Blow-ups of locally conformally Kahler manifolds
aa r X i v : . [ m a t h . AG ] N ov L. Ornea, M. Verbitsky, V. Vuletescu
Blow-ups of LCK manifolds
Blow-ups of locally conformallyK¨ahler manifolds
Liviu Ornea , Misha Verbitsky , and Victor Vuletescu Keywords:
Locally conformally K¨ahler manifold, locally trivial bundle,blow-up.
Abstract
A locally conformally K¨ahler (LCK) manifold is a mani-fold which is covered by a K¨ahler manifold, with the decktransform group acting by homotheties. We show thatthe blow-up of a compact LCK manifold along a com-plex submanifold admits an LCK structure if and onlyif this submanifold is globally conformally K¨ahler. Wealso prove that a twistor space (of a compact 4-manifold,a quaternion-K¨ahler manifold or a Riemannian manifold)cannot admit an LCK metric, unless it is K¨ahler.
Contents Partially supported by CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0118. Partially supported by RFBR grant 10-01-93113-NCNIL-a, RFBR grant 09-01-00242-a, AG Laboratory SU-HSE, RF government grant, ag. 11.G34.31.0023, and Science Foun-dation of the SU-HSE award No. 10-09-0015. – 1 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu
Blow-ups of LCK manifolds A locally conformally K¨ahler (LCK) manifold is a complex manifold M , dim C M >
1, admitting a K¨ahler covering ( ˜
M , ˜ ω ), with the deck trans-form group acting on ( ˜ M , ˜ ω ) by holomorphic homotheties. Unless otherwisestated, we shall consider only compact LCK manifolds.In the present paper we are interested in the birational (or, more pre-cisely, bimeromorphic) geometry of LCK manifolds.An obvious question arises immediately. Question 1.1:
Let X ⊂ M be a complex subvariety of an LCK mani-fold, and M −→ M a blowup of M in X . Would M also admit an LCKstructure?When X is a point, the question is answered in affirmative by Tricerri[Tr] and Vuletescu [Vu]. When dim X >
0, the answer is not immediate.To state it properly, we recall the notion of a weight bundle of an LCKmanifold. Let ( ˜
M , ˜ ω ) be the K¨ahler covering of an LCK manifold M , and π ( M ) −→ Map( ˜
M , ˜ M ) the deck transform map. Since ρ ∗ ( γ )˜ ω = const · ˜ ω ,this constant defines a character π ( M ) χ −→ R > , with χ ( γ ) := ρ ∗ ( γ )˜ ω ˜ ω . Definition 1.2:
Let L be the 1-dimensional local system on M with mon-odromy defined by the character χ . We think of L as of a real bundle witha flat connection. This bundle is called the weight bundle of M .One may think of the K¨ahler form ˜ ω as of an L -valued differential formon M . This form is closed, positive, and of type (1,1). Therefore, for anysmooth complex subvariety Z ⊂ M such that L (cid:12)(cid:12)(cid:12) Z is a trivial local system, Z is K¨ahler.The following two theorems describe how the LCK property behavesunder blow-ups. Theorem 1.3:
Let Z ⊂ M be a compact complex submanifold of an LCKmanifold, and M the blow-up of M with center in Z . If the restriction L (cid:12)(cid:12)(cid:12) Z of the weight bundle is trivial as a local system then M admits an LCK– 2 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu Blow-ups of LCK manifolds metric.
Proof:
See Corollary 2.11.A similar question about blow-downs is also answered.
Theorem 1.4:
Let D ⊂ M be an exceptional divisor on an LCK mani-fold, an M the complex variety obtained as a contraction of D . Then therestriction L (cid:12)(cid:12)(cid:12) D of the weight bundle to D is trivial. Proof:
Theorem 2.9.This result is quite unexpected, and leads to the following theorem abouta special class of LCK manifold called
Vaisman manifolds (Section 2).
Claim 1.5:
Let M be a Vaisman manifold. Then any bimeromorphic con-traction M −→ M ′ is trivial. Moreover, for any positive-dimensional sub-manifold Z ⊂ M , its blow-up M does not admit an LCK structure. Proof:
Corollary 2.13
The proofs of Theorem 1.4 and Theorem 1.3 are purely topological. How-ever, they were originally obtained using a less elementary argument involv-ing positive currents.We state this argument here, omitting minor details of the proof, becausewe think that this line of thought could be fruitful in other contexts too; formore information and missing details, the reader is referred to [D1], [DP]and [D2].A current is a form taking values in distributions. The space of ( p, q )-currents on M is denoted by D p,q ( M ). A strongly positive current is alinear combination X I α I ( z ∧ z ) I where α I are positive, measurable functions, and the sum is taken over allmulti-indices I . An integration current of a closed complex subvariety is astrongly positive current. In the present paper, we shall often omit “strongly”, because we are only interestedin strong positivity. – 3 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu
Blow-ups of LCK manifolds
It is easy to define the de Rham differential on currents, and check thatits cohomology coincide with the de Rham cohomology of the manifold.Currents are naturally dual to differential forms with compact support.This allows one to define an integration (pushforward) map of currents,dual to the pullback of differential forms. This map is denoted by π ∗ , where π : M −→ N is a proper morphism of smooth manifolds.Now, let π : M −→ N be a blow-up of a subvariety Z ⊂ N of codimen-sion k , and ω a K¨ahler form on M . Then ( π ∗ ω ) k has a singular part whichis proportional to the integration current of Z .This follows from the Siu’s decomposition of positive currents ([D1]).Demailly’s results on intersection theory of positive currents ([D2]) are usedto multiply the currents, and the rest follows because the Lelong numbersof π ∗ ω along Z are non-zero.Applying this argument to a birational contraction M ϕ −→ M ′ of anLCK manifold M , and denoting by ˜ M ˜ ϕ −→ ˜ M ′ the corresponding map ofcoverings, we obtain a closed, positive current ξ := ˜ ϕ ∗ ˜ ω on ˜ M ′ , with thedeck transform map ρ acting on ξ by homotheties. Then ρ would also actby homotheties on the current ξ k , k = dim Z , where Z is the exceptionalset of ˜ ϕ .Applying the above result to decompose ξ k onto its absolutely continu-ous and singular part, we obtain that the current of integration [ Z ] of Z ismapped to const [ Z ] by the deck transform action. Since the current of inte-gration of Z is mapped by the deck transform to the current of integrationof ˜ ϕ ( Z ) = Z , the constant const is trivial; this implies that π ( Z ) ⊂ M ′ isK¨ahler, with the K¨ahler metric obtained in the usual way from ˜ ω . A compact complex variety X is said to belong to Fujiki class C if X is bimeromorphic to a K¨ahler manifold. The Fujiki class C manifolds areclosed under many natural operations, such as taking a subvariety, or themoduli of subvarieties, and play important role in K¨ahler geometry.This notion has a straightforward LCK analogue. Definition 1.6:
Let M be a compact complex variety. It is called a locallyconformally class C variety if it is bimeromorphic to an LCK manifold.The importance of the Fujiki class C notion was emphasized by a morerecent work of Demailly and P˘aun [DP], who characterized class C manifoldsin terms of positive currents. Recall that a K¨ahler current is a positive,– 4 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu
Blow-ups of LCK manifolds closed (1,1)-current ϕ on a complex manifold M which satisfies ϕ > ω forsome Hermitian form ω on M .Demailly and P˘aun have proven that a compact complex manifold M belongs to class C if and only if it admits a positive K¨ahler current.For an LCK manifold, an analogue of a K¨ahler current is provided bythe following notion (motivated by Definition 2.2). Definition 1.7:
Let M be a compact complex manifold, θ a closed real 1-form on M , Ξ a positive, real (1,1)-current satisfying d Ξ = θ ∧ Ξ and Ξ > ω for some Hermitian form ω on M . Then Ξ is called an LCK current .It would be interesting to know if an LCK-analogue of the Demailly-P˘auntheorem is true. Question 1.8:
Let M be a complex compact manifold. Determine whetherthe following conditions are equivalent. (i) M belongs to locally conformally class C. (ii) M admits an LCK current. We start by repeating (in a more technical fashion) the definition of anLCK manifold given in the introduction. Please see [DO] for more detailsand several other versions of the same definition, all of them equivalent.
Definition 2.1: A locally conformally K¨ahler (LCK) manifold is a com-plex manifold X covered by a system of open subsets U α endowed with local K¨ahler metrics g α , conformal on overlaps U α ∩ U β : g α = c αβ g β .Note that, in complex dimension at least 2, as we always assume, c αβ arepositive constants. Moreover, they obviously satisfy the cocycle condition.Interpreted in cohomology, the cocycle { c αβ } determines a closed one-form θ , called the Lee form . Hence, locally θ = df α . It is easily seen that e − f α g α = e − f β g β on U α ∩ U β , and thus determine a global metric g whichis conformal on each U α with a K¨ahler metric. One obtains the followingequivalent: – 5 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu Blow-ups of LCK manifolds
Definition 2.2:
A Hermitian manifold M is LCK if its fundamental two-form ω satisfies: dω = θ ∧ ω, dθ = 0 . (2.1)for a closed one-form θ .If θ is exact then M is called globally conformally K¨ahler (GCK).As we work with compact manifolds and, in general, the topology ofcompact K¨ahler manifolds is very different from the one of compact LCKmanifolds, we always assume θ = 0 on X .Let Γ −→ ˜ M π −→ M be the universal cover of M with deck group Γ.As π ∗ θ is exact on ˜ M , π ∗ ω is globally conformal with a K¨ahler metric ˜ ω .Moreover, Γ acts by holomorphic homotheties with respect to ˜ ω . This definesa character χ : Γ −→ R > , γ ∗ ˜ ω = χ ( γ )˜ ω. (2.2)It can be shown that this property is indeed an equivalent definition of LCKmanifolds, see [OV2].Clearly, a LCK manifold M is globally conformally K¨ahler if and only ifΓ acts trivially on ˜ ω ( i.e. im χ = { } ).A particular class of LCK manifolds are the Vaisman manifolds . Theyare LCK manifolds with the Lee form parallel with respect to the Levi-Civitaconnection of the LCK metric. The compact ones are mapping tori overthe circle with Sasakian fibre, see [OV1]. The typical example is the Hopfmanifold, diffeomorphic to S × S n − .On a Vaisman manifold, the vector field θ ♯ − √− J θ ♯ generates a one-dimensional holomorphic, Riemannian, totally geodesic foliation. If this isregular and if M is compact, then the leaf space B is a K¨ahler manifold. Example 2.3:
On a Hopf manifold C n \ { } / h z i z i i , the LCK metric P dz i ⊗ dz i | P z i z i | is Vaisman and regular; the leaf space is C P n − .We refer to [DO] or to the more recent [OV2] for more details aboutLCK geometry.It is known, [Tr, Vu], that the blow-up at points preserve the LCKclass. The present paper is devoted to the blow-up of LCK manifolds along– 6 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu Blow-ups of LCK manifolds subvarieties. In this case, the situation is a bit more complicated and adiscussion should be made according to the dimension of the submanifold.
Definition 2.4:
Let Y j ֒ → M be a complex subvariety. We say that Y is ofinduced globally conformally K¨ahler type (IGCK) if the cohomologyclass j ∗ [ θ ] vanishes, where θ denotes the cohomology class of the Lee formon M . Remark 2.5:
Notice that a IGCK-submanifold of an LCK manifold is al-ways K¨ahler.
Remark 2.6:
By a theorem of Vaisman ([Va2]), any LCK metric on acompact complex manifold Y of K¨ahler type is globally conformally K¨ahlerif dim C Y >
1. Therefore, the IGCK condition above for smooth Y withdim C Y > Y being K¨ahler. Remark 2.7:
Notice that there may exist curves on LCK manifolds whichare not IGCK, despite being obvioulsy of K¨ahler type. For instance, if M isa regular Vaisman manifold, and if Y is a fiber of its elliptic fibration, then Y is not IGCK, as any compact complex subvariety of a compact Vaismanmanifold has an induced Vaisman structure (see e.g. [Ve1, Proposition 6.5]).The main goal of the present paper is to prove the following two theo-rems: Theorem 2.8:
Let M be an LCK manifold, Y ⊂ M be a smooth complexIGCK subvariety, and let ˜ M be the blow-up of M centered in Y . Then ˜ M is LCK. Proof:
See the argument after Lemma 3.4.
Theorem 2.9:
Let M be a complex variety, and ˜ M −→ M the blow-up ofa compact subvariety Y ⊂ M . Assume that ˜ M is smooth and admits anLCK metric. Then the blow-up divisor ˜ Y ⊂ ˜ M is a IGCK subvariety. Proof:
See Remark 3.3.
Remark 2.10:
In the situation described in Theorem 2.9, the variety ˜ Y isof K¨ahler type, because it is IGCK. When Y is smooth, Y is K¨ahler, as– 7 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu Blow-ups of LCK manifolds shown by Blanchard ([Bl2, Th´eor`eme II.6]). Together with Remark 2.6, thisimplies the following corollary.
Corollary 2.11:
Let M be an LCK manifold, and Y ⊂ M a smooth compactsubvariety, such that the blow-up of M in Y admits an LCK metric. Ifdim C ( Y ) > Y is a IGCK subvariety. Remark 2.12:
Note that, from [Ve1, Proposition 6.5], a compact complexsubmanifold Y of a compact Vaisman manifold is itself Vaisman, and θ represents a non-trivial class in the cohomology of Y , so there are no IGCKsubmanifolds of proper dimension dim C ( Y ) > . This implies the followingcorollary.
Corollary 2.13:
The blow-up of a compact Vaisman manifold along a com-pact complex submanifold Y of dimension at least 1 cannot have an LCKmetric.The proofs of these two theorems and of the corollary will be given inSection 3. As a by-product of our proof, we obtain the following: Corollary 2.14: If M is a twistor space, and if M admits a LCK metric,then this metric is actually GCK. Proof:
See Corollary 3.2.Here, by a “twistor space” we understand any of the following construc-tions of a complex manifold: the twistor spaces of half-conformally flat 4-dimensional Riemannian manifolds, twistor spaces of quaternionic-K¨ahlermanifolds, and Riemannian twistor spaces of conformally flat manifolds.
Remark 2.15: ( i ) A similar, weaker result is proven in [KK]. Namely, thetwistor space of half-conformally flat 4-dimensional Riemannian manifoldswith large fundamental group cannot admit LCK metrics with automorphicpotential on the covering. The proof uses different techniques from ours, andwhich cannot be generalized neither to higher dimensions nor to quaternionicK¨ahler manifolds.( ii ) It was known from [Ga, Mu] that the natural metrics (with respectto the twistor submersion) cannot be LCK. Our result refers to any metricon the twistor space, not necessarily related to the twistor submersion. Onthe other hand, as shown by Hitchin, the twistor space of a compact 4-dimensional manifold is not of K¨ahler type, unless it is biholomorphic to– 8 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu Blow-ups of LCK manifolds C P or to the flag variety F [Hi]. Remark 2.16:
So far, we were unable to deal with the reverse statementof Theorem 2.8, namely, to determine whether a smooth bimeromorphiccontraction of an LCK manifold is always LCK. In the particular case whenan exceptional divisor is contracted to a point, this has been proven to betrue by Tricerri, [Tr]; we conjecture that in the general case this is false, butwe are not able to find any example.For GCK (that is, K¨ahler) manifolds, the answer is well known: blow-downs of K¨ahler manifolds can be non-K¨ahler, as one can see from anyexample of a Moishezon manifold.
Remark 2.17:
We summarize the case of blow-up of curves on LCKmanifolds. Since rational curves are simply-connected, they are IGCK sub-manifolds, so blowing-up a rational curve on a LCK manifold always yieldsa manifold of LCK type. The case of the elliptic curves was partially tack-led in Corollary 2.13. If Y is a curve of arbitrary genus contained in anexceptional divisor of a blow-up, then it is also automatically a IGCK sub-variety since the exceptional divisor is so; hence again, blowing it up yieldsa manifold of LCK type.To our present knowledge, the only examples of curves Y on LCK man-ifolds M with genus g ( Y ) > Lemma 3.1:
Let M be an LCK manifold, B a path connected topologicalspace and let π : M −→ B be a continuous map. Assume that either (i) B is an irreducible complex variety, and π is proper and holomorphic. (ii) π is a locally trivial fibration with fibers which are complex subvarietiesof M .Suppose also that the map π ∗ : H ( B ) −→ H ( M )is an isomorphism, and the generic fibers of π are positive-dimensional. Thenthe LCK structure on M is actually GCK.– 9 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu Blow-ups of LCK manifolds
Proof:
Denote by θ the Lee form of M , and let ˜ M be the minimal GCKcovering of X , that is, the minimal covering ˜ M −→ M such that the pullbackof θ is exact. Since H ( B ) ∼ = H ( M ), there exists a covering ˜ B −→ B suchthat the following diagram is commutative, and the fibers of ˜ π are compact:˜ M −−−−→ M ˜ π y y π ˜ B −−−−→ B Let ˜ B ⊂ ˜ B be the set of regular values of ˜ π , and let F b := ˜ π − ( b ) be theregular fibers of ˜ π , dim C F b = k . Since B is connected, all F b represent thesame homology class in H k ( ˜ M ).Denote the K¨ahler form of ˜ M by ˜ ω , conformally equivalent to the pull-back of the Hermitian form on X .Since all F b represent the same homology class, the Riemannian volumeVol ˜ ω ( F b ) := Z F b ˜ ω k is independent from b ∈ B . This gives (recall the definition of the character χ in (2.2))Vol ˜ ω ( F b ) = Z F b ˜ ω k = Z F γ − b ) γ ∗ ˜ ω k = Z F γ − b ) χ ( γ ) k ˜ ω k = χ ( γ ) k Vol ˜ ω ( F b ) , hence the constant χ γ is equal to 1 for all γ ∈ Γ. Therefore, ˜ ω is Γ-invariant,and M is globally conformally K¨ahler.The above lemma immediately implies Corollary 2.14. Corollary 3.2:
Let Z be the twistor space of M , understood in the senseof Corollary 2.14. Assume that Z admits an LCK metric. Then this metricis globally conformally K¨ahler. Proof:
There is a locally trivial fibration Z −→ M , with complex ana-lytic fibers which are compact symmetric K¨ahler spaces, hence Lemma 3.1can be applied. Remark 3.3:
In the same way one deals with the blow-ups: the genericfibers over an exceptional set of a blow-up map are positive-dimensional.Therefore, Lemma 3.1 implies Theorem 2.9.– 10 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu
Blow-ups of LCK manifolds
We can now give
The proof of Corollary 2.13:
If dim C ( Y ) > C ( Y ) = 1 we cannot use this argument directly - see Remark 2.7- so in this case we argue as follows.Assume ˜ M has an LCK metric ˜ ω with Lee form ˜ η . By Theorem 2.9, therestriction ˜ η | Z to the exceptional divisor Z is exact. Hence, after possiblymaking a conformal change of the LCK metric, we can assume ˜ η | V = 0where V is a neighbourhhood of Z . In particular, ˜ η will be the pull-back ofa one-form η on M . On the other hand, ˜ ω gives rise to a current on ˜ M (seealso § ,
1) current Ξ on M with associate Lee form η . Clearly η | Y = 0.Possibly conformally changing now Ξ, we can assume that η is the uniqueharmonic form (with respect to the Vaisman metric of M ) in its cohomologyclass. Possibly η | Y is no longer zero, but remains exact .We now show that η is basic with respect to the canonical foliation F generated on M by θ ♯ − √− J θ ♯ . Indeed, from [Va2], we know that anyharmonic form on a compact Vaisman manifold decomposes as a sum α + θ ∧ β where α and β are basic and transversally (with respect to F ) harmonicforms. In particular, as a transversally harmonic function is constant, wehave η = α + c · θ, (3.1)where c ∈ R and α is basic, transversally harmonic (see [To] for the theoryof basic Laplacian and basic cohomology etc.).Let now S denote the unique homology class in H ( M ) (call it thefundamental circle of θ ) such that R S θ = 1 and R S α = 0 for every basiccohomology class α .As any complex submanifold of a compact Vaisman manifold is tangentto the Lee field and hence Vaisman itself, Y is Vaisman with Lee form θ | Y . Hence we deduce that the fundamental circle of θ is the image of thefundamental circle of θ | Y under the natural map H ( Y ) → H ( M ).We now integrate 3.1 on any γ ∈ H ( Y ) and take into account that η | Y isexact to get c = 0. Hence, η basic. It can then be treated as a harmonic one-form on a K¨ahler manifold (or use the existence of a transversal dd c -lemma).This implies d c η = 0.But then one obtains a contradiction, as follows. Letting J to be thealmost complex structure of M , we see on one hand we have Z M d (Ξ n − ) ∧ J ( θ ) = Z M ( n − n − ∧ θ ∧ J ( θ ) >
0– 11 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu
Blow-ups of LCK manifolds since Ξ is positive. On the other hand, since d ( J ( θ )) = 0 , it follows that d (Ξ n − ) ∧ J ( θ ) is exact so R M d (Ξ n − ) ∧ J ( θ ) = 0 , a contradiction.The following result is certainly well-known, but since we were not ableto find out an exact reference we include a proof here. Lemma 3.4:
Assume (
U, g ) is a K¨ahler complex manifold, Y ⊂ U a compactsubmanifold and let c : ˜ U −→ U be the blow-up of U along Y . Then, forany open neighbourhood V ⊃ Y , there is a K¨ahler metric ˜ g on ˜ U such that˜ g | ˜ U \ c − ( V ) = c ∗ ( g | U \ V ) Proof. (due to M. P˘aun; see also [Vu]).1. There is a (non-singular) metric on O ˜ U ( − D ) (where D is the excep-tional divisor of the blow-up) such that:1.A. Its curvature is zero outside c − ( V ), and1.B. Its curvature is strictly positive at every point of D and in anydirection tangent to D .Indeed, if such a metric is found, everything follows, as the curvature ofthis metric plus a sufficiently large multiple of c ∗ ( g ) will be positive definiteon ˜ U .
2. To finish the proof, we notice that the existence of a metric h withproperty 1.B is clear, due to the restriction of O ˜ U ( − D ) to D .Now let α be its curvature; then α − i∂∂τ = − [ D ] for some function τ , with at most logarithmic poles along D , bounded from above, and non-singular on ˜ U \ D. Consider the function τ := max( τ, − C ) where C is somepositive constant, big enough such that on ˜ U \ c − ( V ) we have τ > − C. Clearly, on a (possibly smaller) neighbourhood of D we will have τ = − C, such that the new metric e − τ h on O ˜ U ( − D ) also satisfies 1.A.Now we can prove Theorem 2.8. Let c : ˜ M −→ M be the blow up of M along the submanifold Y . Let g be a LCK metric on M and let θ be itsLee form. Since Y is IGCK we see θ | Y is exact. Let U be a neighbourhoodof Y such that the inclusion Y ֒ → U induces an isomorphism of the firstcohomology. Then θ | U is also exact, so, after possibly conformally rescaling g , we may assume θ | U = 0 and hence g | U is K¨ahler. In particular, supp( θ ) ∩ U = ∅ . Now choose a smaller neighbourhood V of Y and apply Lemma 3.4.We get a K¨ahler metric ˜ g on ˜ U which equals c ∗ ( g ) outside c − ( V ), so it gluesto c ∗ ( g ) giving a LCK metric on ˜ M .– 12 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu Blow-ups of LCK manifolds
Acknowledgments.
We are indebted to M. Aprodu, J.-P. Demailly, D.Popovici and M. Toma for useful discussions. We are grateful to MihaiP˘aun for explaining us the proof of Lemma 3.4.
References [B] F.A. Belgun,
On the metric structure of non-K¨ahler complex surfaces ,Math. Ann. (2000), 1–40.[Bl1] A. Blanchard,
Espaces fibr´es k¨ahleriens compacts , C.R. Acad. Sci.Paris (1954), 2281–2283.[Bl2] A. Blanchard,
Sur les vari´et´es analitiques complexes , Ann. Sci. E.N.S. (1956), 157–202.[D1] J.-P. Demailly, Analytic methods in algebraic geometry,
Lecture Notes,´Ecole d’´et´e de Math´ematiques de Grenoble “G´eom´etrie des vari´et´esprojectives complexes : programme du mod`ele minimal” (June-July2007).[D2] J.-P. Demailly,
Regularization of closed positive currents and Intersec-tion Theory,
J. Alg. Geom. (1992) 361-409.[DP] J.-P. Demailly, M. P˘aun, Numerical characterization of the K¨ahlercone of a compact K¨ahler manifold , math.AG/0105176 also in Annalsof Mathematics, (2004), 1247-1274.[DO] S. Dragomir, L. Ornea, Locally conformal K¨ahler geometry, Progressin Math. , Birkh¨auser, Boston, Basel, 1998.[Ga] P. Gauduchon,
Structures de Weyl et th´eor`emes d’annulation sur unevari´et´e conforme autoduale , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1991), no. 4, 563–629.[Hi] N. Hitchin, K¨ahlerian twistor spaces , Proc. London Math. Soc. (3) (1981) 133–150.[KK] G. Kokarev, D. Kotschick, Fibrations and fundamental groups ofK¨ahler-Weyl manifolds , Proc. Amer. Math. Soc. (2010), no. 3,997–1010. – 13 – version 2.0, November 14, 2011 . Ornea, M. Verbitsky, V. Vuletescu
Blow-ups of LCK manifolds [Mu] O. Mu¸skarov,
Almost Hermitian structures on twistor spaces and theirtypes , Atti Sem. Mat. Fis. Univ. Modena (1989), no. 2, 285–297.[OV1] L. Ornea, M. Verbitsky, Structure theorem for compact Vaismanmanifolds , Math. Res. Lett., (2003), 799–805.[OV2] L. Ornea, M. Verbitsky, A report on locally conformally K¨ahler man-ifolds , Contemporary Mathematics , 135-150, 2011.[To] P. Tondeur, Foliations on Riemannian manifolds, Universitext,Springer– Verlag 1988.[Tr] F. Tricerri,
Some examples of locally conformal K¨ahler manifolds ,Rend. Sem. Mat. Univ. Politec. Torino (1982), 81–92.[Va1] I. Vaisman, On locally and globally conformal K¨ahler manifolds , Trans.Amer. Math. Soc. (1980), 533–542.[Va2] I. Vaisman,
A geometric condition for an l.c.K. manifold to be K¨ahler ,Geom. Dedicata (1981), 129–134.[Va3] I. Vaisman, Generalized Hopf manifolds , Geom. Dedicata (1982),231–255.[Ve1] M. Verbitsky, Vanishing theorems for locally conformal hyperk¨ahlermanifolds , Proc. of Steklov Institute, (2004),54–79.[Vu] V. Vuletescu,
Blowing-up points on locally conformally K¨ahler mani-folds , Bull. Math. Soc. Sci. Math. Roumanie (100) (2009), 387–390. Liviu OrneaUniversity of Bucharest, Faculty of Mathematics,14 Academiei str., 70109 Bucharest, Romania. and
Institute of Mathematics “Simion Stoilow” of the Romanian Academy,21, Calea Grivitei Street 010702-Bucharest, Romania
[email protected], [email protected]
Misha VerbitskyLaboratory of Algebraic Geometry, Faculty of Mathematics, NRUHSE, 7 Vavilova Str. Moscow, Russia [email protected], [email protected]
Victor VuletescuUniversity of Bucharest, Faculty of Mathematics,14 Academiei str., 70109 Bucharest, Romania. [email protected] – 14 –– 14 –