Moduli of simple holomorphic pairs and effective divisors

Siegmund Kosarew  Paul Lupascu^{∗}

ABSTRACT: In this note we identify two complex structures (one is given by algebraic geometry, the other by gauge theory) on the set of isomorphism classes of holomorphic bundles with section on a given compact complex manifold. In the case of line bundles, these complex spaces are shown to be isomorphic to a space of effective divisors on the manifold.

Introduction

Let (X, O_{X}) be a compact complex analytic space. We denote by Div^{+}(X) the space of all Cartier divisors (including the empty one) on X. This set is a Zariski open subspace in the entire Douady space D(X), parameterizing all compact subspaces of X. If X is smooth (or more generally locally factorial), then Div^{+}(X) is a union of connected components of D(X) [?].

We consider pairs (E, ϕ) consisting of an invertible sheaf E over X coupled with a holomorphic section ϕ in E, which is locally a non zero divisor. Two such pairs (E_{i}, ϕ_{i}) are called equivalent if there exists an isomorphism of sheaves Θ : E_{1} ⟶ E_{2} such that Θ ∘ ϕ_{1} = ϕ_{2}. The set of equivalence classes is called the moduli space of simple holomorphic pairs of rank one on X. This moduli space can be given a structure of a complex analytic space using standard results of deformation theory (see [KO]). There exists a natural bijective map from the moduli space of simple holomorphic pairs of rank one into Div^{+}(X) sending the equivalence class of a pair (E, ϕ) into the divisor, given by the vanishing locus of the section ϕ. In the first part of the paper, we prove that this one-to-one correspondence is in fact an analytic isomorphism with respect to these natural structures on the two spaces.

If X is a smooth compact complex manifold, there is a second possibility of defining an analytic structure on the moduli space of simple holomorphic pairs, namely by using gauge-theoretical methods (compare [OT],[S]). In the last part we show that, in this case, these two structures are isomorphic. This is the ”pair”-version of a previous result due to Miyajima [M] in the case of moduli spaces of simple bundles.

∗Partially supported by SNF, nr. 2000-055290.98/1

Throughout this paper we adopt the following notations: (sets) : the category of sets; (an) : the category of (not necessarily reduced) complex spaces; (an/R) : the category of relative complex spaces over a complex space R; (germs): the category of germs of complex spaces; h_{S} : the canonical contravariant functor h_{S} : C ⟶ (sets), h_{S} = Hom( · , S)

associated to an object S belonging to a category C.

Aknowledgements: The second author would like to thank his advisors Prof. Dr. Ch.Okonek and PD Dr. A.Teleman for their support and for helpful discussions.

1 Main result

The purpose of this part is to prove the following

Theorem 1.1 Let X be a compact complex space with Γ(X, O_{X}) = ℂ. Then there exists a natural complex analytic isomorphism between the moduli space of simple holomorphic pairs of rank one and the space Div^{+}(X).

Remark. Let (X, O_{X}) be a compact complex space with Γ(X, O_{X}) = ℂ. Then:

i.) The Picard functor \underline{Pic}_{X} is representable (cf. [Bi1, p.337]). ii.) There exists a Poincaré line bundle over X × Pic(X).

The existence of a Poincaré bundle can be seen as follows (see [Br, p.55] in the smooth case). The Leray spectral sequence for the projection morphism π : X × S ⟶ S leads to the exact sequence

0 → Pic(S) → Pic(X × S) → H^{0}(R^{1}π_{∗}O_{X×S}^{∗}) → H^{2}(π_{∗}O_{X×S}) → H^{2}(O_{X×S}^{∗}) .

Since Γ(X, O_{X}) = ℂ, the last morphism is injective, hence one gets an exact sequence

0 → Pic(S) → Pic(X × S) → H^{0}(R^{1}π_{∗}O_{X×S}^{∗}) ≅ Hom(S, Pic(X)) → 0 . (1)

For S = Pic(X), this sequence leads to a line bundle P on X × Pic(X) such that P|_{X×{[ξ]}} ≅ ξ for every [ξ] ∈ Pic(X).

A complex structure on the space of simple holomorphic pairs is given as follows: Let P be a Poincaré bundle on X × Pic(X). By [Fl] there exists a linear fiber space H over Pic(X), which represents the functor H : (an/Pic(X)) ⟶ (sets) given by

(S ⟶^{f} Pic(X)) ⟼ Hom(O_{X×S}, (id_{X} × f)^{∗}P) .

(The action of H on morphisms is given by ”pull-back”.) In particular, for every complex space S over P ic(X), there is a bijection

Hom(O_{X×S}, (id_{X} × f)^{∗}P) ≅ Hom_{P ic(X)}(S, H) .  (2)

Let Õ ⊂ H be the subset consisting of zero divisors. The multiplicative group ℂ^{∗} operates on H′ := H \ Õ such that the quotient P(H′) := H′/ℂ^{∗} becomes an open subset of a projective fiber space over Pic(X). The fiber over [ξ] ∈ Pic(X) can be identified with an open subset of P H^{0}(X, ξ). Moreover, P(H′) coincides settheoretically with the moduli space of simple holomorphic pairs, defining a natural analytic structure on it.

In order to prove that P(H′) and Div^{+}(X) are isomorphic as complex spaces, it suffices to prove that the associated functors h_{P(H′)} and h_{Div^{+}(X)} are isomorphic. More precisely, we show that both are isomorphic to the contravariant functor F : (an) ⟶ (sets), defined by S ⟼ F (S), where F (S) denotes the equivalence classes of pairs (E, ϕ), where E is an invertible sheaf on X ×S, and ϕ is a holomorphic section in E whose restriction to each fiber X × {s} is locally a non zero divisor. (Two such tuples are called equivalent if there exists an isomorphism of sheaves Θ : E_{1} ⟶ E_{2} such that Θ ∘ ϕ_{1} = ϕ_{2}.)

Remark. Note that if (E, ϕ) ∼ (E, ϕ), then necessarily Θ = id_{E}. Using this simple observation, it is easy to see that the functor F is of local nature (it is a sheaf), i.e. given any complex space S together with an open covering S = ∪S_{i}, the following sequence is exact:

F (S) ⟶ ∏ F (S_{i})⟶ ∏ F (S_{i} ∩ S_{j}) .

Recall that a relative Cartier divisor in X × S over S is a Cartier divisor Z ⊂ X × S which is flat over S. We denote by Div_{S}^{+}(X) the set of all relative Cartier divisors (including the empty one) over a fixed S.

Lemma 1.2 Let S be a fixed complex space. The map

Z : F (S) ⟶ Div_{S}^{+}(X)

sending the class of (E, ϕ) into the Cartier divisor given by the vanishing locus of the section ϕ is well defined and bijective.

Proof. It is clear that the vanishing locus Z := Z(ϕ) ⊂ X × S depends only on the isomorphism class of (E, ϕ). We need to show that Z is flat over S. Take (x, s) ∈ Z, denote A := O_{S,s}, B := O_{X×S,(x,s)} and let 𝔪 ⊂ A be the maximal ideal. Since the section ϕ restricted to each fiber X × {s} is locally a non zero divisor, we have O_{Z,(x,s)} = B/uB for some nonzero divisor u. The flatness of the morphism A ⟶ B/uB follows then from the general Bourbaki-Grothendieck criterion [Fi, p. 152] since

Tor_{1}^{A}(A/𝔪, B/uB) = ker(B/𝔪B ⟶^{·u} B/𝔪B) = {0}.

We take two simple pairs (E_{i}, ϕ_{i}) over X ×S, defining the same divisor Z := Z(ϕ_{1}) = Z(ϕ_{2}). The invertible sheaf O_{X×S}(Z) admits a canonical section ϕ_{can}, and both pairs (E_{i}, ϕ_{i}) are equivalent to the pair (O_{X×S}(Z), ϕ_{can}). This proves the injectivity. Given Z ∈ Div_{S}^{+}(X), the associated canonical section ϕ_{can} restricted to each fiber X×{s} is locally a non zero divisor since Z lies flat over S. This gives the surjectivity.

The above lemma shows that there exists a natural isomorphism of functors between F and the functor G : (an) ⟶ (sets),

G(S) := Div_{S}^{+}(X) ⊂ D(X × S) .

It follows from the general result of Douady [?] that G is a representable functor and its representation space is exactly Div^{+}(X). In order to prove theorem ?? it suffices to show the following

Theorem 1.3 The functor F is representable by the space of simple holomorphic pairs P(H′).

Proof. Since F is a sheaf, it suffices to prove that F and h_{P(H′)} are isomorphic as functors defined on the category of germs of analytic spaces. The following observation shows that h_{P(H′)} is isomorphic to the sheafified functor associated to the quotient functor h_{H′}/h_{ℂ}^{∗}:

Lemma 1.4 Let M be a complex analytic space, and let G be a complex Lie group acting smoothly and freely on M. Suppose that the quotient M/G exists in the category of analytic spaces such that the canonical morphism M ⟶ M/G is smooth. Then the canonical morphism of functors

(h_{M}/h_{G})^{#} ⟶ h_{M/G} .  (3)

is an isomorphism (The superscript ^{#} denotes here the associated sheafified functor.)

Proof. Since the projection M ⟶ M/G is smooth, one has an epimorphism of sheaves h_{M} ⟶ h_{M/G}, hence the morphism (??) is also an epimorphism. Furthermore, since G acts smoothly and freely on M, the morphism G × M ⟶ M ×_{M/G} M, (g, m) ⟼ (m, gm) is an isomorphism by the relativ implicit function theorem. This shows that (??) is also a monomorphism, i.e. an isomorphism.

Let p : H′ ⟶ Pic(X) be the natural morphism and consider the corresponding tautological homomorphism

u : O_{X×H′} ⟶ (id_{X} × p)^{∗}P

given by the bijection (??). The morphism of functors h_{H′} ⟶ F is defined by sending ϕ : S ⟶ H′ to the isomorphism class of the simple pair

O_{X×S} ⟶^{ϕ^{∗}u} (id_{X} × p ∘ ϕ)^{∗}P .

(Note that ϕ^{∗}u|_{X×{s}} = ϕ(s) ∈ H′ for every s ∈ S.)

\underline{Injectivity:} Let ϕ_{1}, ϕ_{2} be two morphisms from S to H′ such that the associated simple pairs are isomorphic. It follows in particularly that the two sheaves (id_{X} × p ∘ ϕ_{i})^{∗}P are isomorphic via some Θ. The sequence (??) implies p ∘ ϕ_{1} = p ∘ ϕ_{2}. The isomorphism Θ becomes an automorphism, and is given by multiplication with some element a_{Θ} ∈ H^{0}(X × S, O_{X×S}^{∗}) ≅ H^{0}(S, O_{S}^{∗}) (since Γ(X, O_{X}) = ℂ). It follows that ϕ_{1}, ϕ_{2} are conjugate under the action of a_{Θ} : S ⟶ ℂ^{∗}, i.e. the morphism

(h_{H′}/h_{ℂ}^{∗})^{#} ⟶ F

is injective.

\underline{Surjectivity:} Consider a germ (S, 0) of complex space and a simple pair (O_{X×S} ⟶^{α} E). The corresponding morphism f ∈ Hom_{P ic(X)}(S, H′) has the property

E ≅ (id_{X} × p ∘ f)^{∗}P ⊗ L

for some [L] ∈ Pic(S). However, since we are working on the germ S, we may assume that L is trivial. In this way, we obtain a simple pair (O_{X×S} ⟶^{α′} (id_{X} × p ∘ f)^{∗}P), which by (??) leads to a morphism from S to H′. This proves the surjectivity, and completes the proof of theorem ??.

Corollary 1.5 Let (X, O_{X}) be a compact, reduced, connected and locally irreducible analytic space, and let m ∈ H^{2}(X, ℤ) be a fixed cohomology class. Consider the closed subspace of Div^{+}(X) given by

Dou(m) := { Z ∈ Div^{+}(X) | c_{1}(O_{X}(Z)) = m } .

Then there exists an isomorphism of complex spaces

Dou(m) ≅ { (E, ϕ) | E invertible sheaf on X, c_{1}(E) = m, 0 ≠ ϕ ∈ H^{0}(X, E)}/ ∼ .

Proof. One has a commutative diagram

P(H′)  ⟶^{≅}  Div^{+}(X)
↘  ↙
Pic(X)

where the vertical arrows are the natural projective morphisms. The assertion follows by taking the analytic pull-back of the component Pic^{m}(X) ⊂ Pic(X) via these two maps.

Remark. If X is a smooth manifold, then Dou(m) is a union of connected components of Div^{+}(X). Moreover, if X admits a Kähler metric, the spaces Dou(m) are always compact. This follows from Bishop’s compactness theorem, since all divisors in Dou(m) have the same volume (with respect to any Kähler metric). This property fails in the case of manifolds which do not allow Kähler metrics, since a non-Kählerian manifold may have (nonempty) effective divisors which are homologically trivial. (Take for instance a (elliptic) surface X with H^{2}(X, ℤ) = 0.)

2 Gauge-theoretical point of view

When X is a smooth compact, connected complex manifold it is possible to construct a ”gauge-theoretical” moduli space of simple holomorphic pairs (of any rank) on X (compare [OT],[S]).

Let E be a fixed C^{∞} complex vector bundle of rank r on X. We recall the following basic facts from complex differential geometry:

Definition 2.1 A semiconnection (of type (0,1)) in E is a differential operator δ̄: A^{0}(E) ⟶ A^{0,1}(E) satisfying the Leibniz rule

δ̄(f · s) = ¯∂(f) ⊗ s + f · δ̄(s) ∀ f ∈ C^{∞}(X, ℂ), s ∈ A^{0}(E) .

The space of all semiconnections in E will be denoted by D̄(E); it is an affine space over A^{0,1}(EndE). Every δ̄ ∈ D̄(E) admits a natural extension δ̄ : A^{p,q}(E) ⟶ A^{p,q+1}(E) such that

δ̄(α ⊗ s) = ¯∂(α) ⊗ s + (−1)^{p+q}α ∧ δ̄(s) ∀ α ∈ A^{p,q}(X), s ∈ A^{0}(E) .

Moreover, δ̄ induces D̄ : A^{p,q}(EndE) ⟶ A^{p,q+1}(EndE) by

D̄(α) = [δ̄, α] := δ̄ ∘ α + (−1)^{p+q+1}α ∘ δ̄ .

Every holomorphic bundle E over X of differentiable type E induces a canonical semiconnection δ̄:= ¯∂_{E} on E such that δ̄^{2} : A^{0}(E) ⟶ A^{0,2}(E) vanishes identically. Conversely, by [AHS] Theorem (5.1), every semiconnection δ̄ with δ̄^{2} = 0 defines a unique holomorphic bundle E, differentiably equivalent to E such that ¯∂_{E} = δ̄. There exists a natural right action of the gauge group GL(E) ⊂ A^{0}(EndE) of differentiable automorphisms of E on the space D̄(E) × A^{0}(E) by

(δ̄, ϕ) · g := (g^{−1} ∘ δ̄ ∘ g, g^{−1}ϕ) .

Denote by S̄(E) the set of points with trivial isotropy group. After suitable L_{k}^{2}- Sobolev completions, the space B̄^{s.p.}(E) := S̄(E)/GL(E) becomes a complex analytic Hilbert manifold, and S̄(E) ⟶ B̄^{s.p.}(E) a complex analytic GL(E)-Hilbert principal bundle. The map

Υ : D̄(E) × A^{0}(E) ⟶ A^{0,2}(EndE) × A^{0,1}(E)

given by

Υ(δ̄, ϕ) = (δ̄^{2}, δ̄ϕ)

is GL(E)-equivariant, hence it induces a section Υ̂ in the associated Hilbert vector bundle

S̄(E) ×_{GL(E)} [A^{0,2}(EndE) ⊕ A^{0,1}(E)]

over B̄^{s.p.}(E). This section becomes analytic for apropriate Sobolev completions.

Definition 2.2 The gauge-theoretical moduli space M^{s.p.}(E) of simple holomorphic pairs of type E is the complex analytic space given by the vanishing locus of the section Υ̂.

Set-theoretically, M^{s.p.}(E) can be identified with the set of isomorphism classes of pairs (E, ϕ), where E is a holomorphic bundle of type E, and ϕ is a holomorphic section in E, such that the associated evaluation map

ev(ϕ) : H^{0}(X, End(E)) ⟶ H^{0}(X, E)

is injective. This is equivalent to the fact that the only automorphism of (E, ϕ) is the identity. If E is a simple bundle (this happens always if r = 1) and ϕ ∈ H^{0}(X, E), then (E, ϕ) is a simple pair iff ϕ is nontrivial.

Definition 2.3 Fix (δ̄_{0}, ϕ_{0}) ∈ Υ^{−1}(0). A gauge-theoretical family of deformations of (¯∂_{0}, ϕ_{0}) parametrized by a germ (T, 0) is a complex analytic map

ω = (ω_{1}, ω_{2}) : (T, 0) ⟶ (A^{0,1}(EndE), 0) × (A^{0}(E), ϕ_{0})

such that the image of the map (δ̄_{0} + ω_{1}, ω_{2}) is contained in Υ^{−1}(0), and (δ̄_{0} + ω_{1}, ω_{2}) : (T, 0) ⟶ Υ^{−1}(0) is also holomorphic. Two families ω and ω′ over (T, 0) are called equivalent if there exists a complex analytic map

g : (T, 0) ⟶ (GL(E), id_{E})

such that ω′ = ω · g.

Note that, given such a deformation ω = (ω_{1}, ω_{2}), the family ω_{1} induces uniquely a section in the sheaf (A^{0,1}(EndE) × A^{0,0}(E)) ˆ⊗O_{T}, and conversely. In particular, if (T, 0) is an artinian germ, then ω_{1} induces a morphism of sheaves

A^{0,i}(E)_{T} := A^{0,i}(E) ⊗_{ℂ} O_{T} ⟶ A^{0,i+1}(E)_{T}

We denote by h_{gauge} : (germs) ⟶ (sets) the functor which sends a germ (T, 0) into the set of equivalence classes of gauge-theoretical families of deformations over (T, 0).

Theorem 2.4 The functor h_{gauge} has a semi-universal deformation.

Proof. Fix δ̄_{0} ∈ D̄(E) and consider the orbit map β : GL(E) ⟶ D̄(E) × A^{0}(E) given by

β(g) := (δ̄_{0}, 0) · g = (δ̄_{0} + g^{−1}D̄_{0}(g), 0) .

By [BK] Theorem (12.13) and [K] Theorem (1.1), the existence of a semi-universal deformation follows if:

i.) The derivative of Υ at (δ̄_{0}, 0) and the derivative of β at id_{E} are direct linear maps (for apropriate Sobolev completions);

ii.) The quotient ker(dΥ_{(δ̄_{0},0)})/im(dβ_{id_{E}}) is finite dimensional.

One has

ker(dΥ_{(δ̄_{0},0)}) = { (α, ϕ) ∈ A^{0,1}(EndE) × A^{0}(E) | D̄_{0}(α) = 0, δ̄_{0}ϕ = 0 }

and dβ_{id_{E}} : A^{0}(EndE) ⟶ A^{0,1}(EndE)×A^{0}(E) is given by u ⟼ (D̄_{0}(u), 0). Therefore i.) and ii.) follow from standard Hodge theory.

Remark. The existence of a moduli space of simple holomorphic pairs for arbitrary rank can be also deduced from [KO] Theorem (2.1), (2.2) and the proof of Theorem (6.4) of loc.cit. Indeed, one has local semi-universal deformations: Fix (E_{0}, ϕ_{0}) and let E ⟶ X ×(R, 0) be a semi-universal family of vector bundles with E|_{X×{0}} ≅ E_{0}. Similarly as in rank one, the functor H : (an/R) ⟶ (sets) given by

(S ⟶^{f} R) ⟼ Hom(O_{X×S}, (id_{X} × f)^{∗}E)

is representable [Fl] by a linear fibre space p : R̃ ⟶ R. Moreover, there exists a tautological section u : O_{X×R̃} ⟶ (id_{X} × p)^{∗}E arising from the representability of H. The pair ((id_{X} × p)^{∗}E, u^{∗}ϕ_{0}) is a versal deformation of (E_{0}, ϕ_{0}). By [Bi2] there exists then also a semi-universal deformation. Moreover, it can be shown (as in loc.cit. for the case of simple sheaves), that the isomorphy locus of two families of simple holomorphic pairs over S is a locally closed analytic subset of S and as a consequence, this moduli space exists by [KO] (2.1).

The aim of the remaining part is to prove the following

Theorem 2.5 The gauge-theoretical moduli space of simple holomorphic pairs of type E is analytically isomorphic to the complex-theoretical moduli space of simple holomorphic pairs of type E.

Proof. The arguments we use are inspired from [M], where a similar problem is treated (bundles without section). It suffices to show that the associated deformation functors h_{gauge} resp. h_{an} are isomorphic over artinian bases. Note first, that there exists a well defined morphism of functors from h_{an} to h_{gauge}. Moreover, this morphism is injective since

(E_{1}, ϕ_{1}) ∼ (E_{2}, ϕ_{2}) ⟺ (¯∂_{E_{1}}, ϕ_{1}) ∼ (¯∂_{E_{2}}, ϕ_{2}) .

In order to prove the surjectivity, we need to show that every gauge-theoretical family of simple holomorphic pairs determines a complex-theoretical family of simple holomorphic pairs. We will prove this by induction on the length of the artinian base. For n = 0, this follows from the fact that every integrable semi-connection δ̄_{0} determines a holomorphic bundle E_{0} of type E such that one has an exact sequence of sheaves

0 → E_{0} → A^{0,0}(E) →^{¯∂_{E_{0}}} A^{0,1}(E) →^{¯∂_{E_{0}}} A^{0,2}(E) → …

For the induction step, let (T, 0) be a small extension of an infinitesimal neighbourhood (T′, 0) such that ker(O_{T} ⟶ O_{T′}) = ℂ, and let ω = (ω_{1}, ω_{2}) be a gaugetheoretical family of simple holomorphic pairs parametrized by (T, 0). By the induction assumption, we can find a holomorphic vector bundle E′ over X × (T′, 0) which is induced by ω_{1}|_{T′}. Then we have the following exact sequence of sheaves

0  0  0  0
↓  ↓  ↓  ↓
0 → E_{0} → A^{0,0}(E) → A^{0,1}(E) → A^{0,2}(E) → …
↓ ↓ ↓ ↓
0 → E → A^{0,0}(E)_{T} → A^{0,1}(E)_{T} → A^{0,2}(E)_{T} → ⋯
↓ ↓ ↓ ↓
0 → E′ → A^{0,0}(E)_{T′} → A^{0,1}(E)_{T′} → A^{0,2}(E)_{T′} → ⋯
↓  ↓  ↓  ↓
0  0  0  0

In particular, E is locally free, hence it defines a holomorphic vector bundle over X × (T, 0) whose restriction to X × (T′, 0) gives E′. This vector bundle E together with the family of sections ω_{2} gives rise to a complex theoretical family of simple pairs which induces the gauge-theoretical family ω. This completes the proof of the theorem.

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Key words: moduli spaces, simple holomorphic pairs, divisors, semiconnections, holomorphic structures AMS-classification: 14 C 20, 14 J 15, 32 G 05, 32 G 13, 53 C 05

Authors addresses:

Siegmund Kosarew, Institut Fourier, Université de Grenoble 1, F-38402 Saint Martin d’Hères, France Siegmund.Kosarew@ujf-grenoble.fr

Lupascu Paul, Mathematisches Institut, Universität Zürich, Winterthurerstr. 190, 8057-Zürich, Switzerland lupascu@math.unizh.ch