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Quasi-Projectivity of Moduli Spaces

By means of analytic methods the quasi-projectivity of the moduli space of algebraically polarized varieties with a not necessarily reduced complex structure is proven including the case of non-uniruled polarized varieties.

In the theory of algebraic varieties, it is fundamental to study the moduli spaces of algebraic varieties. As for the existence of moduli spaces, it had been known that there exists an algebraic space as a coarse moduli space of non-uniruled polarized projective manifolds with a given Hilbert polynomial. Here an algebraic space denotes a space which is locally a finite quotient of an algebraic variety. Actually the notion of algebraic spaces was introduced to describe the moduli spaces ( [AR1]). According to the theory of algebraic spaces by M. Artin ([AR1,AR2,KT]), the category of proper algebraic spaces of finite type defined over C is equivalent to the category of Moishezon spaces. Hence the moduli spaces of non-uniruled polarized manifolds have abundant meromorphic functions and were considered to be not far from being quasiprojective. Various attempts were made to prove the quasiprojectivity of the moduli spaces of non-uniruled, polarized algebraic varieties (cf. KN,KO1,V]). E. Viehweg ([V]) developed a theory to construct positive line bundles on moduli spaces. He used results on the weak semipositivity of the direct images of relative multicanonical bundles. In particular he could prove the quasiprojectivity of the moduli spaces of canonically polarized manifolds ( [V]). J. Kollár studied the Nakai-Moishezon criterion for ampleness on certain complete moduli spaces in [KO1], with applications to the projectivity of the moduli space of stable curves and certain moduli spaces of stable surfaces under boundedness conditions. However, his approach appears quite different from our present methods, which do not require the completeness of moduli spaces. His result was used to show the projectivity of the compactified moduli spaces of surfaces with ample canonical bundles by V. Alexeev ([AL]).
The main result in this paper is the quasi-projectivity of the moduli space of non-uniruled polarized manifolds. Non-uniruledness is in fact not used here. All we need is the existence of a moduli space.
In fact, given a polarized projective manifold, a universal family of embedded projective manifolds over a Zariski open subspace H of a Hilbert scheme is determined after fixing the Hilbert polynomial.
The identification of points of H, whose fibers are isomorphic as polarized varieties, defines an analytic equivalence relation ∼ such that the set theoretic moduli space is M = H/ ∼. The quotient is already a complex space, if the equivalence relation is proper. Moreover,in this situation, it follows that M is an algebraic space. If the above equivalence relation is induced by the action of a projective linear group G, properness of ∼ means properness of the action of G. In this moduli theoretic case H/ ∼ is already a geometric quotient.
Theorem 1. Let K be a class of polarized, projective manifolds such that the moduli space M exists as a proper quotient of a Zariski open subspace of a Hilbert scheme. Then M is quasi-projective.
The proof of the theorem consists of two steps. The first step is to construct a line bundle on the compactified moduli space with a singular hermitian metric of strictly positive curvature on the interior.
The method is based upon the curvature formula for Quillen metrics on determinant line bundles ( [BGS]), the theory of Griffiths about period mappings ( [GRI]), and moduli of framed manifolds.
The second step is to construct sufficiently many holomorphic sections of a power of the above line bundles in terms of L 2 -estimates of the ∂-operator. The key ingredient here is the theory of closed positive (1, 1)-currents, which controls the multiplier ideal sheaf of a singular hermitian metric. This step can be viewed as an extension of the Kodaira embedding theorem to the quasi-projective case.

Singular hermitian metrics
Definition 1. Let X be a complex manifold and L a holomorphic line bundle on X. Let h 0 be a hermitian metric on L of class C ∞ and ϕ ∈ L 1 loc (X). Then h = h 0 · e −ϕ is called a singular hermitian metric on L.
Following the notation of [DE4] we set and call the real (1, 1)-current the "curvature current" of h. It differs from the Chern current by a factor of 2.
A singular hermitian metric h with positive curvature current is called positive. This condition is equivalent to saying that the locally defined function − log h is plurisubharmonic.
Let W ⊂ C n be a domain, and Θ a positive current of degree (q, q) on W . For a point p ∈ W one defines ν(Θ, p, r) = 1 r 2(n−q) z−p <r Θ(z) ∧ (dd c z 2 ) n−q .
The definition of a singular hermitian metric carries over to the situation of reduced complex spaces.
Definition 2. Let Z be a reduced complex space and L a holomorphic line bundle. A singular hermitian metric h on L is a singular hermitian metric h on L|Z reg with the following property: There exists a desingularization π : Z − → Z such that h can be extended from Z reg to a singular hermitian metric h on π * L over Z.
The definition is independent of the choice of a desingularization under a further assumption. Suppose that Θ h ≥ −c · ω in the sense of currents, where c > 0, and ω is a positive definite, real (1, 1)-form on Z of class C ∞ . Let π 1 : Z 1 − → Z be a further desingularization. Then Z × Z Z 1 − → Z is dominated by a desingularization Z ′ with projections p : Z ′ − → Z and p 1 : Z ′ − → Z 1 . Now p * log h is of class L 1 loc on Z ′ with a similar lower estimate for the curvature. The push-forward p 1 * p * h is a singular hermitian metric on Z 1 . In particular, the extension of h to a desingularization of Z is unique.
In [G-R] for plurisubharmonic functions on a normal complex space the Riemann extension theorems were proved, which will be essential for our application. The relationship with the theory of distributions was treated in [DE].
For a reduced complex space a plurisubharmonic function u is by definition an upper semi-continuous function u : X − → [−∞, ∞) whose restriction to any local, smoothly parameterized analytic curve is either identically −∞ or subharmonic.
A locally bounded function u : X − → [−∞, ∞) from L 1 loc (X) is called weakly plurisubharmonic, if its restriction to the regular part of X is plurisubharmonic.
Differential forms with compact support on a reduced complex space are by definition locally extendable to an ambient subspace, which is an open subset U of some C n . Hence the dual spaces of differential C ∞ -forms on such U define currents on analytic subsets of U. The positivity of a real (1, 1)-current is defined in a similar way as above involving expressions of the form (1).
For locally bounded functions of class L 1 loc the weak plurisubharmonicity is equivalent to the positivity of the current dd c u. It was shown that these functions are exactly those, whose pull-back to the normalization of X are plurisubharmonic. We note Definition 3. Let L be a holomorphic line bundle on a reduced complex space X. Then a singular hermitian metric h is called positive, if the functions, which define locally − log h are weakly plurisubharmonic.
This definition is compatible with Definition 2: Let L be a holomorphic line bundle on a complex space Z equipped with a positive, singular hermitian metric h r on L|Z reg . If π : Z − → Z is a desingularization, and h a positive, singular hermitian metric on π * L, extending h|Z reg , we see that − log h r is locally bounded at the singularities of Z so that h induces a singular, positive metric on L over Z.

Deformation theory of framed manifolds-V -structures
Let X be a compact complex manifold and D ⊂ X a smooth (irreducible) divisor. Then (X, D) is called a logarithmic pair or a framed manifold.
For any m ∈ N an associated V -structure X m on X is defined, in terms of local charts π : W − → U, U ⊂ X, W ⊂ C n such that π is just an isomorphism, if U ∩ D = ∅ or a Galois covering of order m with branch locus U ∩ D (and group G = Z m ).
By definition, the differential forms and vector fields on X with respect to the V -structure, which are V -differentiable or V -holomorphic, are defined on X\D with the property that the local lifts under π|W ∩ π −1 (U\D) : W ∩ π −1 (U\D) − → U\D can be extended in a holomorphic or differentiable way to W .
With m being fixed, we denote by T V X and A V,q X (T V X ) resp. the sheaves of V -holomorphic vector fields and V -differentiable q-forms with values in T V X resp.
is well-defined and exact.
By definition, a family (X s , D s ) s∈S of framed manifolds, parameterized by a complex space S is given by a smooth, proper, holomorphic map f : X − → S together with a divisor D ⊂ X , such that f |D is proper and smooth, such that X s = f −1 (s) and D s = D ∩ X s . A local deformation of a framed manifold (X, D) over a complex space S with base point s 0 ∈ S is a deformation of the embedding i : D ֒→ X, i.e. induced by a family D ֒→ X − → S together with an isomorphism (X, D) ∼ − → (X s 0 , D s 0 ), where two such objects are identified, if these are isomorphic over a neighborhood of the base point. The existence of versal deformations (i.e. complete and semi-universal deformations) of these objects is known. We denote by T • (X) ≃ H • (X, T X ) and T • (X, D) resp. the tangent cohomology of X and (X, D) resp. Corollary 1. The space of infinitesimal deformations of (X, D) equals ). It can also be computed in terms of Cech cohomology as We have the following exact sequence.: We denote by T 1 1 (X) ⊂ T 1 (X) the image of T 1 (X, D). The composition of H 1 (X, T X )) − → H 1 (D, O D (D)) with the natural map H 1 (D, O D (D) − → H 2 (X, O X ) equals the map induced by the cupproduct with the Chern class of D. The latter is induced by the Atiyah sequence for the pair (X, O X (D)), and its kernel T 1 0 (X) consists of those infinitesimal deformations for which the isomorphism class of the line bundle [D] extends. Assume that D is an ample divisor on X, and λ X = c 1 (D) its (real) Chern class. Then the pair (X, λ X ) is a polarized variety, and T 1 0 (X) is the space of infinitesimal deformations of (X, λ X ). Studying moduli spaces of polarized varieties, we are free to replace the ample divisor D by a uniformly chosen multiple, in which case T 1 0 (X) and T 1 1 (X) can be identified.
Remark 1. The group of infinitesimal automorphisms T 0 (X, D) vanishes, if K X + [D] is positive. Like in the case of canonically polarized manifolds, in a family of such framed manifolds the relative automorphism functor (or more generally isomorphism functor) is represented by a space such that the natural map to the base is finite and proper. Moreover, general deformation theory implies that any versal deformation is universal.

Cyclic coverings
Let X be a compact complex manifold, and D, D ′ effective divisors such that D ∼ m · D ′ for some m ∈ N. Denote by E and E ′ resp. bundle spaces for the corresponding line bundles. Let be the morphism over X, which sends a bundle coordinate α to α m .
Let σ be a canonical section of π. Then we define If D is a smooth divisor, the subspace X m ⊂ E is a manifold, and π ′ |X m : X m − → X is a Galois covering with branch locus D ⊂ X.
We assume now that D is very ample, providing an embedding Φ : X − → P N . We denote by P the dual projective space, and by Σ ⊂ P N × P the tautological hypersurface with divisor We have flat families over X × P and P resp.
Here the bundle E comes from the globally defined divisor D. The bundle E ′ is first defined locally with respect to P , which gives rise to a cocycle with coefficients from C * , i.e. a line bundle on P .
Proposition 1. The total space X m is smooth. In particular, the dualizing sheaf ω X m /P equals the relative canonical sheaf K X m /P := K X m ⊗ π * K −1 P .
Proof. As X m ⊂ E ′ is of codimension one, it is sufficient, to find a local function for any x 0 ∈ X m , which vanishes at x 0 , and whose gradient at this point is non-zero. Let again σ be a canonical section of the line bundle E over X × P . We denote by t 0 the image of x 0 in P , and take local coordinates t of P around t 0 . Let α be a local bundle coordinate of E ′ around t 0 , and z a local coordinate on X so that x 0 is given by (z 0 , α 0 , t 0 ). Now t 0 ∈ P corresponds to a section σ t 0 (z) of Since D is very ample on X, we find a section of E|X×{t 0 }, which does not vanish at x 0 . This section gives some t 1 ∈ P , i.e. some σ t 1 . Let σ t(τ ) = σ t 0 + τ σ t 1 be the line through t 0 and t 1 . Then (∂g/∂τ )| τ =0 = 0.
Remark 2. The analogous statement is true for smooth families f : X − → S of manifolds over complex spaces, assuming the existence of families of divisors D and D ′ , with D ∼ m·D ′ , and a relative embedding and the induced map X m − → S is smooth. In particular, the canonical and dualizing sheaves K Xm/S×P = K Xm ⊗ f * m K −1 S×P and ω Xm/S×P resp. are isomorphic, if S is smooth.
Let (X, D) be a framed manifold, and D ∼ mD ′ for some effective D ′ as above. Again, let G = Z m denote the Galois group, X is isomorphic to the quotient X m /G, and the group G acts on The average over the group defines a retraction. Next, we identify H 1 (X m , T Xm ) G with the V -tangent cohomology group ∨ H 1 (U, T V X ) in the sense of Section 3: The . This argument avoids any smoothing of invariant differential forms.
Remark 3. The infinitesimal deformations of a framed manifold (X, D) can be identified with T 1 (X,

Canonically polarized framed manifolds
Definition 4. A framed manifold (X, D) is called In the sequel we always assume condition ( * ) m for some fixed m. We note that for the Galois covering µ : X m − → X with smooth X m the relation holds. In our applications the divisor D will always be ample so that condition (ii) is slightly stronger than (i). We will still use the term "canonically polarized framed manifold" in this case. This will also be justified later.
Proposition 2. Let D ′ ⊂ X be a very ample divisor as above, and m > 2. Let D ⊂ X be a smooth divisor D ∼ m · D ′ such that is very ample. Then the canonical bundle K Xm is very ample.
) is a direct summand. Let Z m ≃ G ֒→ Aut(X m ) be the group of deck transformations with a generator γ, and denote by ζ a primitive m-the root of unity. Let ⊕ m j=1 E j be an eigenspace decomposition of the space of global sections of K Xm with respect to the eigenvalues ζ j of γ. It follows that the spaces E j can be identified with the space of global sections of K X +(m−j)·D ′ , again with j = 1, . . . , m. The pull-backs of sections of such a space are sections of K Xm − (j − 1)A, where A ⊂ X m , A ≃ D ′ , is the branching divisor of µ, so that the identification Γ(X, O X (K X +(m−j)·D ′ )) ≃ E j is a multiplication with a canonical section of [(j − 1)A].
The space E 1 clearly separates points, whose image under µ are different.
Let p, q ∈ X m with µ(p) = µ(q) = x. Then there exist sections of [K X + (m − 2)D ′ ] and [K X + (m − 1)D ′ ] which do not vanish at x. A suitable linear combination of the induced elements of E 1 and E 2 separates p and q. The argument is also applicable to tangent vectors. Now we consider the situation given in diagram (5), where S need not be smooth. Let A ⊂ S × P be the locus of singular divisors D. Over its complement the direct image of the relative canonical sheaf is certainly locally free.
We write X ′ m := X m \f −1 m (A), T := P × S, T ′ := T \A, and f ′ m for the restriction of the map f m . In a similar way we restrict f := f × id to T ′ and get f ′ : Proposition 3. The locally free sheaf f ′ m * K X ′ m /T ′ possesses a natural, locally free extension.
) from the proof of Proposition 2. Now for the family (X × P ) ′ − → T ′ , with relatively (very) ample divisor D ′ , the Kodaira-Nakano vanishing theorem and the Grothendieck-Grauert comparison theorem show that for j > 0 the sheaves f * (K (X ×P )/T + j · [D ′ ]) are locally free on T (Here the divisor D ′ corresponds to the line bundle E ′ ). Let j = 0. Since f ′ m * (K X ′ m /T ′ ) is locally free on T ′ , also f * (K X ×P/T ) is locally free, when restricted to T ′ . On the other hand, it does not involve the divisor D ′ , and f * (K X ×P/T ) is the pull-back of the direct image of K X /S , so it is constant along all fibers of T − → S, and locally free in the interior, hence also f * (K K X ×P /T ) is locally free.
Next, we want to recover the above extension of the relative canonical sheaf. We have the diagram (5). The fibers of f m are branched along the D s with singularities over the singularities of the branching divisors. By definition the map f m is flat with Cohen-Macaulay fibers. According to results of Kleiman [KL] for such morphisms relative dualizing sheaves commute with base change. Again, we denote by the letter ω dualizing sheaves.
In particular, the extension from Proposition 2 equals f m * (ω Xm/T ) (which is compatible with further pull-backs).
Later we will consider this sheaf from a Hodge theoretic viewpoint.

Singular Hermitian metrics for families of canonically polarized framed manifolds
We first recall some facts concerning the period map in the sense of Griffiths [GRI] for families f : Y − → S of manifolds with very ample canonical bundle. We will apply the results to families of the form f m : X m − → S with relative dimension n from Section 4. The direct image under f of the relative canonical sheaf K Y/S is also called Hodge bundle E 0 . It is equipped with the flat metric from R n f * C. Explicitly, for any two holomorphic n-forms φ and ψ on a manifold Y s 0 we have Let ∂/∂s be a tangent vector at a point s 0 Then the contraction with the Kodaira-Spencer class [ ).
The natural metric on the latter space is again induced by the integration of exterior products of differential forms, after we provide the fibers with a family of auxiliary K"ahler structures (e.g. of K"ahler-Einstein type). Following Griffiths [GRI,Theorem (5.2)] the curvature Θ 0 of this hermitian metric is given by the formula which is defined in terms of cohomology classes. (Here H denotes the harmonic projection). So Θ 0 is semi-positive as well as its trace tr(Θ 0 ). If tr(Θ 0 )( ∂ ∂s , ∂ ∂s )| s 0 = 0, also Θ 0 ( ∂ ∂s , ∂ ∂s )| s 0 vanishes. The auxiliary K"ahler metric is only needed to show the positivity of the curvature, the metric on the relative canonical bundle is independent of the choice. The sheaf R 1 f * Ω n−1 Y/S is usually denoted by E 1 . Denote by D the period domain of Hodge structures, and by Φ : is a subspace of the tangent space of D at the point Φ(s), and it carries the natural L 2 -inner product (cf. (7)). We denote this metric by (7), the trace of the curvature of the flat metric restricted to a bundle E 0 gives exactly ds 2 0 . This argument shows: Lemma 3. Let Y − → S, S = ∆ * k × ∆ ℓ be a holomorphic family of canonically polarized manifolds. Let h S be the natural C ∞ hermitian metric on det f * K Y/S . Then the curvature Θ S is semi-positive (in the sense of C ∞ -forms), and dominated by a constant multiple of the K"ahler form ω S induced by ds 2 P oinc .
For effectively parameterized families f m : X m − → T and large m the Xs 0 )) is in fact injective. This was shown in a general setting by Ivinskis, and attributed to Griffiths in [IV] for the special case of cycling coverings. (The cohomological assumption for [IV,Theorem 2.4] is easily satisfied for sufficiently high powers m of the line bundles [D s ], which can be chosen uniformly on the moduli space from the beginning by Noether induction.) Now the base S is equipped with the line bundle λ f r = det f m * K Xm/T (which equals the determinant line bundle in the sense of the derived category, because of the Kodaira vanishing theorem). Then the curvature of the induced hermitian metric h on λ is Θ h = tr(Θ 0 ). Altogether: Proposition 4. The curvature Θ h of (λ f r , h) is semi-positive. It is strictly positive in all directions, where the family is effectively parameterized (i.e. where the Kodaira-Spencer map is non-zero).
Now we return to the notation of Section 5. The main theorem is stated for non-singular base spaces.
Theorem 2. The determinant (invertible) sheaf det f m * K Xm/T carries a natural positive hermitian metric, whose Lelong numbers vanish everywhere. Moreover, for all p ∈ N, the exterior powers Θ p h of its curvature form Θ h are well-defined (p, p)-currents, whose Lelong numbers vanish everywhere as well.
We shall apply the theorem in two different situations: Over the interior of the moduli space we deal with families of manifolds of the type X m , where in the limit we have singular Galois coverings X m − → X (cf. Section 5). Here the key point is that the total space X m is already smooth according to Proposition 1 so that we can identify the relative dualizing sheaf with the relative canonical sheaf. The other situation occurs at the boundary of the moduli space, where we are free to modify the boundary.
The theorem follows from the known results in the theory of mixed Hodge structures. We show here an upper estimate for a singular Hermitian metric. Together with the positivity of this metric the vanishing of the Lelong numbers follows.
Concerning singular base spaces of holomorphic families, we observe that the L 2 -inner products (for tangent vectors of the base) are welldefined for singular bases spaces. For our applications we will need that the construction is functorial, i.e. compatible with base changes like restrictions to closed subspaces and desingularizations in the view of Definition 2.
For a family f m : X m − → T (T is smooth), we denote by A ⊂ T the set of points with singular fibers. Let ν : T − → T be given by a sequence of blow-ups with regular centers so that the preimage B of A is a normal crossings divisor. Let X m − → X m × T T be a desingularization of the of the component of X m × T T that dominates T , with the property that the preimage of B is a normal crossing divisor. Let be the induced commutative diagram. We denote by a prime accent the restriction of fiber spaces to the resp. complements of normal crossing divisors.
An argument of Deligne shows that the local monodromy of R n f m * C on T ′ is unipotent around generic points of A, i.e. in codimension one. And since it is locally abelian on T ′ , this holds everywhere. For our purpose the unipotent reduction is sufficient. We need a local statement with respect to the base T . The argument is known: Around each component of the normal crossings divisor B the eigenvalues of the local monodromy transformation on R n f ′ m * C are certain roots of unity [B]. After taking a finite morphism κ :Ť − → T , branched over B the local monodromy groups become unipotent. We consideř The canonical extension of R nf ′ m * CX′ m ⊗ OŤ′ toŤ ( [DL]) is a coherent sheaf. By a theorem of W. Schmid [S], the subsheaff ′ m * KX′ m /Ť ′ extends to a locally free sheaf onŤ . Kawamata's theorem [KA] states that this locally free extension is equal tof m * KX m . It is known that also f m * K Xm is locally free: Namely as κ ′ is a proper holomorphic map of equidimensional complex manifolds, K Xm ⊂ κ ′ * KX is a direct summand, and hence f m * K Xm ⊂ f m * κ ′ * KX m = κ * fm * KX m is a direct summand. Now the latter is locally free, asf m * KX m is a locally free OŤ -module, and κ is a finite proper map of complex manifolds. We have K Xm = ν ′ * K Xm on the manifold X m so that f m * K Xm is locally free.
Next, we use W. Schmid's description of sections off m * KX m around points of the normal crossing divisor. Let ∆ k ≃ U ⊂Ť be an open subset such that the complement of the normal crossing divisor is Let φ be a section of KX m overf −1 m (U). Over U ′ it can be expressed in terms of a basis {s 1 , . . . , s M } of multivalued (locally constant) sections of R nf m * CX m over U ′ . So φ = f ν · s ν for certain multivalued holomorphic functions on U ′ . According to [S, (4.17)], the holomorphicity of φ in points of the normal crossing divisor is equivalent to the f ν having at most logarithmic singularities. Next the L 2 -norm is computed at points t ∈ U ′ . (We identify KX with KX /Ť ).
The latter integrals are independent of t, because the s i are locally flat sections. So for some constants c j > 0.
The unipotent reduction preserves such estimates so that a similar estimate (with different constants) also holds for sections of f m * K Xm .
This implies an estimate for sections of f m * K Xm . We only note the following rough estimate: Let W ⊂ T be an open subset. Then for any for certain positive constants α j and holomorphic functions τ j , which vanish on A. This proves the following lemma: for certain β j > 0.
In Proposition 4 we showed that h is a positive singular hermitian metric so that the corollary implies the theorem for p = 1.

The curvature form Θ satisfies a Poincaré growth condition on ∆
In particular all powers Θ p define closed (p, p)-currents. These estimates hold for the Hodge metrics overŤ , T , and since T − → T is a modification of complex manifolds, also the Θ p h on T are closed currents.
We show the last statement of Theorem 2.
Let x ∈ P × S be a point and z 1 , . . . , z k local coordinates such that x = 0. Let (locally) h = e −u , with u plurisubharmonic, and define ϕ = log z 2 . For any positive (p, p)-current R and small r > 0 the quantity ν(R, x, r) is defined by and in terms of Demailly's generalized Lelong numbers In a straightforward way a generalized Jensen formula can be proved: It is known that for any fixed r 1 So we need only to show that the right hand side of the above Jensen formula is bounded for fixed r 1 , independent of r. This follows immediately, because (ii) As a plurisubharmonic function u is (locally) bounded from above (iii) dd c u satisfies a Poincaré growth condition on T .

Convergence property of generalized Petersson-Weil metrics
We include the definition of generalized Petersson-Weil metrics, which can also be part of a conceptual approach to an analytic theory of moduli spaces, relating canonically polarized manifolds to framed manifolds and m-framed manifolds. However, we will use later families of m-framed manifolds and Galois coverings instead, and properties of the period map.
In the first place, generalized Petersson-Weil metrics are intrinsically defined K"ahler metrics on the base spaces of universal deformations. Due to functoriality these will be seen to descend to moduli spaces.
In this section, we will assume that for all ε ∈ Q, 0 ≤ ε ≤ ε 0 the divisor is positive. This condition is satisfied for ε 0 = 1/m 0 in our basic situation, where (X, D) is m 0 -framed and D positive. The methods of [TS1,K1,K2, T-Y] yield unique K"ahler-Einstein metrics η X,m on the V -manifolds X m (cf. Section 3) of Ricci-curvature −1. As in the smooth compact case we can see that the V -K"ahler-Einstein metrics define generalized Petersson-Weil metric on the moduli space of framed manifolds as follows: the representative of the Kodaira-Spencer class of v according to Remark 3 in T 1 (X, D), which is harmonic with respect to η X,m .
Definition 5. Let v, w ∈ T s 0 S, and A m,v , A m,w corresponding harmonic Kodaira-Spencer forms. Then the Petersson-Weil inner product The K"ahler property of the induced form ω P W,m on S can be shown in the same way as for the case of smooth, canonically polarized varieties. Also a fiber integral formula holds for the Petersson-Weil form, and a line bundle equipped with a Quillen metric can be constructed, whose curvature form equals ω P W,m up to a constant [BGS].
On the other hand the tangent cohomology T 1 (X, D) can be computed in terms of the complete K"ahler-Einstein metric ω X ′ on X ′ = X\D as the H 1 (2) (X ′ , T X ′ ) the L 2 -cohomology group of the sheaf of holomorphic vector fields T X ′ [SCH]. The L 2 -structure on the tangent cohomology defines a Petersson-Weil metric ω P W,f r on M f r .
Let Ω X /S be the relative volume form, i.e. a hermitian metric on n T X /S , induced by all η Xs,m , and denote by η X ,m the negative of its curvature form on the total space. Its restrictions to all fibers are the K"ahler-Einstein forms on the fibers. Let v = ∂/∂s ∈ T s S be a tangent vector, and ∂ ∂s + a α ∂ ∂z α the horizontal lift with respect to η X ,m . Also in the case of V -structures, its exterior derivative restricted to the fiber X ∂(a α ) = ∂a α ∂zβ ∂ ∂z α dzβ equals the harmonic Kodaira-Spencer form A m,v . For a more detailed discussion of the Petersson-Weil inner product and Petersson-Weil forms for singular base spaces cf. also [F-S].
Denote by η Xs the usual K"ahler-Einstein metrics, and by η X the negative of its Ricci form on the total space.
Measuring convergence in C k,α (X ′ )-spaces with respect to quasicoordinates on X ′ = X\D the η X,m tend to the complete K"ahler-Einstein metric ω X ′ on X ′ [TS2]. In a holomorphic family of framed manifolds, this convergence yields a convergence of the relative volume forms Ω X /S,m to the relative volume form Ω X ′ /S of the smooth K"ahler-Einstein metrics in the spaces C k,α (X ′ ), X ′ = X \D. Together with the above fact about the characterization of harmonic Kodaira-Spencer forms we see immediately that the harmonic Kodaira-Spencer forms A m,v converge to the harmonic L 2 -integrable Kodaira-Spencer forms A f r,v on X ′ with respect to the complete K"ahler-Einstein metrics on X ′ .
Proposition 5. For the generalized Petersson-Weil metrics on moduli spaces of framed manifolds We show the second claim: We have the V -structures on the fibers X s , and map the usual K"ahler-Einstein metrics to K"ahler-Einstein V -metrics on the quotients X m,s /Z m . Any harmonic Kodaira-Spencer V -form lifts to a harmonic Kodaira-Spencer form on X m,s . the factor 1/m is due to the integration over m sheets as opposed to the integration over the V -manifold.

Moduli spaces of framed manifolds
In the analytic case, a polarization of a framed manifold (X, D) is the assignment of a K"ahler class λ X ∈ H 2 (X, R). Polarizations, which are images of integer valued cohomology classes, coincide with inhomogeneous polarizations in the sense of Mumford (cf. [M-F-K]). (Here, we can also allow rational coefficients and consider Q-divisors.) The following definition is also sensible for inhomogeneously polarized framed projective varieties (X, D, λ X ) (over C).
Definition 6. (i) A compact K"ahler manifold X is called uniruled over a smooth divisor D, if there exists a surjective meromorphic map ϕ : P 1 × Y − → X with the following properties: The map ϕ does not allow a meromorphic factorization over pr 2 : P 1 × Y − → Y . The restriction of pr 2 to the proper transform of D under ϕ is a modification. (ii) A polarized framed manifold (X, D, λ) is called non-uniruled, if the K"ahler manifold D is non-uniruled, and if X is not uniruled over D.
In the analytic category, the (coarse) moduli space of non-uniruled polarized K"ahler manifolds exists.
For non-uniruled, polarized, projective framed manifolds (X, D, λ X ), first the Hilbert polynomials P (x) for λ X on X and Q(x) for λ X |D are of interest. (If the polarization λ X is represented by D, we have Let λ X be represented by a basic polar divisor and corresponding ample line bundle L X . As usual, Matsusaka's big theorem ( [MA, L-M]) is applied to (X, L X ): There exists an integer c > 0 only depending on P (x), such that for all m ≥ c the sheaves m · L X are very ample.
Theorem 3. There exists an algebraic space M f r in the sense of Artin, which is the coarse moduli space of isomorphism classes of nonuniruled, polarized, framed projective manifolds (X, D, λ X ) with fixed Hilbert polynomials P (x)and Q(x).
As non-uniruledness is an open and closed condition for polarized varieties, we can also impose the condition that both X, and D are nonuniruled. Then the assignment (X, D, λ X ) → (X, λ X ) (with Hilbert polynomials fixed) defines a natural map M f r − → M of algebraic spaces, where M denotes the moduli space of uniruled polarized manifolds. If the divisors D are very ample and represent the polarization λ X (and X is non-uniruled), D may also be singular giving rise to a moduli space M equipped with a natural morphism ν : M − → M.
Proof. First, c > 0 as above is taken and m ≥ c fixed and for all polarized varieties X with Hilbert polynomial P (x) a corresponding projective embedding X ֒→ P N induced by global sections of m · L X considered. As subvarieties of P N these X have P (m · x) as Hilbert polynomials. We denote by Hilb P P N the Hilbert scheme of all subvarieties with P (m · x) in the sense of Grothendieck [GRO]. The locus H ⊂ Hilb P P N of all smooth subvarieties is quasi-projective. Let The graph Γ ⊂ H × H of the equivalence relation identifying embedded manifolds with singular framings is mapped properly to the graph Γ ⊂ H × H, which defines the moduli space M of polarized projective manifolds. By assumption, the natural map Γ − → H is proper, so Γ also defines a proper equivalence relation. This ensures the existence of a natural complex structure on M. (Observe that this statement can also be proved in the non-reduced category). Finally M carries the structure of an algebraic space. The construction is compatible with the restriction to H f r . If the above equivalence relations are given by the action of G = P GL(N + 1, C) on H and H resp. the moduli spaces M, M f r and M are eventually geometric quotients. In the analytic case the statement of the Matsusaka-Mumford theorem is also valid (cf. [SCH]) for framed polarized manifolds.
Later we will consider compactifications of the algebraic spaces M and M by normal crossings divisors with a morphism M − → M. We can assume that it is induced by a flat morphism H − → H of compactified Hilbert schemes of the similar type.
The moduli space M is induced by a smooth family of the form (11) with hyperplane section D ′ ⊂ X , such that the very ample divisors D ′ s represent a fixed multiple of the polarizations on X s . Let n = dim X s as before. According to Fujita's theorem [FU], the divisors K Xs + mD ′ s are ample for m ≥ n + 2. We fix m > dim X s + 3 and represent m[D ′ s ] by all possible divisors D s . This gives rise to a diagram of the form (12). We pull back the divisor D ′ to X and obtain a bundle space E ′ − → X . Let E − → X be the bundle associated to D. Like in Section 4 we construct a family of cyclic coverings f m : X m − → H and a diagram where the branch locus of µ is D ⊂ X . The fibers X m,s are smooth for s ∈ H f r . Let (X, D) be a fixed framed manifold with branched covering X m − → X as above, and let R and R resp. denote base spaces of universal deformations. Then there exists a closed holomorphic embedding κ : R − → R which induces the map κ in a neighborhood of the corresponding moduli point, where it is of the form R/Aut(X, D) − → R/Aut(X m ).
Proof. We show the properness of κ. Let (X j , D j ), j ∈ N be framed manifolds such that the images under κ of the isomorphism classes are represented by canonically polarized manifolds X m,j . Assume that the isomorphism classes of X m,j converge to the class of some canonically polarized manifold Z. We consider a universal deformation Z − → S of Z and can assume that the X m,j are fibers. The space of automorphisms Aut S (Z) − → S is mapped properly to the base S with reduced, 0-dimensional fibers. Hence, for a subsequence of the j, the group of deck transformations of the X m,j converges to a subgroup of Aut (Z). Its fixed point set is a smooth divisor, and a subsequence of the (X j , D j ) converges to a framed manifold (X, D). The bundle K X + [D] is positive, since K Z is positive.
Let (X, D) be a general framed manifold, and ( R, r 0 ) the base of a universal deformation with fibers (X r , D r ). We have [D] = m · [D ′ ] by assumption. As for all r the [D r ] are divisible by m, the given family of polarized manifolds gives rise to a family X m − → R with Galois covering X m − → X . This family is induced by a universal deformation Y − → R of X m . The base change R − → R must be an embedding by Remark 3.
Let Z m ⊂ Aut(X m ) be the group of deck transformations. It acts on R leaving the subspace R ⊂ R pointwise fixed, since the group action can be lifted to all of its fibers. Furthermore, there is a natural morphism Aut(X, D) − → Aut(X m )/Z m so that we have a holomorphic map R/Aut(X, D) − → R/(Aut(X m )/Z m ) ⊂ R/Aut(X m ), which is given by the action of a finite group.

Fiber integrals and determinant line bundles for morphisms
We will use the method of generalized determinant line bundles. Let F : Z − → S be a proper, holomorphic map of complex spaces and L a coherent O Z -module.
The direct image R • F * L of L under the proper map F in the derived category can be locally represented by a sequence F • of finite, free O Smodules, which is bounded to the right. If the morphism is flat, the sequence can be chosen as bounded, and the tensor product of the determinant sheaves of the F i with alternating exponents ±1 is by definition the determinant line bundle λ = det(L), and the latter is globally well-defined.
Let L = O Z (L) be a holomorphic line bundle equipped with a hermitian metric of class C ∞ . According to Bismut, Gillet and Soulé, [BGS], under the assumption of F being a smooth K"ahler morphism of complex manifolds (or reduced complex spaces [F-S]), the Chern form of the Quillen metric h Q on det(L) is equal to the component of degree two of a fiber integral: where td and ch resp. define the Todd and Chern character resp. (This holds also, when L is replaced by a hermitian vector bundle.) By functoriality and universal properties, this equation extends to L replaced by an element of the Grothendieck group, i.e. a virtual holomorphic vector bundle. If n denotes the fiber dimension, the virtual bundle (L − L −1 ) n+1 has rank zero, and the lowest term in ch((L − L −1 ) n+1 ) is 2 n+1 c 1 (L) so that the only contribution of the Todd character in (14) is equal to 1. Hence the Chern form of det((L − L −1 ) n+1 ) equals Now we return to the situation of moduli spaces like in Section 8. The Hilbert scheme H f r carries the determinant line bundle λ f r with singular hermitian metric h f r according to Proposition 4 and Lemma 2. It is important that the line bundle λ f r on H f r was extended to the line bundle λ on H. Let π : H r − → H be a desingularization with fiber product ν r : H r − → H r and pull-back λ r of λ. Since ν r is a smooth map, we can apply the above methods. We consider the determinant bundle det(( λ r − λ −1 r ) N +1 ).
So far we are given a smooth holomorphic map ν : H − → H and a holomorphic line bundle on λ on H, whose restriction λ f r to H f r carries the C ∞ hermitian metric h f r with curvature form Θ f r .
We use the above arguments to extend the determinant line bundle det(( λ − λ −1 ) N +1 ) as a coherent sheaf from H to H. We denote by Θ the curvature current of λ. Let ℓ = dim H. In order to define a fiber integral for any (ℓ − 1, ℓ − 1)-form ϕ of class C ∞ with compact support, we set At this point, we may blow up H with exceptional set in H\H f r and realize H f r as a complement of a divisor with only normal crossings singularities so that the assumptions of Lemma 3 are satisfied. The upper Poincaré growth estimate for Θ f r implies that the above integral is finite, and it vanishes, if ϕ is d-exact. So Θ Q is well-defined as a dclosed (1, 1)-current. Also Lemma 3 implies that Θ Q is positive (in the sense of currents).
Proposition 7. At all points H the Lelong numbers of Θ Q vanish.
The above statement also holds after descending to the moduli space at points of the boundary, as we can always achieve the situation of Section 2 after blowing up the boundary.
Proof. The proof follows immediately from Theorem 2.
Proof. We use an auxiliary C ∞ hermitian metric h a on λ with curvature form Θ a . Then the fiber integral H/H Θ N +1 a exists and represents up to a numerical constant the Chern class c 1 (det(( λ − λ −1 ) N +1 )) on H. On H f r the difference Θ f r −Θ a is (globally) of the form √ −1∂∂u. Now Basic properties of the L 2 -Dolbeault-complex on ∆ * k × ∆ l (cf. [Z]) show that u, and ∂u can be chosen as locally L 2 -integrable (with respect to metrics with Poincaré growth condition). So H f r /H √ −1 ∂u ∧ Ω defines actually a current. We claim that in the sense of currents holds.
In fact, the right hand side applied to a C ∞ -form with compact support equals Corollary 3. There exists a singular hermitian metric h Q for det(( λ− λ −1 ) N +1 ) on H, whose curvature is positive in the sense of currents.
Remark 4. Furthermore, it follows from the construction that for any subspace of H, in particular for any curve in H, the restriction of h Q and Θ Q resp. exist as singular metric and d-closed current resp. If C ⊂ H is a local analytic curve through a point p, representing a direction, where X − → H is effective, the current is strictly positive in this direction.
The latter fact follows immediately, because the form Θ f r is strictly positive on the preimage of C in H f r .
After blowing up the boundary λ Q possesses a line bundle extension λ Q on H. The result of this paragraph concerning Hilbert schemes is so far: Theorem 4. The compactified Hilbert scheme H ⊃ H carries a line bundle λ Q with a singular hermitian metric h Q whose curvature Θ Q is positive. The Lelong numbers vanish everywhere, and Θ Q is strictly positive in effective directions of the family X − → H. Moreover, on H the construction is functorial with respect to base changes of families concerning the line bundle and its curvature.
In a final step we descend to the moduli space M.
The automorphism groups of the polarized manifolds act on local universal deformation spaces in a finite way (with uniformly bounded orders). By functoriality, a certain power (λ Q ) µ descends from H to some λ M on M. On M the line bundle λ Q gives rise to a coherent sheaf. As µ · Θ Q is invariant under the action of the projective linear group on H, it descends to a current Θ M on M. We look at the natural map u : H − → M extended to u : H − → M. The current Θ M will now be extended to M: Let ϕ be a C ∞ differential form of degree (dim M−1, dim M−1) with compact support. We take a closed subvariety S ⊂ H, so that the map S − → M is generically finite, and dominant. The following definition is independent of the choice of S: is well-defined, and Θ h | Z ≥ η Z in the sense of currents, where η Z denotes some C ∞ hermitian form on Z.

L 2 -methods
In this section, we gather some results based upon H"ormander's techniques (cf. also the result by Ohsawa and Takegoshi [O-T]).
Let (Y, ω Y ) be a complete K"ahler manifold, and (L, h) be a hermitian line bundle on Y . We write and use the semi-colon notation for covariant derivatives with respect to the metric tensor. Moreover the components of the connection form of the line bundle are and we use Θ αβ = θ α;β for the coefficients of the curvature tensor. We use ∇ α to denote covariant derivatives of L-valued tensors, and .. , ..(p) resp. denote norms and pointwise norms resp. Let ϕ = ϕ β dz β be any L-valued (0, 1)-form of class C ∞ . Then is the formal adjoint of the ∂-operator.
The rough Laplacian is defined by and the Bochner-Kodaira-Weitzenboeck formula for this situation reads where R αβ denotes the Ricci-tensor of ω X . (The contribution R αβ cancels out, if we replace L by L + K Y ). The formula implies for all C ∞ -forms with compact support. According to Andreotti and Vesentini [A-V] the estimate (17) holds (use cut-off functions) for all square integrable forms ϕ, for which ∂ϕ and ∂ * ϕ, taken in the distributional sense, are square integrable. Let H 1 and H 2 resp. be the Hilbert spaces of square integrable L-valued (n, 0)-and (n, 1)-forms resp. Then the exterior derivative ∂ is a densely defined closed operator T : H 1 − → H 2 whose adjoint T * is given by ∂ * (cf. [A-V]).
Proposition 8. Let (Y, ω Y ) be a K"ahler manifold, which possesses also a complete K"ahler metric, and let (L, h) be a holomorphic line bundle, with a singular hermitian metric, whose Lelong numbers vanish everywhere. Suppose that for some continuous, everywhere positive function c(p) on Y . Then for any L-valued (n, 1)-current v with ∂v = 0, and Proof. We assume first that h is of class C ∞ and that ω Y is complete. We follow the argument of H"ormander and Demailly. The closed subspace F ⊂ H 2 of all ∂ closed forms contains the range of T , and T * vanishes on the orthogonal complement of F so that we can consider T as an operator from H 1 to F , and T * as an operator from F to H 1 . Now (17) implies for all g ∈ F , contained in the domain of T * with (1/c(p)) v(p) 2 < ∞ that For any g, v ∈ F , with g in the domain of T * and (1/c(p)) v(p) 2 < ∞, we have For any such v there is a continuous linear functional on the range of T * sending T * g to (g, v). The Hahn-Banach theorem implies the existence of some u ∈ H 1 such that (T * g, u) = (g, v) for all g in the domain of This result was extended by Demailly to arbitrary K"ahler metrics in [DE1]. The generalization to singular hermitian metrics is due to Nadel [NA].

Multiplier ideal sheaves
Let (L, h) be a singular hermitian line bundle on a complex manifold M. The sheaf L 2 (L, h) of square-integrable sections with respect to h is defined by for open subsets U ⊂ M. There exists an ideal sheaf I(h), called multiplier ideal sheaf such that holds. If we write h = e −ϕ · h 0 , where h 0 is a hermitian metric of class C ∞ , and ϕ ∈ L 1 loc (M) is the weight function, we see that holds. We also use the notation I(ϕ) for this sheaf.
For any modification π : M − → M of complex manifolds, and any plurisubharmonic function χ the following identity of multiplier ideal sheaves is known (cf. [DE4,Prop. 5.8]): Definition 7. A plurisubharmonic function ϕ on a complex manifold is said to have analytic singularities, if locally where the f i denote holomorphic functions, ϕ 0 is a C ∞ -function, and α ∈ R + .
If σ i are global sections of a line bundle L, defines a singular hermitian metric of positive curvature. In the above sense it will be called a metric with analytic singularities or algebraic singularities resp. (In the latter case α ∈ Q + is also required.) In the above situation the holomorphic functions f i define some ideal J ⊂ O M . We blow up M along the ideal J , desingularize in a way such that the exceptional set of the blow-up becomes a normal crossing divisor D = D I . We call the resulting modification π : M − → M. Now where R = ρ j D j , ρ i ∈ N is the exceptional divisor of π on M .
The pull-back of |f i | 2 to M is of the same type, with zero-set equal to D, so it is of the form where {τ i } are defining functions of {D i }, β i ∈ R ≥0 , and ϕ 0 is some C ∞ -function. In this case the multiplier ideal sheaf can be computed explicitly as where ⌊β i ⌋ is the Gaussian bracket. Together with (18) this implies In particular, I(ϕ • π) is locally free.
Proposition 9. Let ∆ n ⊂ C n be a polydisk, ϕ a plurisubharmonic function with analytic singularities on ∆ n , and ψ a plurisubharmonic function such that √ −1∂∂ψ is absolutely continuous on any local holomorphic curve C ⊂ ∆ n with ϕ|C ≡ −∞.
Then, after replacing ∆ n by any smaller, relatively compact polydisk, there exists real numbers γ arbitrarily close to 1 such that holds.
Proof. In the sequel, we always allow ∆ n to be replaced by a slightly smaller polydisk. We first apply the above modification to M = ∆ n with respect to ϕ. Then we perform a further sequence of blow-ups and get a modification π : ∆ − → ∆ n so that also J = I((ϕ + ψ) • π) is locally free, and such that with M = ∆ the exceptional divisor is of the above form D = D i . We still have (20,21) for ϕ.
For any point x ∈ ∆\D the function ϕ • π is of class C ∞ , and ψ • π is absolutely continuous, when restricted to curves through x. Hence the Lelong number ν((ϕ + ψ) • π, x) vanishes. By [BO, SK] we have J Next, we use (19) as above and get for all α > 0.
The latter fact allows us to compute the multiplier ideal sheaf from the Lelong number: As it is locally free and the space is smooth, it is sufficient to compute it for points on the regular part of the normal crossings divisor D. Let D i be the zero set of a coordinate function τ i .
at some x ∈ D i \ j =i D i . It follows from [BO, SK] that We need to see that no lower power τ k i is contained in this multiplier ideal sheaf.
From the Lelong number of h, we get the known lower estimate We use this estimate on a local analytic curve C x , which intersects D i in x transversally. So Cx h|τ i | 2k dV Cx = ∞. The same argument is used for points on D i near x. By Fubini's theorem τ k i is not in the multiplier ideal sheaf. Now equation (18) implies the claim.
Remark 5. The above proposition is still valid for the wider class of those plurisubharmonic functions, which differ from a plurisubharmonic function with analytic singularities, by a function, which is bounded by c · log(− log δ(x))), where c > 0 is a constant, and δ is the distance of x from the singular set.

A criterion for quasi-projectivity
Let X be a not necessarily reduced algebraic space with compactification X in the sense of algebraic spaces, and let L be a holomorphic line bundle on X with a positive singular hermitian metric h in the sense of Section 2.
Condition (P). We say that the positivity condition (P) holds, if (i) for all p ∈ X and any holomorphic curve C ⊂ X through p with C ∩ X = ∅ the (positive, d-closed) current Θ h |C is well-defined, and the Lelong number ν(Θ h |C, p) vanishes, (ii) for any smooth locally closed subspace Z ⊂ X the current Θ h | Z is well-defined, and Θ h | Z ≥ γ Z in the sense of currents, where γ Z denotes some C ∞ hermitian form on Z.
Now we state the criterion.
Theorem 6. Let X be an irreducible, not necessarily reduced algebraic space with a compactification X. Let L be a holomorphic line bundle on X. The map where N(m) = dim |mL|, defines an embedding of X for sufficiently large m, if it satisfies condition (P).
We will prove the theorem by induction over n = dim X. The case n = 1 is obvious: Let X be an algebraic curve. If X is smooth, the assumption implies that deg(L) > 0. Let X be a singular curve and π : X − → X be the normalization. Then deg(π * L) > 0 from the assumption so that L ⊗ℓ defines an embedding of X into a projective space.
13. Bigness of L Let X be a reduced, compact complex space of dimension n, and L = O X (L) ∈ Coh(X) an invertible sheaf. In the sequel we denote by ν : Y − → X The normalization of the (not necessarily locally irreducible) space X, and by ρ : Z − → Y a modification such that Z is smooth. If X is a Moishezon space, we assume also that Z is projective. Let π = ρ • ν.
The other implications are obvious.
For any m > 0 with h(X, L ⊗m ) > 0 we denote by Φ L ⊗m : X − → P N , N = N(m) the meromorphic map induced by global sections.
Proposition 11. Let X be a (reduced) Moishezon space then the following are equivalent: Proof. We need to show that (i) implies (ii), the remaining implications are clear.
We return to the situation of Theorem 6. To show that L is big on the reduced space, we do not need that the Lelong numbers of h vanish. However, we give the more general argument.
Let U ⊂ X reg be a Zariski open subset, which is quasi-projective. We can find a projective compactification U together with a modification γ : U − → X such that the divisor D = U \U has only normal crossings singularities and such that the singular hermitian metric h extends from U to U as a singular hermitian metric on γ * L (cf. Section 2. As usual one can construct a complete K"ahler form η U on U with Poincaré growth near the boundary from a K"ahler form on U and a canonical section of D.
Lemma 6. Let x ∈ U be a point. Then there exists some m 0 > 0 so that for any m ≥ m 0 there is a section with σ(x) = 0.
Proof. We use Kodaira's argument. Let W = {(z 1 , . . . , z n )} ⊂ U be a coordinate neighborhood and ρ a cut-off function with support in W , which is identically equal to one a relatively compact neighborhood of x contained in W and has values between 0 and 1. We set There exists some m 0 > 0 and a continuous strictly positive function c(p) on U so that Let m ≥ m 0 . We chose a local section σ x ∈ H 0 (W, K X + mL) on W with σ x (x) = 0 and set f = ∂(ρσ x ).
By assumption that h has absolutely continuous curvature on U, so the Lelong numbers vanish on all curves, and by the argument from Section 11 the multiplier ideal sheaf is equal to the structure sheaf, and h is in L 1 loc (U). Moreover f vanishes identically in a neighborhood of and Proposition 8 implies the existence of a (n, 0)-form u of class C ∞ (since f is of class C ∞ ) with values in mL such that The finiteness of the above integral (plus the fact that h is bounded from below, and that u is holomorphic on some neighborhood of x) imply that u(x) = 0. Now we can see that In a similar way, by taking two points and directions at a point we obtain the following lemma.
Lemma 7. For any compact set K ⊂ U there exists a number m(K) > 0 so that for all m ≥ m(K) the linear system |H 0 (2) (U, O U (K U + mL))| gives an embedding of K into a projective space.
We have the following extension property: Lemma 8. There is a canonical embedding: Proof. In our situation h possesses an extension h as a singular metric on γ * L with positive curvature in the sense of Definition 2. In particular, h is locally bounded from blow by a positive constant. For any So γ * σ extends holomorphically to U.
We are given a singular hermitian metric on L over the singular complex space X, which amounts to a singular hermitian metric h on X reg , which can be extended from U as a singular metric h on γ * L over U . The latter defines a multiplier ideal sheaf I( h m ) ⊂ O U which is defined by the following property: For all open subsets W ⊂ U the space For any such bundle, the pull-back L of L to the normalization X − → X satisfies lim sup m− →∞ m −n h 0 ( X, O X (m L)) > 0. According to Proposition 10, any such bundle is big in the usual sense, and Proposition 11 guarantees that associated linear systems embed certain Zariski open subsets.
We claim: Proposition 12. The above line bundle (L, h) is big.
, be a section, which we extend to U . We denote by D 0 ⊂ U the zero divisor. Next, we consider the restriction morphism Since Since h is locally bounded from below by some (positive) constant (in the appropriate measure theoretic sense) We state the following general fact, which implies that the above line bundle L, pulled back to a desingularization is nef.
Proposition 13. Let Y be a projective manifold and (L, h) a positive, singular hermitian line bundle, whose Lelong numbers vanish everywhere. Then L is nef.
Proof. Let A be an ample line bundle on X. For any y ∈ Y one considers a finite, locally free resolution of the maximal ideal at y. Then we chose a multiple ℓ(y) · A so that all P j ⊗ K −1 Y (ℓ(y) · A) are positive. The value for ℓ(y) can be taken uniformly in a neighborhood of y. So we choose ℓ 0 uniformly on Y with this property. As the Lelong numbers vanish, the multiplier ideal sheaves I(h m ) are equal to O Y for all m > 0. So H q (Y, P r ⊗ O(ℓ 0 A + mL)) = 0 for all r and q, m > 0 by the Nadel vanishing theorem. Now H 1 (Y, m Y,y ⊗ O Y (ℓ 0 A + mL)) = 0 for m > 0, and the sheaves O Y (ℓ 0 A + mL) are globally generated. So O Y (ℓ 0 A + mL) is globally generated, in particular nef. Hence L + ℓ 0 m A is nef for all m > 0. With m − → ∞ the claim follows.
14. Embedding of non-reduced spaces Proposition 14. Let X be a compact complex space, which possesses a holomorphic line bundle L, whose restriction to the reduction X red is ample. Then L is ample.
Proof. Let O X red = O X /I, and X j = (X red , O X /I j+1 ) so that X red = X 0 and X = X k for some k. Let L = O X (L), and L 0 = L|X 0 . Now We can assume from the beginning that L 0 is very ample on X 0 , and that H 1 (X 0 , F j ⊗ L ⊗ℓ ) = 0 for all ℓ > 0, and j = 0, . . . k − 1, furthermore that F j ⊗ L ⊗ℓ is globally generated for all j, and all ℓ > 0. Now for all ℓ > 0 the map H 0 (X, L ⊗ℓ ) − → H 0 (X 0 , L ⊗ℓ 0 ) is surjective. So we have a holomorphic map Φ |L| : X − → P N , whose restriction Φ 0 to X 0 is an embedding, i.e O P N − → O X followed by O X − → O X 0 is surjective. We show by induction over j the existence of compatible maps α j : O P N(j) − → O X j . (In each step the number N will have to be raised.) We assume that O P N − → O X j is surjective. The pull-back of the small extension (E j ) of sheaves of analytic C-algebras with respect to this map induces the direct sum of spaces X j+1 ⊕ X j P N . As the surjective morphism α j can be lifted to the morphism α j+1 , the induced small extension is trivial: we have the following diagram.
We need the above statement in a more general situation.
Let Z be a non-reduced complex space equipped with a holomorphic line bundle L, L = O Z (L). Let X = red (Z), and let X ⊂ X be a Zariski open subset. We denote by Z the restriction of the non-reduced structure to X. We consider the meromorphic map Φ = Φ |L| X | : X ֒→ P N .
Proposition 15. Assume that Φ| X : X − → P N is an embedding, and let L be also nef. Then for some multiple ℓ 0 the meromorphic map Φ |ℓ 0 L| : Z − → P M defines an embedding of an open subspace of Z.
Proof. First, we take a (projective) desingularization of X, and pull back the meromorphic map Φ. Then we eliminate the indeterminacy set by a sequence of blow-ups with smooth centers. This procedure is locally done by embedding the space in a smooth ambient space, blowing up the ambient space along smooth centers several times, and by taking in each step the proper transform of X. We take locally embeddings of X, which extend to embeddings of Z. However, in general there exist no proper transforms of non-reduced spaces. So in all steps, we take the full pre-image under the monoidal transformation, and arrive at a space ρ ∨ : Z ∨ − → Z together with the restriction π : X − → X, which allows a morphism Ψ : X − → P N . We denote by Z k ⊂ Z the k-th infinitesimal neighborhood of the compact manifold X in Z so that Z 0 = X and Z k 0 = Z for some k 0 > 0. We consider the k-th infinitesimal neighborhoods Z k of X in Z ∨ , and we define Z := Z k 0 , and we denote the map to Z by ρ. The infinitesimal neighborhoods give rise to short exact sequences where the F j are coherent O X -modules.
Let n = dim X. Denote by L the pull-back of L to Z. We claim that (for any fixed j) h 1 ( X, F j (ℓ · L| X ))O(ℓ n−1 ): The bundle L| X is big. After replacing L by a multiple we write L| X = A + E, where A is ample and E is effective by Kodaira's lemma. As X is projective and L is nef and big, we can fix a K"ahler form η X and find hermitian metrics h r on L| X such that the curvature of h r is greater or equal to −(1/r)η X for all r > 0. This shows the existence of some m 0 such that H 1 ( X, F j ⊗ O X (mA + ℓ L)) = 0 for all m ≥ m 0 and ℓ > 0.
Let mE denote the non-reduced space with support E, induced by the divisor mE. Then as well.
We fix m = m 0 and look at ℓ ≫ 0. Then h 1 ( X, F j ((m 0 + ℓ) L)) = O(ℓ n−1 ). Now is exact, and h 0 ( X, O X j (m 0 +ℓ) L) grows like ℓ n , because we can assume by induction that high powers of L embed a Zariski open subset of X j , so h 0 ( X, O X j+1 ((m 0 + ℓ) L)) ∼ ℓ n . This means that the sections of H 0 ( X, O X j+1 ((m 0 + ℓ) L)) define a meromorphic map Ξ : X j+1 − → P M , which embeds an open subset W ∩ X, W ⊂ X j+1 of the reduction X(cf. Proposition 11). We assume that W is closed in some open set P M \A. The sets A and W can also be chosen in a way that is still exact. As in the proof of Proposition 14 we consider the fibered sum of complex spaces W ⊕ W ∩ X j (P M \A), which is isomorphic to the trivial extension (P M \A)[F j |W ], which is clearly quasi-projective. The rest follows like in the proof of Proposition 11.

Proof of the quasi-projectivity criterion
We first need the following fact: Lemma 9. Let π : Y − → X be a proper holomorphic map of reduced, compact not necessarily normal complex spaces. Let S ⊂ X be a closed subspace such that π is an isomorphism over X\S. Let I = I S ⊂ O X be the vanishing ideal of S. Then for any coherent sheaf F on X there is a number m > 0 and morphism µ : I m S · (π * π * F ) − → F , which is an isomorphism over X\S.
Proof. We consider the short exact sequence 0 − → F − → π * π * F − → C − → 0, where supp(C) ⊂ S. Now the zero set of the annihilator ideal V (Ann O X (C)) is contained in S so that I m · C = 0 for some m > 0.
We consider the situation of Theorem 6 and assume that X is reduced and irreducible. Let S = {x ∈ X; |mL| does not define an embedding around x for all m > 0}.
From Proposition 12, we know that the line bundle L in X is big, and by Proposition 11 the linear system |mL| provides an embedding of some Zariski open subspace. Using Noether induction, we see that there is a number m 0 > 0 such that Φ |m 0 L| embeds X\S. In particular, it embeds X, if S is empty.
Lemma 10. Let F be a coherent O X -module. Then there exists some ℓ 0 > 0 such that for all ℓ ≥ ℓ 0 the sheaf F ⊗ O X (ℓm 0 L) is generated by global sections at all points x ∈ X\S.
From the construction, we have a morphism of sheaves Ψ * O P N (1) − → π * O X (m 0 L), which is an isomorphism over Γ Φ \π −1 ( S). We have the following morphisms.
Here, we chose m 1 ≫ 0 large enough so that a corresponding m 2 satisfies I m 2 S π * π * F ⊂ F . Over X\S the above morphism of sheaves are isomorphisms so that we can produce enough global sections, which generate F ⊗ O X (m 0 ℓL) over X\S.
In the above situation we also need the case, where F is an ideal in I ⊂ O X . If X is smooth or normal, we have automatically π * π * I = I.
where W runs through the open sets of Y .
If α : Y − → Y is a desingularization such that the singular locus of Y corresponds to a normal crossings divisor in Y , we have ω Y is coherent.

Now we set
X ⊗L ⊗m is not generated by global sections at x for all m}, and S − = {x ∈ X; ω (2)∨ X ⊗L ⊗m is not generated by global sections at x for all m}. Now, by definition S + ∪ S − ⊃ S, and the converse inclusion follows from Lemma 10, so Let R be the non-normal locus of X. We denote by I S∪R ⊂ O X the ideal of functions that vanish on S ∪ R. Lemma 10 implies that there exists m 1 > 0 such that O X (m 0 m 1 L) ⊗ I S∪R is generated by global sections at all points x ∈ X\S. Let {σ 0 , . . . , σ N (m 0 m 1 ) } be a C-basis of Γ(X, O X (m 0 m 1 L) ⊗ I S∪R ). Then defines a singular hermitian metric on m 0 m 1 L over the space X, whose singularities are contained in S ∪ R. Let m + and m − resp. be integers such that ω (2) X ⊗ O X (m + m 0 L) and ω (2)∨ X ⊗ O X (m − m 0 L) are generated by global sections over X\S. Let {σ +,k } be a basis for the space of sections of the former sheaf over C. We define a hermitian metric h + on ω (2) In a similar way a metric h − on ω (2)∨ X ⊗ O X (m − m 0 L) is constructed. These metrics are so far only well-defined, where the sheaves are locally free, i.e. at least on (X) reg . We chose a desingularization π : X − → X in such a way that also π * ω (2) X is invertible (after dividing by the torsion part). Then the pull-backs of the sections σ +,k and σ −,k define singular hermitian metrics over X on the corresponding line bundles. We impose a further condition: Let U ⊂ X reg be a Zariski open subset, which is quasi-projective. Let U be a projective compactification that dominates X with modifications ρ : U − → X and γ : U − → X. We pull back h, h + , and h − back to U , and we assume as above that U\U is a divisor with normal crossings singularities. We denote by η U a K"ahler form on U and by η U a complete K"ahler form on U with Poincaré growth condition near the boundary as above. By Proposition 12 the line bundle γ * L is big on U . Kodaira's lemma provides an effective Q-divisor A such that the Q-divisor γ * L − A is ample, giving rise to a strictly positive hermitian metric h ′ of class C ∞ on the Q-line bundle γ * L−A. Let a·A be a divisor with a ∈ N, and σ A a section of O U (a·A). Then h ′ |σ A | 2/a defines a singular hermitian metric on γ * L, whose curvature Θ satisfies Θ ≥ α · η U for some α > 0 on U . Let D j be the components of the normal crossings divisor D = U \U. We equip the bundles [D j ] with a C ∞ hermitian metric. We can find canonical sections τ j and some β > 0 such that the curvature of the modified hermitian metric The following considerations apply to the above line bundles on U .
Although as a singular hermitian metric on a line bundle h is only defined over X reg , a coherent multiplier ideal sheaf I( h) ⊂ O X can be given a meaning as follows: For W ⊂ X open, we define (I( h) ⊗ O X (L ⊗e ))(W ) = {σ ∈ Γ(W, O X (L ⊗e )); V ∩(X)reg |σ| 2 h < ∞ for all V ⊂⊂ W }.
Remark 6. We can see from the definition of h on X that for sufficiently large p the zero set V (I( h)) ∩ X = S. Furthermore for large p the embedding dimension of the non-reduced space defined by I( h) is equal to dim X.
Proof. For large p the contribution of h p−1 0 to γ * (h p−δ 0 h r + h r+1 − ) dominates the rest, so that the zero set of the multiplier ideal sheaf is contained in S ∪ R and contains S ∪ R. Next δ > 0 is chosen small enough: The term 1/|σ A | 2 is equipped with a small exponent so that the L 2 -integrability condition for holomorphic sections is not affected, and V (I( h)) ∩ X = S ∪ R still holds. For large p also the second statement is satisfied.
Proposition 16. The canonical map is surjective.
So 0 = ∂(γ * τ −u), i.e. v = γ * τ −u ∈ H 0 (U, γ * (L ⊗e ⊗ω (2)∨ X )⊗O U (K U )). We claim that v is square-integrable: By (24) u 2 h is integrable over (X) reg . Since h is a singular hermitian metric of positive curvature, it is locally bounded from below by a positive constant. Moreover τ is of class C ∞ , and U carries the complete metric η U (with Poincaré growth condition). So v extends holomorphically to U. Then v gives rise to a holomorphic section of L ⊗e on X, which coincides with τ , when restricted to the subspace (S, O X /I( h)) (cf. equation (24)).
Proof of the Theorem. We know from the induction hypothesis that m 0 L|S defines an embedding of S. In particular, m 0 L|S is big. Now e is a sufficiently high multiple of m 0 , and also eL|S embeds S. In the last step, we need to raise the power of L without affecting the multiplier ideal sheaf I( h). Here, we replace L ⊗e by L ⊗(e+t) , where the factor L ⊗t is equipped with the original metric h t . Since the curvature of h is absolutely continuous, by Proposition 9, we may assume that I( h) = I( h · h t ) holds over X, if we perturb δ by a small amount (i.e. we perturb h). However, this metric is not of analytic singularity, but a singularity of type log(− log(t)) is negligible (cf. Remark 5). We chose t large enough so that L ⊗(e+t) defines an embedding of a Zariski open subspace of (S, O X /I( h)) by Proposition 15. Now L ⊗(e+t) embeds X\S as well as a non-empty open subset of S, and it also separates normal directions of this set in X. This contradicts the choice of S. This proves Theorem 6 for reduced spaces X.
For non-reduced spaces again we use induction over the dimension. Let L be a line bundle on X with the above assumptions. We know that for some m > 0 the meromorphic map Φ |mL|red(X)| embeds red(X). By Proposition 15 we can choose m > 0 so that Φ |mL| embeds a Zariski open subspace X ′ ⊂ X. Let T = red(X)\red(X ′ ). According to the above proof, a multiple of L, restricted to a high infinitesimal neighborhood T inf of T gives rise to a linear system, which embeds a Zariski open subspace of T inf . Finally, by Proposition 16 global sections over T inf can be extended to all of X contradicting the choice of T .