Examples of non-Kähler Calabi-Yau manifolds with arbitrarily large b_2
aa r X i v : . [ m a t h . AG ] F e b EXAMPLES OF NON-K ¨AHLER CALABI-YAU MANIFOLDSWITH ARBITRARILY LARGE b TARO SANO
Abstract.
We construct non-K¨ahler Calabi-Yau manifolds of dimen-sion ≥ Contents
1. Introduction 11.1. Comments on the construction 22. Preliminaries 33. Construction of examples 83.1. Properties of the smoothings 9Acknowledgement 13References 131.
Introduction
In this paper, we say that a compact complex manifold X is a Calabi-Yaumanifold if its canonical bundle ω X ≃ O X and H i ( X, O X ) = H ( X, Ω iX ) = 0for 0 < i < dim X . Moreover, we say that X is a Calabi-Yau n -fold if itsdimension is n .Projective Calabi-Yau manifolds form an important class of algebraic va-rieties. Non-K¨ahler Calabi-Yau manifolds are well investigated in complexdifferential geometry including those without the condition on the coho-mology groups (cf. [FLY12], [Tos15]). Reid’s fantasy [Rei87] suggests thepossible importance of non-K¨ahler Calabi-Yau manifolds in the classificationof projective ones.One of the important problems on projective Calabi-Yau manifolds iswhether their topological types are finite or not. Inspired by these back-ground, we construct infinitely many non-K¨ahler Calabi-Yau manifolds asfollows. Theorem 1.1.
For positive integers m and N ≥ , there exists a simplyconnected non-K¨ahler Calabi-Yau N -fold X = X ( m ) such that b ( X ) = ( m + 10 N = 4 m + 2 N ≥ , a ( X ) = N − , where b ( X ) is the 2nd Betti number and a ( X ) is the algebraic dimensionof X . The topological Euler number e ( X ) can also be computed (Proposition3.6). Since the 2nd Betti number can be arbitrarily large, the examples giveinfinitely many topological types of non-K¨ahler Calabi-Yau manifolds ofdimension N ≥
4. As far as we know, these are the first examples of Calabi-Yau manifolds with infinitely many topological types in a fixed dimension N ≥ b and b = 0 are constructed byClemens and Friedman (cf.[Fri91], [LT96]). [HS19] constructed non-K¨ahlerCalabi-Yau 3-folds with arbitrarily large b by smoothing normal crossing va-rieties. The idea of constructing projective Calabi-Yau manifolds by smooth-ing SNC varieties goes back to papers [KN94] and [Lee10]. In this paper,we construct examples by the same method as [HS19].1.1. Comments on the construction.
The idea of the construction in[HS19] was to consider distinct SNC varieties by gluing smooth varietiesalong their anticanonical divisors through an automorphism of infinite order.The point is that we use smooth varieties which are blow-ups of anothervarieties and the number of blow-up centers increases when we change thegluing isomorphisms.In this paper, we consider an SNC variety of the form X = X ∪ X ,where X = P × T and X is its blow-up and T ⊂ P × P n is a hypersurfaceof bi-degree (1 , n + 1). The essential ingredient in this paper is the Schoentype Calabi-Yau manifold with infinite automorphisms (cf. [MOF]). When n = 2, the intersection of two irreducible components is the Schoen Calabi-Yau 3-fold which is a fiber product of two rational elliptic surfaces over P .We actually use isomorphisms of such Calabi-Yau ( N − P × P of bidegree (3 , Remark . In [HS19], examples of SNC Calabi-Yau 3-folds are constructedby using automorphisms of (2 , , X (2 , , ⊂ P × P × P induced by the covering involutions of double covers X → P × P . Notethat, for X = X (2 ,..., ⊂ ( P ) n , the covering involutions of projections X → ( P ) n − are birational maps with indeterminacies when n ≥
4, thus we neednew gluing isomorphisms to construct examples in higher dimensions. It is
ON-K ¨AHLER CALABI-YAU MANIFOLDS 3 also not clear that the construction of Clemens–Friedman can be generalizedto higher dimensions.After finishing the manuscript, the author received an e-mail from Nam-Hoon Lee and he constructed a non-K¨ahler Calabi-Yau 4-fold by smoothingSNC varieties [Lee21]. 2.
Preliminaries
We can construct an SNC variety by gluing two smooth proper varietiesalong their smooth divisors as follows.
Proposition 2.1.
Let X and X be a smooth proper varieties and D i ⊂ X i be a smooth divisor for i = 1 , with an isomorphism φ : D → D . Thenthere exists an SNC variety X with a closed immersion ι i : X i ֒ → X for i = 1 , in the Cartesian diagram D i ◦ φ (cid:15) (cid:15) i / / X ι (cid:15) (cid:15) X ι / / X , and we write X =: X ∪ ψ X .Moreover, if D is connected and D i ∈ |− K X i | , then ω X ≃ O X .Proof. See [HS19, Proposition 2.1, Corollary 2.2] and references therein. (cid:3)
Definition 2.2.
Let X be an SNC variety and X = S Ni =1 X i be the decom-position into its irreducible components. Let D := Sing X = S i = j ( X i ∩ X j )be the double locus and let I X i , I D ⊂ O X be the ideal sheaves of X i and D on X . Let O D ( X ) := ( N O i =1 I X i /I X i I D ) ∗ ∈ Pic D be the infinitesimal normal bundle as in [Fri83, Definition 1.9].We say that X is d -semistable if O D ( X ) ≃ O D . If X = X ∪ X forsmooth varieties X and X , then O D ( X ) ≃ N D/X ⊗ N D/X .The following result on smoothings of a SNC variety is essential for theconstruction. Theorem 2.3.
Let X be an n -dimensional proper SNC variety whose dual-izing sheaf ω X is trivial. Assume that X is d-semistable. Then there existsa semistable smoothing φ : X → ∆ of X over a unit disk, that is, a propersurjective morphism such that X is smooth, X ≃ φ − (0) and X t := φ − ( t ) is smooth for t = 0 .Proof. This follows from [KN94], [CLM19], [FFR19], [FP20]. (cid:3)
We shall use the following description of general rational elliptic surfacesas hypersurfaces in P × P . TARO SANO
Proposition 2.4.
Let S ⊂ P × P be a general hypersurface of bidegree (3 , , that is a member of | p ∗ O P (3) ⊗ p ∗ O P (1) | , where p : S → P and p : S → P are the projections. (i) S is a rational elliptic surface without ( − -curve. Moreover, p induces an anticanonical elliptic fibration and p is the blow-up at 9points which appear as intersection of two cubic curves. (ii) Let C ⊂ S be an irreducible curve which is not a ( − -curve. Then C ≥ and | C | is base point free.Proof. (i) Note that S is defined as( sF + tG = 0) ⊂ P × P , where [ s : t ] ∈ P is the coordinates and F, G ∈ C [ x , x , x ] are generalhomogeneous polynomials of degree 3. By this description, we see that S issmooth and p is the blow-up at 9 points ( F = G = 0) ⊂ P . We see that − K S = p ∗ O P (1) and it induces an elliptic fibration, thus S is a rationalelliptic surface with a section.Since − K S is nef, an irreducible curve C such that C < − − C ⊂ S is a ( − C is p -verticaland contained in a singular fiber. It is well-known that a generic pencil ofcubic curves contains only irreducible curves (cf. [Tot08, Lemma 3.1]), thus S can not contain a ( − C ≥ K S · C ≤ C is neither a ( − − p ( C ) is a point, then C is a fiber of p since all fibers areirreducible and reduced. Thus | C | = |− K S | is a free linear system.If p ( C ) = P , then we see that C > − K S · C > C is not a( − h ( S, C ) ≥ χ ( S, C ) = χ ( S, O S ) + C ( C − K S )2 > C > . Thus | C | has no fixed part and C is nef and big. We also check that | C | isfree. Indeed, we have an exact sequence H ( S, O S ( C )) → H ( f, O f ( C )) → H ( S, O S ( C − f ))for a smooth fiber f ∈ |− K S | , H ( S, O S ( C − f )) = H ( S, K S + C ) = 0 bythe vanishing theorem and f · C ≥ C is not a section. (cid:3) We also need the following isomorphism of a rational elliptic surface S induced by a quadratic transformation of P . (ii) and (iii) are technical, butessential in the construction of our Calabi-Yau manifolds. Proposition 2.5.
Let S ⊂ P × P be a general (3 , -hypersurface. Let p , . . . , p ∈ P be the points on which the birational morphism µ = p : S → P induced by the 1st projection is not an isomorphism. Let E i := µ − ( p i ) for i = 1 , . . . , be ( − -curves and H := µ ∗ O P (1) . Then we have thefollowing. ON-K ¨AHLER CALABI-YAU MANIFOLDS 5 (i)
For ≤ i < j < k ≤ , there exist a hypersurface S ijk ⊂ P × P andan isomorphism φ ijk : S → S ijk over P such that φ ∗ ijk ( H ijk ) = 2 H − E i − E j − E k , where H ijk is the pull-back of O P (1) to S ijk by the 1st projection. (ii) For a positive integer m , there exist a hypersurface S m ⊂ P × P and an isomorphism φ m : S → S m over P such that (1) φ ∗ m H S m = (27 m + 1) H − (9 m − m ) F − m F − (9 m + 3 m ) F , where H S m is the pull-back of O P (1) to S m and F i := E i − + E i − + E i for i = 1 , , . (iii) In (ii), the divisor L m := H + φ ∗ m H S m + mK S is ample and free on S .Remark . In (iii), the essential part used later is that the divisor is ef-fective and can be written as a sum of smooth curves. However, it is niceto show the freeness since the linear system contains a smooth irreduciblemember. By this property, we can determine the 2nd Betti number of ourCalabi-Yau manifolds obtained as smoothings.
Proof. (i) We see that p , . . . , p ∈ P is in “Cremona general position” inthe sense of [Tot08, pp.1178], thus can consider the quadratic transformation q ijk : P P at p i , p j , p k . This q ijk induces an isomorphism φ ijk : S → S ijk onto a hypersurface S ijk ⊂ P × P . By this construction, we see that φ ijk is an isomorphism over P .Let E ′ i , E ′ j , E ′ k ∈ Pic S ijk be the images of H − E j − E k , H − E i − E k , H − E i − E j ∈ Pic S by φ ijk . For l = i, j, k , let E ′ l := φ ijk ( E l ) ⊂ S ijk . Weidentify Pic S ≃ Z and Pic S ijk ≃ Z by the basis ( H, E , . . . , E ) and( H ijk , E ′ , . . . , E ′ ). Here Z is the lattice with a bilinear form( a, b , . . . , b ) · ( a ′ , b ′ , . . . , b ′ ) := aa ′ − X l =1 b l b ′ l . Let h, e i , e j , e k ∈ Z be the images of H, E i , E j , E k via the identificationPic S → Z . Then we see that φ ∗ ijk : Pic S ijk → Pic S is the reflectionorthogonal to H − E i − E j − E k , that is, it induces φ ∗ ijk : Z → Z determinedby φ ∗ ijk ( x ) = x + ( x · α ijk ) α ijk , where α ijk := h − e i − e j − e k ∈ Z . Hence we obtain the required equalityby substituting x = h .(ii) Let ψ S := φ ◦ φ ◦ φ : S → S ′ be a composite of three isomorphismsas in (i), that is, φ is the quadratic transformation at p , p , p , and φ and φ are quadratic transformation at the images of { p , p , p } and { p , p , p } respectively. Then let ψ := ψ S ′ ◦ ψ S : S → S ′′ , TARO SANO where ψ S ′ is the same operation on S ′ and let S := S ′′ (and S := S ).We can perform this operation for any positive integer i and construct anisomorphism ψ i : S i − → S i as a composite of six quadratic transformations.Now let φ m := ψ m ◦ · · · ◦ ψ : S → S m .On each surface T = S i , we have an isomorphism Pic T → Z determinedby the basis ( H T , E T, , . . . , E T, ), where H T is the pull-back of O P (1) and E T,i ’s are the exceptional divisors. Note that the labelling for the exceptionaldivisors are determined as (i). We also use the same symbol for isomorphismsof the Picard group and Z , that is, for ψ : S → S ′ , we write ψ ∗ : Pic S ′ → Pic S and ψ ∗ : Z → Z , for example. We are reduced to show the followingto obtain the equality (1). Claim . Let h, e , . . . , e ∈ Z be the elements corresponding to ( H, E , . . . , E )or ( H S m , E S m , , . . . , E S m , ) as before. Let f i := e i − + e i − + e i for i = 1 , ,
3. Then we have(2) φ ∗ m ( h ) = (27 m + 1) h − (9 m − m ) f − m f − (9 m + 3 m ) f . Proof of Claim.
We check the required equality by induction on m . Recallthat φ ∗ ijk : Z → Z is the reflection for α ijk = h − e i − e j − e k ∈ Z . Thenwe compute ψ ∗ S ( h ) = φ ∗ ( φ ∗ ( φ ∗ ( h ))) = φ ∗ ( φ ∗ (2 h − f ))= φ ∗ (4 h − f − f ) = 8 h − f − f − f . Then we compute similarly ψ ∗ ( h ) = φ ∗ ( φ ∗ ( φ ∗ (8 h − f − f − f ))) = 28 h − f − f − f , thus obtain the equality for φ = ψ . Suppose that we have the equality (2)for φ m . By a similar computation, we obtain φ ∗ m +1 ( h ) = ψ ∗ m +1 φ ∗ m ( h )= ψ ∗ m +1 ((27 m + 1) h − (9 m − m ) f − m f − (9 m + 3 m ) f )= (27( m +1) +1) h − (9( m +1) − m +1)) f − m +1) f − (9( m +1) +3( m +1)) f . Thus we obtain the claim by induction. (cid:3)
This finishes the proof of (ii).(iii) We have L m · ( − K S ) = 6 > L m = ( H + φ ∗ m H S m ) + 2 m ( K S · ( H + φ ∗ m H S m ))= (2 + 2(27 m + 1)) − m = 54 m − m + 4 > . By these and the Riemann-Roch formula, we see that L m is effective.Since S is general, it is enough to show L m · C > − C . Wewrite C = αH − P i =1 β i E i for some non-negative integers α, β , . . . , β such ON-K ¨AHLER CALABI-YAU MANIFOLDS 7 that α − P i =1 β i = − α − P i =1 β i = 1. Let γ i := β i − + β i − + β i for i = 1 , ,
3. We may assume that α ≤ m since, if α > m , then we have L m · C ≥ ( H + mK S ) · C = α − m > . We may also assume that β i ≥ i since we have L m · E i = φ ∗ m ( H S m ) · E i + mK S · E i ≥ (9 m − m ) − m > . Then we compute L m · C = α (27 m + 2) − (9 m − m ) γ − m γ − (9 m + 3 m ) γ − m = α (27 m + 2) − (9 m − m )(3 α − − m ( γ + 2 γ ) − m = (9 m + 2) α + (9 m − m ) − m ( γ + 2 γ ) − m ≥ (9 m + 2) α + (9 m − m ) − m (2(3 α − − m = − mα + 2 α + 9 m + 2 m = 9 m ( m − α ) + 2 m + 2 α ≥ m + 2 α > , where we used γ + 2 γ ≤ α −
1) for the 1st inequality and m ≥ α forthe 2nd inequality. Thus we see that L m is ample by the Nakai–Moishezoncriterion. Since L m · ( − K S ) = 6, we see that L m | F is free for a generalsmooth element F ∈ |− K S | . By this and the exact sequence0 → O S ( K S + L m ) → O S ( L m ) → O F ( L m ) → , we check that | L m | is free as in the proof of Proposition 2.4(ii). (cid:3) We have the following description of Calabi-Yau manifolds of “Schoentype” arising from general rational elliptic surfaces. The author learned (i)in the following proposition in [MOF] when n = 2. Proposition 2.8.
Let S ⊂ P × P be a general (3 , -hypersurface and T ⊂ P × P n a general (1 , n + 1) -hypersurface for some n ≥ . Let X := P × T and X := S × P n be divisors in P × P × P n and X := X ∩ X . Let p S : X → S and p T : X → T be the surjective morphisms induced by theprojections. (i) X is a Calabi-Yau ( n + 1) -fold and there is a natural isomorphism ϕ : S × P T → X , where the fiber product is defined by the projections φ S : S → P and φ T : T → P . (ii) (cf. [GM93, Corollary 3.2] ) We have Pic X ≃ (Pic S ⊕ Pic T ) / Z ( − K S , K T ) . Thus
Pic X ≃ Z when n = 2 and Pic X ≃ Z when n ≥ .Proof. (i) We see that X is Calabi-Yau by the adjunction formula. Wehave the natural closed immersion ϕ : S × P T ֒ → ( P × P ) × P ( P × P n ) ≃ P × P × P n TARO SANO and see that its image is X . Since S × P T is reduced, we see that ϕ inducesthe required isomorphism.(ii) By the same argument as the 1st paragraph of the proof of [Nam91,Proposition 1.1], we see that the naturally induced homomorphism p ∗ S ⊕ p ∗ T : Pic S ⊕ Pic T → Pic X is surjective. By the same argument as the proof of [GM93, Proposition 3.1],we can write ( L , L ) ∈ Ker( p ∗ S ⊕ p ∗ T ) as( L , L ) = ( A , A ) + m ( − K S , K T )for some numerically trivial A ∈ Pic S , A ∈ Pic T and m ∈ Z . Since H ( S, O S ) = H ( T, O T ) = 0, we see that A and A are linearly trivial.Thus we see that Ker( p ∗ S ⊕ p ∗ T ) = Z ( K S , − K T ) and obtain the requiredisomorphism.Since the projection morphism T → P n is the blow-up along the intersec-tion ( F = G = 0) ⊂ P n of general divisors ( F = 0) , ( G = 0) of degree n + 1,we see that Pic T ≃ Z if n = 2 and Pic T ≃ Z if n ≥
3. Thus we obtainthe latter statement. (cid:3)
Remark . We can also consider the case n = 1. In this case, T ⊂ P × P is also P and X is a double cover of S . We compute that Pic X ≃ Z .3. Construction of examples
We first explain the construction of examples X ( m ) by smoothing SNCvarieties X ( m ) in the following. Example 3.1.
Let m, n be positive integers. Let S ⊂ P × P be a general(3 , S m ⊂ P × P be the hypersurface in Proposition2.5(ii) with the isomorphism φ m : S → S m over P .Let T ⊂ P × P n be a general (1 , n + 1)-hypersurface and Y := P × T ⊂ P × P × P n . Let D := Y ∩ ( S × P n ) ⊂ Y . Then there is a natural isomorphism ϕ : D → S × P T as in Proposition 2.8(i). Note that D ∈ |− K Y | and the normalbundle N D /Y ≃ p ∗ S O S (3 h + f ), where p S : D → S is the projection, f ∈|− K S | and h := µ ∗ S O P (1) for the birational morphism µ S : S → P inducedby the projection.Let Y := P × T and D := Y ∩ ( S m × P n ) ⊂ Y . Then there is anatural isomorphism ϕ : D → S m × P T as above. Let Φ m : D → D bethe isomorphism which fits in the following diagram: D ϕ (cid:15) (cid:15) Φ m / / D ϕ (cid:15) (cid:15) S × P T φ m × id / / S m × P T. ON-K ¨AHLER CALABI-YAU MANIFOLDS 9
For i = 1 , . . . , m , let F i := p − S ( f i ) ⊂ D for a smooth general fiber f i ∈|− K S | . Let Γ m := p − S ( C m ) for a smooth member C m ∈ | H + φ ∗ m (3 H S m ) + ( m − K S | = | L m + 2( H + φ ∗ m H S m ) + ( − K S ) | of an ample and free linear system guaranteed by Proposition 2.5(iii). Nowlet ν : ˜ Y → P × T = Y be the blow-up along F , . . . , F m , and ν : X → ˜ Y be the blow-up along the strict transform ˜Γ m ⊂ ˜ Y of Γ m ⊂ D . Thus wehave a composition µ := ν ◦ ν : X → Y . Let X := Y = P × T .Let ˜ D ⊂ X be the strict transform of D and µ D : ˜ D → D be theinduced isomorphism. Then we can glue X and X along the compositionisomorphism Ψ m : ˜ D µ D −−→ D m −−→ D and construct an SNC variety X ( m ) := X := X ∪ Ψ m X by Proposition 2.1. We check that X is d-semistable since we have N D /Y ⊗ Φ ∗ m N D /Y ≃ O D ( p ∗ S (3( H + φ ∗ m H S m )+2 f ) ≃ O D ( F + · · · + F m +Γ m )and the blow-up centers of µ are chosen so that this becomes trivial. Wealso see that ω X ≃ O X since ˜ D ∈ |− K X | and D ∈ |− K X | . Thus wecan apply Theorem 2.3 and construct a semistable smoothing X ( m ) → ∆ .Let X ( m ) be its general smooth fiber. Thus we obtain a compact complexmanifold X ( m ). We also write X := X ( m ) for short in the following.3.1. Properties of the smoothings.
We have the following basic proper-ties of X ( m ) in Example 3.1. Proposition 3.2.
The above X = X ( m ) satisfies the following. (i) The Hodge to de Rham spectral sequence H q ( X, Ω pX ) ⇒ H p + q ( X, C ) degenerates at E . (ii) We have H i ( X, O X ) = 0 and H ( X, Ω iX ) = 0 for < i < dim X .We also have ω X ≃ O X , thus X is a Calabi-Yau manifold. (iii) The 2nd betti number is b ( X ) = m + ρ T , where ρ T := rk Pic T .Proof. (i) This is [PS08, Corollary 11.24].(ii) Let X := X ∩ X . We compute H i ( X , O X ) = 0 for 0 < i < dim X by the exact sequence · · · → H i − ( X , O X ) → H i ( X , O X ) → M j =1 H i ( X j , O X j ) → · · · . By this and the upper semi-continuity theorem, we obtain H i ( X, O X ) = 0for 0 < i < dim X . Since we have an exact sequence H ( X, O X ) → H ( X, O ∗ X ) → H ( X, Z ) → H ( X, O X )from the exponential exact sequence, we see that H ( X, Z ) = 0. By thisand (i), we obtain H ( X, Ω X ) = 0. Claim . We have H ( X, Ω iX ) = 0 for 2 ≤ i ≤ dim X − Proof of Claim.
For the semistable smoothing
X → ∆ and i ≥
0, we havethe locally free sheaf Λ iX := Ω i X / ∆ (log X ) | X which is defined in [Fri83, pp.92]. It is enough to show H (Λ iX ) = 0 for2 ≤ i ≤ dim X −
1. By applying [Fri83, Proposition 3.5] to Λ iX for X = X ∪ X , we have an exact sequence0 → V → Λ iX → V /V → , where V and V /V are described as V ≃ Ker(Ω iX ⊕ Ω iX ( ι ∗ , − ι ∗ ) −−−−−→ Ω iX ) , V /V ≃ Ω i − X . By this and H ( X i , Ω iX i ) = 0 = H ( X , Ω i − X ) for i = 2 , . . . , dim X −
1, weobtain H ( V ) = 0 and H ( V /V ) = 0. Thus we obtain H (Λ iX ) = 0 for2 ≤ i ≤ dim X − (cid:3) We also see that ω X ≃ O X as in the proof of [HS19, Theorem 3.4]. Hencewe see that X is a Calabi-Yau manifold in our sense.(iii) Note that Pic X ≃ H ( X , Z ) and Pic X ≃ H ( X, Z ) by H i ( X , O X ) =0 and H i ( X, O X ) = 0 for i = 1 ,
2. We compute b ( X ) = m + ρ T + 1 asfollows. We have an exact sequence0 → Pic X → Pic X ⊕ Pic X R −→ Pic X , where R = ( ι ∗ , − ι ∗ ) for the closed immersion ι i : X ֒ → X i for i = 1 ,
2. Byusing the isomorphismPic X ≃ (Pic S ⊕ Pic T ) / Z ( K S , − K T )as in Proposition 2.8, we see that the image Im R ⊂ Pic X is generated bythe image of Pic T and p ∗ S ( H ) , p ∗ S ( φ ∗ m ( H S m )). Thus we see thatIm R ≃ Z ρ T +2 . Since Pic X ≃ Z ρ T + m +1 = Z m + ρ T +2 and Pic X ≃ Z ρ T +1 , we see thatPic X ≃ Z m + ρ T +1 by the above exact sequence, thus obtain b ( X ) = m + ρ T + 1.Then, by the Clemens map γ : X → X (cf. [Cle77], [HS19, Theorem2.9]), we have the exact sequence Z ≃ H ( X , R γ ∗ Z ) → H ( X , Z ) → H ( X, Z ) → H ( R γ ∗ Z ) = 0 . Thus we see that H ( X, Z ) ≃ Z m + ρ T as in [HS19, Claim 3.6(ii)]. (cid:3) The following lemma is useful to see the non-projectivity of X and com-pute the algebraic dimension of X ON-K ¨AHLER CALABI-YAU MANIFOLDS 11
Lemma 3.4.
Let X = X ( m ) be the SNC Calabi-Yau variety as in Example3.1. Let N := dim X . Let L ∈ Pic X be a line bundle such that L i := L | X i is effective for i = 1 , . We may write L = µ ∗ ( O P ( a ) ⊠ O T ( H )) ⊗ O X ( m X j =1 b j E j + cF ) , L = O P ( a ′ ) ⊠ O T ( H ) , where E j := µ − ( F j ) ⊂ X for j = 1 , . . . , m and F := µ − (Γ m ) are theexceptional divisors of µ : X → P × T and H , H ∈ Pic T .Then we have a = a ′ = 0 and κ ( L ) ≤ N − .Proof. Note that L | ˜ D ≃ Ψ ∗ m L | D , where Ψ m : ˜ D → D is the isomor-phism used to construct X . We have a natural surjection π S : Pic ˜ D ≃ −→ (Pic S ⊕ Pic T ) / Z ( K S , − K T ) pr −→ Pic S/ Z ( K S )by Proposition 2.8(ii), where pr is the projection. We see that π S ( L | ˜ D ) = [ aH + c (3( H + H ′ ))] = ( a + 3 c )[ H ] + 3 c [ H ′ ] ∈ Pic S/ Z ( K S ) , where [ · ] means the image of Pic S → Pic S/ Z K S and H ′ := φ ∗ m ( H S m ). Wealso see that π S (Ψ ∗ m L | D ) = a ′ [ H ′ ] . By comparing the above two terms, we see that a ′ = 3 c and a + 3 c = 0since [ H ] and [ H ′ ] are linearly independent. Since a, a ′ ≥
0, we obtain a = a ′ = 0. This implies that κ ( L i ) ≤ dim T for i = 1 ,
2, thus κ ( L ) ≤ dim T = N − (cid:3) Proposition 3.5.
For m > , X = X ( m ) is not projective. Moreover, wehave a ( X ) = n = dim X − for a very general t ∈ ∆ , where a ( X ) is the algebraic dimension.Proof. Let pr T : P × T → T be the projection. We see that, for a veryample H T ∈ Pic T , the line bundles H := µ ∗ (pr ∗ T O T ( H T )) ∈ Pic X and H := pr ∗ T O T ( H T ) induce a line bundle H ∈ Pic X such that H | X i ≃ H i .Since we have an isomorphism Pic X ≃ Pic X as in [HS19], H induces aline bundle H t and this induces a fiber space X → T . Its general fiber is aK3 surface since the general fiber of X → T is a SNC surface which is aunion of P and its blow-up at 18 points. Thus the fiber space X → T is aK3 fibration.We see that there is no line bundle L t on X such that κ ( L t ) ≥ dim T + 1by the same argument as [HS19, Proposition 3.19(iii)]. Indeed, if such a linebundle exists, then there exists L ∈ Pic X such that L | X i is effective for i = 1 , κ ( L ) ≥ dim T + 1. This contradicts Lemma 3.4. (cid:3) We can also compute the following topological invariants of X . Proposition 3.6. (i) X is simply connected. (ii) The topological Euler number of X is (3) e ( X ) = ( γ m −
12) ( − n ) n +1 + n + 2 nn + 1 + 24( n + 1)( − n ) n , where γ m := − m − m + 5) .Proof. (i) We show this by following the proof of [HS19, Proposition 3.10].As in [HS19, Proposition 3.10], we see that(4) π ( X ) ≃ π ( X ′ ) ∗ π ( ˜ X ) π ( X ′ ) , where X ′ i := X i \ X for i = 1 , X := γ − ( X ) for the Clemens map γ : X → X . We see that π ( X ′ ) = { } by the Gysin exact sequence H ( X , Z ) → H ( X , Z ) → H ( X ′ , Z ) → H ( X , Z ) . Indeed, for a section C ⊂ T of the fiber space T → P , the class [ { p } × C ] ∈ H ( X , Z ) is sent to a generator of H ( X ). For example, a fiber of T → P n over the exceptional locus can be taken as C . We also see that π ( X ′ ) = { } as in [HS19] (In fact, the argument is easier since π ( X ′ ) = { } ). By theseand (4), we see that π ( X ) = { } .(ii) As in the proof of [HS19, Claim 3.7], we see that e ( X ) = e ( X ) + e ( X ) − e ( X )by the Mayer-Vietoris exact sequence and the Clemens map. Note that T → P has singular fibers with only one nodes over points q , . . . , q d n ∈ P ,where d n := ( n + 1) n n is the degree of the discriminant hypersurface in |O P n ( n + 1) | (cf. [L¨09,Lemma 2.1]). Note also that a smooth hypersurface H n +1 ⊂ P n has theEuler number e ( H n +1 ) = ( n + 1) + ( − n ) n +1 − n + 1 = ( − n ) n +1 + n + 2 nn + 1 =: σ n . Thus we compute e ( T ) = e ( P ) e ( H n +1 ) + d n ( − n = 2 σ n + δ n , where we put δ n := ( − n d n . We compute that e ( X ) = e ( P × T ) = e ( P ) e ( T ) = 3(2 σ n + δ n ) = 6 σ n + 3 δ n . To compute e ( X ), note that X → P × T is the blow-up along F , . . . , F m and (the strict transform of) Γ m . Note that e ( F i ) = 0 since F i is a productof an elliptic curve and a Calabi-Yau hypersurface. Thus we see that e ( X ) = e ( P × T ) + e (Γ m ) = 3(2 σ n + δ n ) + e (Γ m ) . Note also that Γ m → C m is a Calabi-Yau fibration and the discriminantlocus consists of 18 d n points since C m · ( − d n K S ) = d n ( − K S · (3 H + 3 φ ∗ H )) = 18 d n . ON-K ¨AHLER CALABI-YAU MANIFOLDS 13
Indeed, Γ m → C m is singular at the intersection of C m and the discriminantlocus D S ⊂ S of X → S , and D S consists of smooth fibers over the d n points which is the discriminant locus of T → P . We also check e ( C m ) = − ( K S + C m ) · C m = − m − m + 5) =: γ m . Thus we compute e (Γ m ) = e ( C m ) · e ( H n +1 ) + 18 d n ( − n = γ m σ n + 18 δ n and e ( X ) = (6 σ n + 3 δ n ) + ( γ m σ n + 18 δ n ) = ( γ m + 6) σ n + 21 δ n . To compute e ( X ), note that X → P has a fiber with non-zero Eulernumber only at the discriminant locus p , . . . , p of S → P . Then wecompute e ( X ) = 12(1 · e ( H n +1 )) = 12 σ n . By these, we obtain e ( X ) = (( γ m + 6) σ n + 21 δ n ) + (6 σ n + 3 δ n ) − σ n = ( γ m − σ n + 24 δ n . (cid:3) Remark . The above implies that the Euler number can be arbitrarilynegative when n is odd and can be arbitrarily positive when n is even (except n = 2). If n = 2, then we compute e ( X ) = 288. We check that the formula(3) also holds when n = 1. Remark . It would be interesting whether our examples X ( m ) satisfiesthe ∂ ¯ ∂ -lemma and the hard Lefschetz property (cf.[Fri19], [QW18]). Wehope to seek these elsewhere. Acknowledgement
The author is grateful to Kenji Hashimoto for useful discussions. He is alsograteful to Nam-Hoon Lee for useful information. This work was partiallysupported by JSPS KAKENHI Grant Numbers 17H06127, JP19K14509.
References [Cle77] C. H. Clemens,
Degeneration of K¨ahler manifolds , Duke Math. J. (1977),no. 2, 215–290. MR 444662[CLM19] Kwokwai Chan, Naichung Conan Leung, and Ziming Nikolas Ma, Geom-etry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties ,https://arxiv.org/pdf/1902.11174.pdf (2019).[FFR19] Simon Felten, Matej Filip, and Helge Ruddat,
Smoothing toroidal crossingspaces , https://arxiv.org/pdf/1908.11235.pdf (2019).[FLY12] Jixiang Fu, Jun Li, and Shing-Tung Yau,
Balanced metrics on non-K¨ahlerCalabi-Yau threefolds , J. Differential Geom. (2012), no. 1, 81–129.MR 2891478[FP20] Simon Felten and Andrea Petracci, The logarithmic Bogomolov-Tian-Todorovtheorem , https://arxiv.org/pdf/2010.13656.pdf (2020).[Fri83] Robert Friedman,
Global smoothings of varieties with normal crossings , Ann. ofMath. (2) (1983), no. 1, 75–114. MR 707162 (85g:32029) [Fri91] ,
On threefolds with trivial canonical bundle , Complex geometry and Lietheory (Sundance, UT, 1989), Proc. Sympos. Pure Math., vol. 53, Amer. Math.Soc., Providence, RI, 1991, pp. 103–134. MR 1141199[Fri19] ,
The ∂∂ -lemma for general Clemens manifolds , Pure Appl. Math. Q. (2019), no. 4, 1001–1028. MR 4085665[GM93] Antonella Grassi and David R. Morrison, Automorphisms and the K¨ahler coneof certain Calabi-Yau manifolds , Duke Math. J. (1993), no. 3, 831–838.MR 1240604[HS19] Kenji Hashimoto and Taro Sano, Examples of non-K¨ahler Calabi-Yau 3-foldswith arbitrarily large b , https://arxiv.org/abs/1902.01027 (2019).[KN94] Yujiro Kawamata and Yoshinori Namikawa, Logarithmic deformations of nor-mal crossing varieties and smoothing of degenerate Calabi-Yau varieties , Invent.Math. (1994), no. 3, 395–409. MR 1296351[L¨09] Michael L¨onne,
Fundamental groups of projective discriminant complements ,Duke Math. J. (2009), no. 2, 357–405. MR 2569617[Lee10] Nam-Hoon Lee,
Calabi-Yau construction by smoothing normal crossing vari-eties , Internat. J. Math. (2010), no. 6, 701–725. MR 2658406[Lee21] Nam-Hoon Lee, An example of non-K¨ahler Calabi-Yau fourfold ,https://arxiv.org/pdf/2102.12656.pdf (2021).[LT96] P. Lu and G. Tian,
The complex structures on connected sums of S × S ,Manifolds and geometry (Pisa, 1993), Sympos. Math., XXXVI, Cambridge Univ.Press, Cambridge, 1996, pp. 284–293. MR 1410077[MOF] https://mathoverflow.net/questions/352878/calabi-yau-threefold-with-an-automorphism-of-infinite-order.[Nam91] Yoshinori Namikawa, On the birational structure of certain Calabi-Yau three-folds , J. Math. Kyoto Univ. (1991), no. 1, 151–164. MR 1093334[PS08] Chris A. M. Peters and Joseph H. M. Steenbrink, Mixed Hodge structures , Ergeb-nisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Sur-veys in Mathematics [Results in Mathematics and Related Areas. 3rd Series.A Series of Modern Surveys in Mathematics], vol. 52, Springer-Verlag, Berlin,2008. MR 2393625 (2009c:14018)[QW18] Lizhen Qin and Botong Wang,
A family of compact complex and symplecticCalabi-Yau manifolds that are non-K¨ahler , Geom. Topol. (2018), no. 4, 2115–2144. MR 3784517[Rei87] Miles Reid, The moduli space of -folds with K = 0 may nevertheless be irre-ducible , Math. Ann. (1987), no. 1-4, 329–334. MR 909231[Tos15] Valentino Tosatti, Non-K¨ahler Calabi-Yau manifolds , Analysis, complex geom-etry, and mathematical physics: in honor of Duong H. Phong, Contemp. Math.,vol. 644, Amer. Math. Soc., Providence, RI, 2015, pp. 261–277. MR 3372471[Tot08] Burt Totaro,
Hilbert’s 14th problem over finite fields and a conjecture on thecone of curves , Compos. Math. (2008), no. 5, 1176–1198. MR 2457523
Department of Mathematics, Faculty of Science, Kobe University, Kobe,657-8501, Japan
Email address ::