aa r X i v : . [ m a t h . AG ] F e b AN EXAMPLE OF NON-K ¨AHLER CALABI–YAUFOURFOLD
NAM-HOON LEE
Abstract.
We show that there exists a non-K¨ahler Calabi–Yau four-fold, constructing an example by smoothing a normal crossing variety. Introduction
In this note, a
Calabi–Yau manifold is a simply-connected compact com-plex manifold with trivial canonical class and H i ( M, O M ) = 0 for 0 < i < dim M . K K K K K K Mathematics Subject Classification.
Key words and phrases. non-K¨ahler Calabi–Yau fourfold, semistable smoothing,semistable degeneration.
In dimension four, the situation is even more obscure. A huge numberof non-homeomorphic K¨ahler Calabi–Yau fourfolds can be constructed ascomplete intersections in toric varieties. However, to the best of the author’sknowledge, not a single example of non-K¨ahler Calabi–Yau fourfold has beenfound. The purpose of this note is to construct an example of non-K¨ahlerCalabi–Yau fourfold. Hence, we establish the following:
Theorem 1.1.
There exists a non-K¨ahler Calabi–Yau fourfold.
We note that some examples of simply-connected non-K¨ahler compactholomorphic symplectic fourfolds with trivial canonical class and H ( M, O M ) = 0were constructed ([4, 10]). We also remark that interests in non-K¨ahlerCalabi–Yau manifolds in other directions also have been growing rapidly([6, 9, 20, 23]).We shall construct our example by smoothing a normal crossing variety.By smoothing, we mean the reverse process of the semistable degenerationof a manifold to a normal crossing variety. If a normal crossing variety isthe central fiber of a semistable degeneration of Calabi–Yau manifolds, itcan be regarded as a member in a deformation family of those Calabi–Yaumanifolds. So building a normal crossing variety smoothable to a Calabi–Yau manifold can be regarded as building a deformation type of Calabi–Yaumanifolds. The construction by smoothing is intrinsically up to deformation.The structure of this note is as follows.We start Section 2 by introducing two background materials – a smoothingtheorem and a Calabi–Yau threefold. The smoothing theorem will be usedto smooth a normal crossing variety to a non-K¨ahler Calabi–Yau fourfoldand the Calabi–Yau threefold will be used as a building block in constructingthe normal crossing variety in the next sections.In Section 3, we build a smoothable normal crossing variety. The con-struction, starting from the Calabi–Yau threefold introduced in Section 2,involves several steps of taking quotients and blow-ups of varieties.In Section 4, we showed that the fourfold, which is a smoothing of thenormal crossing variety constructed in Section 3, is a non-K¨ahler Calabi–Yaufourfold and calculate its topological Euler number.2. A smoothing theorem and a Calabi–Yau threefold
By a variety, we mean a reduced complex analytic space. Let X = X ∪ X be a variety whose irreducible components are two smooth varieties X and X . X is called a normal crossing variety if, near any point p ∈ X ∩ X , X is locally isomorphic to { ( x , x , · · · , x n ) ∈ C n +1 | x n − x n = 0 } with p corresponding to the origin and X , X locally corresponding to thehypersurfaces x n − = 0 , x n = 0 respectively in C n +1 . Note that the variety D X := X ∩ X is smooth. Suppose that there is a proper map ς : X → ∆ from a complex manifold X onto the unit disk ∆ = { t ∈ C |k t k ≤ } N EXAMPLE OF NON-K ¨AHLER CALABI–YAU FOURFOLD 3 such that the fiber X t = ς − ( t ) is a smooth manifold for every t = 0 and X = X . We denote a generic fiber X t ( t = 0) by M X and we say that X is a semistable degeneration of a smooth manifold M X and that M X is a semistable smoothing (simply smoothing) of X .We will use the following result from [11] (Theorem 2.8 in [11] ) as apartial generalization of results in [12]. Theorem 2.1.
Let X = X ∪ X be a normal crossing variety of dimension n ≥ whose irreducible components are two smooth compact varieties X and X such that D X is smooth. Assume that (1) ω X ≃ O X , (2) H n − ( X , O X ) = 0 , H n − ( X i , O X i ) = 0 for i = 1 , , (3) N D X /X ⊗ N D X /X on D X is trivial.Then X is smoothable to a complex manifold M X . This smoothing theorem was proved firstly in [12] with an assumption ofK¨ahlerness condition and the condition was removed as in the present formin [11]. We will also need the following lemma (Lemma 3.14 in [11]) in theproof of Theorem 4.1.
Lemma 2.2.
With conditions in Theorem 2.1, assume further that X , X are projective and M X , D X are a projective Calabi–Yau n -fold and a Calabi–Yau ( n − -fold respectively. Then there exists a big line bundle on X . We will build a normal crossing variety and smooth it to a Calabi–Yaufourfold by applying Theorem 2.1 and show, using Lemma 2.2, that theCalabi–Yau fourfold is non-K¨ahler. To make the normal crossing variety X = X ∪ X , we need to construct the varieties X , X first. X and X will be built from Beauville’s Calabi–Yau threefold and two involutions onit. We briefly recall Beauville’s Calabi–Yau threefold. Let ζ = e π √− . By E ζ , we denote the elliptic curve whose period is ζ and by E ζ / h ζ i the quotientof the product manifold E ζ by the scalar multiplication by ζ . Let Q = 0 , Q = 1 − ζ , Q = − ζ E ζ . These are exactly the fixed points of the scalar multiplication by ζ on E ζ . For i, j, k = 0 , ,
2, let Q i,j,k = ( Q i , Q j , Q k ) ∈ E ζ and let Q i,j,k be its image in E ζ / h ζ i . Then Y = E ζ / h ζ i has singularities oftype (1 , ,
1) at Q i,j,k ’s and the blow-up Y → Y at these 27 singular pointsgives a Calabi–Yau threefold Y . This is a K¨ahler Calabi–Yau threefold withHodge numbers h , ( Y ) = 0 and h , ( Y ) = 36, which was originally foundby Beauville ([3]). NAM-HOON LEE Construction of a normal crossing variety X = X ∪ X Consider two 3 × ( Z [ ζ ]) A = − , A = − . Note that A i is the 3 × A i ) = − i = 1 , A i induces an involution σ i on E ζ . Note that σ ∗ i acts as multi-plication by − H , (cid:16) E ζ (cid:17) . Noting that the subgroup of SL ( Z [ ζ ]) thatis generated by A , A is infinite, we make the following remark which willplay a key role in showing the non-K¨ahlerness in the proof of Theorem 4.1. Remark 3.1.
The group of automorphisms of E ζ that is generated by σ , σ is infinite.There is a unique involution ρ i on Y such that the following diagramcommutes: E ζ σ i −−−−→ E ζ y y Y Y x x Y ρ i −−−−→ Y We note that ρ ∗ i also acts as multiplication by − H , ( Y ) and thefixed locus S i of ρ i is a smooth surface (a disjoint union of irreducible sur-faces). Now we move on to the construction of smooth varieties X , X . Weborrow a method of construction from [15] ( § K ψ : P → P be any involution fixing two distinct points and considera quotient X i = ( Y × P ) / ( ρ i , ψ ). Then the singular locus of X i is a productof smooth surfaces and ordinary double points, resulting from the fixed locusof ρ i . Let X i → X i be the blow-up along the singular locus of X i . It iselementary to check that X i is smooth. Choose a point p ∈ P such that p = ψ ( p ). Let D ′ i be the image of Y × { p } in X i and D i be the inverse imageof D ′ i in X i . Then D i is isomorphic to Y and it is an anticanonical divisorof X i whose normal bundle N D i /X i in X i is trivial. Since Y is projective, allthe varieties Y × P , X i and X i are projective.We summarize our notations, including ones to be defined: • ζ = e π √− . • E ζ = C / ( Z ⊕ Z ζ ) is the elliptic curve with period ζ . • Y = E ζ / h ζ i . • φ : e Y → E ζ is the blow-up at the 27 points of Q i,j,k ’s. N EXAMPLE OF NON-K ¨AHLER CALABI–YAU FOURFOLD 5 • η : e Y → Y is the map induced by E ζ → Y . • Y → Y is the blow-up of Y at its singular points. Y is a K¨ahlerCalabi–Yau threefold. • σ i : E ζ → E ζ is the involution induced by the matrix A i for i = 1 , • ρ i : Y → Y is the involution induced by σ i for i = 1 , • S i = Y ρ i is the fixed locus of ρ i for i = 1 , • ˇ S i = φ ( η − ( S i )), the fixed locus of σ i for i = 1 , • ψ : P → P is an involution fixing two distinct points q , q ∈ P . • X i = ( Y × P ) / ( ρ i , ψ ) for i = 1 , • X i → X i is the blow-up along the singular locus of X i for i = 1 , • f X i → Y × P is the blow-up of Y × P along the surface S i × { q , q } . • D ′ i : the image of Y × { p } in X i for a point p ∈ P with p = ψ ( p ) for i = 1 , • D i : the inverse image of D ′ i in X i for i = 1 , D i ≃ Y and D i ∈ |− K X i | . • X ∗ i = X i − D i . • X = X ∪ X is the normal crossing variety of X , X , made bygluing along their isomorphic smooth anticanonical sections D , D . • D X = X ∩ X . • M X is a smoothing of a normal crossing variety X = X ∪ X .We make a normal crossing variety X = X ∪ X by gluing transversallyalong D and D , (see § D X := X ∩ X is a copy of Y . Since D X = X ∩ X is an anticanonical divisor of both X and X , ω X ≃ O X . Notethat N D X /X ⊗ N D X /X is trivial. Let X ∗ i = X i − D i . For varieties X , X and X = X ∪ X constructed in this section, we gather some of theirproperties: Proposition 3.2. (1) X and X are projective. (2) ω X ≃ O X . (3) N D X /X ⊗ N D X /X on D X is trivial. (4) Both X i and X ∗ i are simply-connected for i = 1 , . (5) H k ( X i , O X i ) = 0 for i = 1 , , k = 1 , , , . (6) H k ( X , O X ) = 0 for k = 1 , , .Proof. The properties (1), (2), (3) are already shown.We show the property (4). One can obtain X i differently. Let q , q bethe fixed points of ψ . Let f X i be the blow-up of Y × P along the surface S i × { q , q } . Then the involution on Y × P induces an involution on f X i ,whose fixed locus is the exceptional divisor over S i × { q , q } . The quotientof f X i by the involution is isomorphic to X i . This may be summarized by NAM-HOON LEE the diagram, f X i −−−−→ X i y y Y × P −−−−→ X i Let c X i → X i be the universal covering. The fourfold f X i is simply-connected, therefore the quotient map f X i → X i lifts to a map f X i → c X i so there is a commutative diagram: c X i (cid:15) (cid:15) f X i / / ? ? ⑦⑦⑦⑦⑦⑦⑦⑦ X i . Since ρ i has a fixed point, f X i → c X i cannot be a map of degree one and so itis a map of degree two. Then c X i → X i is of degree one and c X i is necessarilyisomorphic to X i , thus X i is simply-connected.There is a natural projection: X i = ( Y × P ) / ( ρ i , ψ ) → Y /ρ i . Let ν bethe composition of X i → X i and ( Y × P ) / ( ρ i , ψ ) → Y /ρ i . Let x ∈ Y /ρ i be a point in the branch locus of the map Y → Y /ρ i . Then ν − ( x ) is aunion of three smooth rational curves, one of which (denoted by l ) crosses D i transversely at a single point and the other two are disjoint from D i ,resulting from the blow-up. Since X i and D i are simply connected, thefundamental group π ( X ∗ i ) of X ∗ i is generated by a loop around D i . We canassume that the loop is contained in l ∗ = l − D i . Since the loop can becontracted to a point in l ∗ , X ∗ i is simply-connected.We move on to the property (5). Sincedim H k ( X i , O X i ) ≤ dim H k ( f X i , O f X i ) = dim H k ( Y × P , O Y × P ) = 0for k = 1 ,
2, we have H k ( X i , O X i ) = 0for k = 1 , X i has an effective anticanonical divisor D i , which is a Calabi–Yau threefold. Hence, we have H ( X i , O X i ) ≃ H ( X i , Ω X i ) = 0 . Taking the cohomology of the structure sheaf sequence,0 → O X i ( K X i ) → O X i → O D i → , we obtain an exact sequence H ( X i , O X i ( K X i )) → H ( X i , O X i ) → H ( D i , O D i ) → H ( X i , O X i ( K X i )) → H ( X i , O X i ) = 0 . N EXAMPLE OF NON-K ¨AHLER CALABI–YAU FOURFOLD 7
Since, by Serre duality, H ( X i , O X i ( K X i )) ≃ H ( X i , O X i ) = 0 , dim H ( X i , O X i ( K X i )) = dim H ( X i , O X i ) = 1and dim H ( D i , O D i ) = 1 , we have dim H ( X i , O X i ) = 0.Finally, we show the property (6). From the exact sequence of sheaves0 → O X → O X ⊕ O X → O D X → , we obtain an exact sequence H k − ( D X , O D X ) → H k ( X , O X ) → H k ( X , O X ) ⊕ H k ( X , O X ) . Since H k − ( D X , O D X ) = H k ( X , O X ) = H k ( X , O X ) = 0for k = 2 ,
3, we have H ( X , O X ) = H ( X , O X ) = 0 . Moreover, the exact sequence0 → H ( X , O X ) → H ( X , O X ) ⊕ H ( X , O X ) → H ( D X , O D X ) → H ( X , O X ) → H ( X , O X ) ⊕ H ( X , O X ) = 0gives H ( X , O X ) = 0. (cid:3) The example
By Theorem 2.1 with the properties in Proposition 3.2, one can show thatthe normal crossing variety X , constructed in Section 3, is smoothable toa smooth fourfold M X with trivial canonical class. We can also check that H i ( M X , O X ) = 0 for i = 1 , , Theorem 4.1. M X is a non-K¨ahler Calabi–Yau fourfold.Proof. We only need to show that M X is simply-connected and non-K¨ahler.First, we show the simply-connectedness. We can obtain the topologicaltype of M X by pasting X ∗ and X ∗ . One can regard the normal bundle N D X /X i as a complex manifold containing D X . Then N ∗ D X /X i := N D X /X i − D X is a C ∗ -bundle over D X , where C ∗ := C − { } . The triviality property on N D X /X ⊗ N D X /X implies the map ϕ : N ∗ D X /X → N ∗ D X /X , locally defined by ( x ∈ C ∗ , y ∈ D X ) (1 /x, y ) , NAM-HOON LEE is globally well-defined and an isomorphism. Note that D X in X i has aneighborhood U i that is homeomorphic to N D X /X i . Let U ∗ i = U i − D X .Then the map ϕ induces a homeomorphism between U ∗ and U ∗ . One canconstruct a manifold M ′ by pasting together X ∗ and X ∗ along U ∗ and U ∗ with the homeomorphism. The manifold M ′ is homeomorphic to M X . Notethat X ∗ , X ∗ are simply-connected (the property (4) in Proposition 3.2).Hence, by Seifert–van Kampen theorem, M ′ is simply-connected.For the non-K¨ahlerness, suppose that M X is K¨ahler, then it is necessarilyprojective. Note that D X , X and X are all projective (the property (1)in Proposition 3.2). Hence, by Lemma 2.2, there exists a big line bundle L on X . Let h i be the big divisor class in Pic( X i ) corresponding to L| X i ,then h | D X is linearly equivalent to h | D X . Note that D X is a copy of Y .Let us denote the divisor class in Pic( Y ) of h | D X , h | D X by ˆ h . Chasingthe construction of X and X , one can check that ˆ h belongs to Pic( Y ) ρ ∗ ∩ Pic( Y ) ρ ∗ , where Pic( Y ) ρ ∗ i is the subgroup of Pic( Y ) that consists of theclasses invariant under ρ ∗ i .The linear system | D X | is base-point free and it gives a fibration X i → P and D X is one of its generic fibers. Hence ˆ h is a big divisor of Y (see, forexample, Corollary 2.2.11 in [17]). Let φ : e Y → E ζ be the blow-up at the27 points of Q i,j,k ’s and η : e Y → Y be the map induced by E ζ → Y suchthat the diagram commutes: e Y η −−−−→ Y φ y y E ζ −−−−→ Y It is not hard to check that ˇ h = φ ∗ ( η ∗ (ˆ h )) is a big divisor of E ζ andthe class ˇ h belongs to NS( E ζ ) σ ∗ ∩ NS( E ζ ) σ ∗ . Note that any big divi-sor of the abelian variety E ζ is ample. However, the group of automor-phisms of E ζ that is generated by σ , σ is infinite (Remark 3.1) and soNS( E ζ ) σ ∗ ∩ NS( E ζ ) σ ∗ does not contain an ample class. Therefore, we havea contradiction and M X should be non-K¨ahler. (cid:3) Topological invariants of M X can be calculated from the topological man-ifold M ′ in the proof of Theorem 4.1. For example, the topological Eulernumber χ ( M X ) of M X is χ ( M X ) = χ ( M ′ ) = χ ( X ∗ ) + χ ( X ∗ ) − χ ( U ) . Note χ ( X i ) = χ ( X ∗ i ) + χ ( D i ) = χ ( X ∗ i ) + χ ( Y )and χ ( U ) = χ ( S ) χ ( D ) = 0. Hence, χ ( M X ) = χ ( X ) + χ ( X ) − χ ( Y ) . N EXAMPLE OF NON-K ¨AHLER CALABI–YAU FOURFOLD 9
The topological Euler characteristic of e X i is χ ( e X i ) = 2 χ ( Y ) + 2 χ ( S i )On the other hand, 2 χ ( X i ) − χ ( S i ) = χ ( e X i )and so χ ( X i ) = 12 (cid:16) χ ( S i ) + χ ( e X i ) (cid:17) = 2 χ ( S i ) + χ ( Y ) + χ ( S i )= χ ( Y ) + 3 χ ( S i ) . Let ˇ S i = φ ( η − ( S i )). Note that ˇ S i is the fixed locus of σ i . Let Θ = { Q i,j,k | i, j, k = 0 , , } . One can easily check χ ( S i ) = 2 (cid:12)(cid:12) ˇ S i ∩ Θ (cid:12)(cid:12) = 18 , where (cid:12)(cid:12) ˇ S i ∩ Θ (cid:12)(cid:12) is the number of points in ˇ S i ∩ Θ.Therefore, the topological Euler number χ ( M X ) of M X is χ ( M X ) = χ ( X ) + χ ( X ) − χ ( Y )= χ ( Y ) + 3 χ ( S ) + χ ( Y ) + 3 χ ( S ) − χ ( Y )= 3 χ ( S ) + 3 χ ( S )= 108 . The pair of matrices A , A in Section 2, which can be used in the con-struction, is obviously not unique. Any pair of 3 × A , A thatsatisfies the following conditions,(1) A i ∈ SL ( Z [ ζ ]),(2) A i is the 3 × A i ) = − ρ i is not fixed-free and its fixed locus is a smooth surface and(5) The subgroup of SL ( Z [ ζ ]) containing both A , A is infinite,gives rise to a non-K¨ahler Calabi–Yau fourfolds through the construction ofSections 3 and 4, where the conditions guarantee, respectively,(1) A i induces an automorphism of E ζ which induces an automorphismof Y .(2) A i induces an involution of E ζ which induces an involution of Y ,(3) ρ ∗ i acts as multiplication by − H , ( Y ) so that X i has an anti-canonical section isomorphic to Y ,(4) both X i and X ∗ i are simply-connected, which leads to the simply-connectedness of M X and(5) the group of automorphisms of E ζ that is generated by σ , σ isinfinite, which eventually leads to the non-K¨ahlerness of M X . The author could not obtain other non-K¨ahler Calabi–Yau fourfolds of dif-ferent topological Euler numbers although he tried many pairs of such ma-trices. The author suspects that all the pairs of such matrices may give riseto non-K¨ahler Calabi–Yau fourfolds of the same topological Euler number(= 108).
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Department of Mathematics Education, Hongik University 42-1, Sangsu-Dong, Mapo-Gu, Seoul 121-791, KoreaSchool of Mathematics, Korea Institute for Advanced Study, Dongdaemun-gu, Seoul 130-722, South Korea
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