, ϕ is asmooth form in D Z such that ι ∗ ϕ = φ , and ˜ f : D X → D Z is a smooth map such that ˜ f | i ( X ) = f , then (2.1) f ∗ φ = i ∗ ˜ f ∗ ϕ. Assume that g : X → Y and h : Y → Z are smooth maps such that f = h ◦ g . Let j : Y → D Y be a local embedding and ˜ g : D X → D Y and ˜ h : D Y → D Z smooth mapssuch that ˜ g | i ( X ) = g and ˜ h | j ( Y ) = h , respectively. Notice that the restriction of ˜ h ◦ ˜ g to i ( X ) is f . If φ ∈ E ( Z ) and φ = ι ∗ ϕ it follows by Theorem 2.1 that h ∗ φ = j ∗ ˜ h ∗ ϕ and g ∗ h ∗ φ = i ∗ ˜ g ∗ ˜ h ∗ ϕ = f ∗ φ . Hence,(2.2) f ∗ φ = g ∗ h ∗ φ, φ ∈ E ( Z ) . Remark 2.2.
An a priori different definition of smooth forms on X is given in [6,Section 3.3]. If i : X → D is a local embedding, then the space of smooth forms on X is defined in [6, Section 3.3] as E ( D ) / N ( D ), where N ( D ) is the space of smoothforms ϕ in D such that for any smooth manifold W and any smooth map g : W → D with g ( W ) ⊂ X one has g ∗ ϕ = 0.It is clear that N ( D ) ⊂ Ker i ∗ and it is in fact claimed in [6, Section 3.3] that N ( D ) = Ker i ∗ , but we have not been able to find a proof in the literature. It followsfrom Theorem 2.1 that the claim indeed is true: If g : W → D is a smooth map with g ( W ) ⊂ X , then g = i ◦ γ for a smooth map γ : W → X . In view of (2.2) thus g ∗ ϕ = γ ∗ i ∗ ϕ = 0 if i ∗ ϕ = 0.The space of currents on X , C ( X ), is the dual of the space of test forms, i.e.,compactly supported forms in E ( X ), cf. [7, Section 4.2]. Let f : X → Z be as in NOTE ON SMOOTH FORMS ON ANALYTIC SPACES 3
Theorem 2.1 and assume that f is proper. Then f ∗ φ is a test form on X if φ is atest form on Z . If µ is a current on X thus f ∗ µ is a current on Z defined by(2.3) f ∗ µ.φ = µ.f ∗ φ. By Theorem 2.1 and (2.2) we get
Corollary 2.3.
Let f : X → Z be a smooth proper map between reduced analyticspaces. Then the induced mapping f ∗ : C ( X ) → C ( Z ) has the property that if f = h ◦ g , where g : X → Y and h : Y → Z are smooth proper maps, then f ∗ µ = h ∗ g ∗ µ, µ ∈ C ( X ) . Example 2.4.
Suppose that i : X → D is an embedding and consider the inducedmapping i ∗ : C ( X ) → C ( D ). It follows from (2.3) and the definition of test forms on X that i ∗ is injective. Thus C ( X ) can be identified with its image i ∗ C ( X ). In viewof the definition of C ( X ) and (2.3) it follows that i ∗ C ( X ) is the set of currents µ in D such that µ.ϕ = 0 if i ∗ ϕ = 0. Notice in particular that i ∗ i ( X )].3. Proofs
We will prove Theorem 2.1 by showing that the right-hand side of (2.1) is indepen-dent of the choices of embeddings i , ι and extensions ˜ f and ϕ of f and φ , respectively.The technical part is contained in Proposition 3.2, cf. [4, Proposition III 2.4.10] and[5, Proposition 1.0.1]. We begin with the following lemma. Lemma 3.1.
Let M be a reduced analytic space, N a complex manifold, and p : M → N a proper holomorphic map. If dim N = d ≥ and rank x p < d for all x ∈ M reg ,then p is not surjective.Proof. If M is smooth it follows from the constant rank theorem that p cannotbe surjective. If M is not smooth, let π : f M → M be a Hironaka resolution ofsingularities. Then f M is smooth and ˜ p := p ◦ π is a proper holomorphic map withthe same image as p . Since π is a biholomorphism outside the exceptional divisor E = π − ( M sing ) we have rank x ˜ p < d for all x ∈ f M \ E . By semi-continuity of therank it follows that rank x ˜ p < d for all x ∈ f M . By the constant rank theorem thus ˜ p cannot be surjective. (cid:3) Proposition 3.2.
Let D ⊂ C N be an open set and let ϕ be a smooth form in D . (i) Let W ⊂ V be analytic subsets of D . If the pullback of ϕ to V reg vanishes,then the pullback of ϕ to W reg vanishes. (ii) Let W be a smooth not necessarily complex submanifold of D , let V ⊂ D bean analytic subset, and assume that W ⊂ V . If the pullback of ϕ to V reg vanishes, then the pullback of ϕ to W vanishes.Proof of part (i). We may assume that W is irreducible of dimension d . We mayalso assume that ϕ has positive degree since a smooth function vanishing on V reg must vanish on W by continuity. The case d = 0 is then clear since the pullback ofa form of positive degree to discrete points necessarily vanishes. Let ˜ π : V ′ → V bea Hironaka resolution of singularities. Suppose that W ′ ⊂ V ′ is analytic and suchthat ˜ π ( W ′ ) = W . Let π = ˜ π | W ′ and let φ be the pullback of ϕ to W reg . Since thepullback of ϕ under W ′ ֒ → V ′ → V ֒ → D is 0, it follows that π ∗ φ = 0. We will findsuch W ′ and π such that π ∗ φ = 0 implies φ = 0. MATS ANDERSSON & H˚AKAN SAMUELSSON KALM
To begin with we set W ′ = ˜ π − ( W ). If ˜ π ( W ′ sing ) = W , replace W ′ by W ′ sing .Possibly repeating this we may assume that ˜ π ( W ′ sing ) * W . Thus ˜ π ( W ′ sing ) is aproper analytic subset of W . Set π = ˜ π | W ′ and notice that π : W ′ → W is properand surjective.Let M = W ′ \ π − ( W sing ∪ π ( W ′ sing )) , N = W reg \ π ( W ′ sing ) , and let p = π | M . Since M is smooth and p : M → N is proper and surjective itfollows from the constant rank theorem that there is x ∈ M such that rank x p = d .Since d is the optimal rank of p this holds for x in a non-empty Zariski-open subsetof M . Let f M = { x ∈ M ; rank x p ≤ d − } be the complement of this set. Thenrank x p | f M ≤ d − x ∈ f M reg . By Lemma 3.1, p ( f M ) * N and thus p ( f M ) is aproper analytic subset of N .Now, N \ p ( f M ) is a dense open subset of W reg and so it suffices to show that φ = 0there. However, M \ p − p ( f M ) is a (non-empty) open subset of M and in this set p has constant rank = d = dim W . Thus, p is locally a simple projection and it followsthat if p ∗ φ = 0, then φ = 0. Proof of part (ii).
We use induction over dim V . The case dim V = 0 is clear sosuppose that dim V > w ∈ W . If w ∈ V reg , then there is a neighborhood U ⊂ W of w contained in V reg . Then clearly the pullback of ϕ to U vanishes. Assume now that w ∈ V sing . If there is a neighborhood U ⊂ W of w contained in V sing , then thepullback of ϕ to U vanishes in view of the induction hypothesis and part (i) of thisproposition. If not, then there is a sequence of points w j ∈ W converging to w suchthat w j ∈ V reg . Then there are neighborhoods U j ⊂ W of w j contained in V reg . Thepullback of ϕ to U j vanish. Now, ϕ ( w ) is a multilinear mapping on T w D dependingcontinuously on w . Since the pullback of ϕ to U j vanish the restriction of ϕ ( w j ) to T w j W vanish. By continuity thus the restriction of ϕ ( w ) to T w W vanishes.Hence, for any w ∈ W , the restriction of ϕ ( w ) to T w W vanishes; thus the pullbackof ϕ to W vanishes. (cid:3) Proof of Theorem 2.1.
Let φ ∈ E ( Z ) and let f ∗ φ be the form on X reg defined by theright-hand side of (2.1). Clearly f ∗ φ is smooth on X . As mentioned, we will showthat it is independent of the choices of extensions ˜ f and ϕ as well as of the localembeddings.First assume that X is smooth. The set X ⊂ X of points where f has maximalrank is open. By the constant rank theorem each point x ∈ X has a neighborhood U x such that f | U x is a submersion onto a smooth submanifold f ( U x ) of D Z containedin Z . By Proposition 3.2 (ii), if ϕ is a smooth form in D Z such that ι ∗ ϕ = φ , thenthe pullback of ϕ to f ( U x ) only depends on the pullback of ϕ to Z reg , i.e., onlyon φ . Thus, f ∗ φ is well-defined in U x . Hence, f ∗ φ is well-defined in X and so,by continuity, well-defined in the closure X . Repeating the argument with X and f replaced by X \ X and f | X \ X it follows that f ∗ φ is well-defined in X , where X ⊂ X \ X is the set of points where f | X \ X has maximal rank. Notice thatthis rank is strictly less than the rank of f in X . Thus, continuing the process ofconstructing such open sets X k , after a finite number of steps we get X k = ∅ . Since X = ∪ j X j , f ∗ φ is well-defined in X . NOTE ON SMOOTH FORMS ON ANALYTIC SPACES 5
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M. Andersson, H. Samuelsson Kalm, Department of Mathematical Sciences, Divisionof Algebra and Geometry, University of Gothenburg and Chalmers University ofTechnology, SE-412 96 G¨oteborg, Sweden
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