aa r X i v : . [ m a t h . AG ] S e p NOTE ON QUASI-POLARIZED CANONICAL CALABI-YAU THREEFOLDS
JIE LIUA
BSTRACT . Let ( X , L ) be a quasi-polarized canonical Calabi-Yau threefold. In this note,we show that | mL | is basepoint free for m ≥
4. Moreover, if the morphism Φ | L | is notbirational onto its image and h ( X , L ) ≥
2, then L =
1. As an application, if Y is a n -dimensional Fano manifold such that − K Y = ( n − ) H for some ample divisor H , then | mH | is basepoint free for m ≥ Φ | H | is not birational onto itsimage, then Y is either a weighted hypersurface of degree 10 in the weighted projectivespace P ( · · · , 1, 2, 5 ) or h ( Y , H ) = n −
1. I
NTRODUCTION
A normal projective complex threefold X is called a canonical Calabi-Yau threefold if O ( K X ) ∼ = O X , h ( X , O X ) = X has only canonical singularities. We say that X is a minimal Calabi-Yau threefold , if, in addition, X has only Q -factorial terminal singu-larities. A pair of a normal projective variety X and a line bundle L is called a polarizedvariety if the line bundle L is ample, and a quasi-polarized variety if the line bundle L is nef and big. For a given quasi-polarized canonical Calabi-Yau threefold ( X , L ) , thefollowing questions naturally arise. (1) When Φ | mL | (the rational map defined by | mL | ) is birational onto its image?(2) When | mL | is basepoint free? These two questions have already been investigated by several mathematicians in var-ious different settings [6, 13, 14] etc. Our first result can be viewed as a generalizationof [13, Theorem 1.1] and [14, Theorem 1].
Let ( X , L ) be a quasi-polarized canonical Calabi-Yau threefold. Then | mL | isbasepoint free when m ≥ . Moreover, if Φ | L | is not birational onto its image, then eitherL = or h ( X , L ) = . The estimate is sharp as showed by a general weighted hypersurface of degree 10in the weighted projective space P (
1, 1, 1, 2, 5 ) . We remark also that we have always h ( X , L ) ≥ Φ | L | is always birational ontoits image by [6, Theorem 1.7]. The basepoint freeness of | H | is an easy consequence of[12, Theorem 24] and the existence of semi-log canonical member in | H | (cf. [8, Proposi-tion 4.2]), and for the second part of the theorem, our proof basically goes along the line Date : September 5, 2018.2010
Mathematics Subject Classification.
Key words and phrases. birationality, Calabi-Yau threefolds, Fano manifolds, freeness. f [14, Theorem 1]. As the first application of Theorem 1.2, we generalize our previousresult in [11, Theorem 1.7]. Let X be a weak Fano fourfold with at worst Gorenstein canonical singularities.Then(1) the complete linear system | − mK X | is basepoint free for m ≥ ;(2) the morphism Φ |− mK X | is birational onto its image for m ≥ . As above, the estimates in Corollary 1.3 are both optimal as showed by a generalweighted hypersurface of degree 10 in the weighted projective space P (
1, 1, 1, 1, 2, 5 ) .As the second application, in higher dimension, using the existence of good ladder onFano manifolds with coindex four proved in [11] and the work of Fujita on polarizedprojective manifold with small ∆ -genus and sectional genus (cf. [4]), we derive the fol-lowing theorem which can also be viewed as a generalization of [13, Theorem 1.1] inhigher dimension. Let X be a n-dimensional Fano manifold such that − K X = ( n − ) H for someample divisor H. Then(1) the complete linear system | mH | is basepoint free when m ≥ ;(2) the morphism Φ | mH | is birational onto its image when m ≥ .Moreover, if the morphism Φ | H | is not birational onto its image, then one of the following holds.(1) X is a weighted hypersurface of degree in the weighted projective space P ( · · · , 1, 2, 5 ) .(2) h ( X , H ) = n − . As in dimension 4, the same example given in Theorem 1.4 guarantees that the esti-mates given in Theorem 1.4 are best possible, and we have always h ( X , H ) ≥ n − X is a general weightedcomplete intersection of type (
6, 6 ) in the weighted projective space P ( · · · , 1, 2, 2, 3, 3 ) and H ∈ |O X ( ) | , then we have h ( X , H ) = n −
1. This leads us to ask the followingnatural question. [4, 2.14][10, Problems 2.4]
Is there an example of Fano n-fold X such that − K X = ( n − ) H for some ample divisor H and h ( X , H ) = n − ? Acknowledgements.
I want to thank Andreas H ¨oring and Christophe Mourouganefor their constant encouragements and supports.2. P
ROOF OF THE MAIN RESULTS
Throughout the present paper, we work over the complex numbers and we adopt thestandard notation in Koll´ar-Mori [9], and will freely use them. We start by selectingsome results in minimal model program, and we shall use them in the sequel.
Let ( X , L ) be a quasi-polarized projective variety with at most Gorenstein canon-ical singularities.(1) There exists a projective variety Y with only Q -factorial terminal singularities and a propersurjective birational morphism ν : Y → X such that K Y = ν ∗ K X . Moreover, in this case,M : = ν ∗ L gives a quasi-polarization on Y.
2) Assume moreover that aL − K X is nef and big for some positive integer a. Then | mL | isbasepoint free for any large m and gives a proper surjective birational morphism µ : X → Zsuch that L = µ ∗ H for some ample line bundle H on Z.Proof.
The assertion ( ) is a consequence of [2, Corollary 1.4.4], and Y is called a ter-minal modification of X . The statement ( ) is an easy corollary of the Basepoint-freetheorem. In fact, applying Basepoint-free theorem (cf. [9, Theorem 3.3]), | mL | is base-point free for all large m and we define ϕ : X → Z to be the Stein factorization of themorphism Φ | mH | . Clearly ϕ is independent of the choice of m . In particular, there existstwo ample line bundles H and H on Z such that mL = ϕ ∗ H and ( m + ) L = ϕ ∗ H .Set H = H − H . It follows that L = µ ∗ H . (cid:3) Let X be a reduced equi-dimensional algebraic scheme and B an effective R -divisor on X. The pair ( X , B ) is said to be SLC (semi-log canonical) if the following conditionsare satisfied.(1) X satisfies the Serre condition S , and has only normal crossing singularities in codimensionone.(2) The singular locus of X does not contain any irreducible component of B.(3) K X + B is an R -Cartier divisor.(4) For any birational morphism µ : Y → X from a normal variety, if we write K Y + B Y = µ ∗ ( K X + B ) , then all the coefficients of B Y are at most .Moreover, ( X , B ) is called a stable log pair if in addition(5) K X + B is ample.A stable variety is a log stable pair ( X , B ) with B = , and we will abbreviate it as X. Let ( X , L ) be a n-dimensional quasi-polarized projective manifold.(1) The ∆ -genus ∆ ( X , L ) of ( X , L ) is defined to be n + L n − h ( X , L ) .(2) The sectional genus g ( X , L ) of ( X , L ) is defined to be (cid:0) K X · L n − + ( n − ) L n (cid:1) /2 + . Now we give the proof of Theorem 1.2.
Proof of Theorem 1.2.
Recall that canonical singularities are normal rational Cohen-Macaulay singularities. By Lemma 2.1 (2), there exists a proper surjective birationalmorphism µ : X → Z such that L = µ ∗ H for some ample line bundle H on Z . More-over, as µ ∗ K X = K Z , we have O ( K Z ) = O X . In particular, we get µ ∗ K Z = K X . Itfollows that Z has only canonical singularities. Thus, Z has only rational singularitiesand R i µ ∗ O X = i >
0. This implies h ( Z , O Z ) ∼ = h ( X , O X ) =
0. As a conse-quence, ( Z , H ) is a polarized canonical Calabi-Yau threefold. On the other hand, usingthe projection formula, we get µ ∗ O X ( mL ) = O Z ( mH ) and R i µ ∗ O X ( mL ) = i > µ ∗ : H ( Z , mH ) → H ( X , mL ) is an isomor-phism for all m . In particular, | mL | is basepoint free if and only if | mH | is basepointfree and Φ | mL | is birational onto its image if and only if Φ | mH | is birational noto its im-age. According to [8, Proposition 4.2], there exists an member S ∈ | H | such that S isa stable surface with K S = H | S . By Kawamata-Viehweg vanishing theorem and ourassumption, the natural restriction H ( Z , mH ) −→ H ( S , mH | S ) s surjective for all m ∈ Z . Thanks to [12, Theorem 24], | mK S | is basepoint free for all m ≥
4. Consequently, | mH | is also basepoint free for all m ≥ Φ | L | is not birational onto its image. By Lemma 2.1(1), there exists a terminal modification ν : Y → X such that ( Y , M ) is a quasi-polarizedminimal Calabi-Yau threefold where M = ν ∗ L . As above, we see that L = M andthe induced morphism ν ∗ : H ( X , mL ) → H ( Y , mM ) is an isomorphism for all m . Inparticular, Φ | mL | is birational onto its image if and only if Φ | mM | is birational onto itsimage. Thus, after replacing ( X , L ) by ( Y , M ) , we may assume that ( X , L ) itself is aquasi-polarized minimal Calabi-Yau threefold. In particular, X is actually factorial by[7, Lemma 5.1]. As mentioned in the introduction, we have always h ( X , L ) ≥ h ( X , L ) ≥ Φ | L | ( X ) = dim Φ | L | ( X ) ≥ . By Hironaka’s resolution theorem, there exists a smoothprojective threefold Y and a proper surjective birational morphism π : Y → X and adecomposition | π ∗ L | = | F | + B such that | F | is basepoint free. Let T ∈ | F | be a general smooth member. By the proof of[14, Theorem 1], Φ | ( m + ) L | is birational onto its image if Φ | π ∗ mL | T + K T | is birational ontoits image. Thus, if ( π ∗ L | T ) ≥
2, by [16, Theorem 1 (ii)], the complete linear system | π ∗ mL | T + K T | is birational onto its image if m ≥
3. If ( π ∗ L | T ) =
1, by the projectionformula, we get L · π ∗ T = T is a general member in the movable family | F | .Thanks to [14, Lemma 1.1 (4)], we see that L = dim Φ | L | ( X ) = . Since h ( X , O X ) =
1, there exists a smooth projective three-fold Y and a proper surjective birational morphism µ : Y → X and a decomposition | µ ∗ L | = n | F | + B such that | F | is a free pencil. Let T be a general smooth element in | F | . Then Φ | ( m + ) L | is birational onto its image if Φ | π ∗ mL | T + K T | is birational onto its image. Using the sameargument as in the 1st case, we obtain L = Φ | L | is not birational onto its image. (cid:3) Corollary 1.3 is an immediate consequence of Theorem 1.2 and the existence of gooddivisor on weak Fano fourfolds established in [8, Theorem 5.2].
Proof of Corollary 1.3.
The statement (2) was proved in [11, Theorem 1.7]. By Lemma 2.1(2), there exists a surjective proper birational map µ : X → Z and an ample line bundle H on Z such that µ ∗ H = − K X . Moreover, as µ ∗ K X = K Z , it follows that − K Z = H and µ ∗ K Z = K X . According to [8, Theorem 5.2], there exists a member Y ∈ | − K Z | such that Y has only Gorenstein canonical singularities. As a consequence, ( Y , − K Z | Y ) is a polarized canonical Calabi-Yau threefold. Thanks to Kawamata-Viehweg vanishingtheorem, the natural restriction map H ( Z , − mK Z ) −→ H ( Y , − mK Z | Y ) is surjective for all m ∈ Z . Then, by Theorem 1.4, we see that | − mK Z | is basepointfree if m ≥
4. On the other hand, the same argument as in Theorem 1.2 shows that theinduced morphism µ ∗ : H ( Z , − mK Z ) → H ( X , − mK X ) is an isomorphism for all m .Hence, | − mK X | is basepoint free for all m ≥ (cid:3) ext we give the proof of Theorem 1.4. Proof of Theorem 1.4.
By [11, Theorem 1.2] and [3, Theorem 1.1], there exists a descend-ing sequence of subvarieties of XX = X n ) X n − ) · · · ) X such that X i + ∈ | H | X i | and X i has only Gorenstein canonical singularities. Moreover,it is easy to see that ( X , H | X ) is a polarized canonical Calabi-Yau threefold. Thanks toTheorem 1.2, | mH | X n − | is basepoint free if m ≥
4. By Kawamata-Viehweg vanishingtheorem, it is easy to see that the natural restriction H ( X , mH ) −→ H ( X , mH | X ) is surjective for all m ∈ Z . Thus | mH | is basepoint free if m ≥
4. On the other hand, if Φ | H | is not birational onto its image, since we can choose all X i to be general, Φ | H | X | isnot birational onto its image (cf. [14, Lemma 1.3]). If h ( X , H ) = n −
2, by [11, Theorem1.2], we get h ( X , H ) ≥ n −
1. As a consequence, we obtain h ( X , H | X ) = h ( X , H ) − ( n − ) ≥ H n = ( H | X ) =
1. Then, by definition, we have g ( X , L ) : = ( K X · H n − + ( n − ) H n ) /2 + = H n + = ∆ ( X , H ) : = H n + n − h ( X , H ) ≤ n + − ( n − ) = ∆ ( X , H ) ≥ g ( X , L ) = ∆ ( X , H ) = X is isomorphic to either a weighted hypersur-face of degree 10 in the weighted projective space P ( · · · , 1, 2, 5 ) or a weighted com-plete intersection of type (
6, 6 ) in the weighted projective space P ( · · · , 1, 2, 2, 3, 3 ) .However, if X is a weighted complete intersection of type (
6, 6 ) in the weighted projec-tive space P ( · · · , 1, 2, 2, 3, 3 ) , then the group H ( X , mH ) ( m ≥ ) contains the mono-mials { x x m − , · · · , x n − x m − , x n − x m − , x n x m − , x n + x m − , x n + x m − } ,where x i are the weighted homogeneous coordinates of P ( · · · , 1, 2, 2, 3, 3 ) in order.This shows that Φ | mH | ( m ≥ ) is one-to-one on the non-empty Zariski open subset { x = } ∩ X and this case is excluded. (cid:3)
3. F
URTHER DISCUSSIONS
Let ( X , L ) be a quasi-polarized canonical Calabi-Yau threefold such that h ( X , L ) = ( Y , M ) be the terminal modification of ( X , L ) . Then Y is smooth in codimensiointwo. By Riemann-Roch formula and the projection formula, we obtain χ ( X , mL ) = χ ( Y , mM ) = M m + M · c ( Y ) + χ ( Y , O Y ) . s h ( X , O Y ) =
0, by Serre duality, we get χ ( Y , O Y ) =
0. Thus, using Kawamata-Viehweg vanishing theorem, we obtain1 = h ( X , L ) = h ( Y , M ) = M + M · c ( Y ) .Moreover, thanks to [15, Thereom 0.5], we have M · c ( X ) ≥
0. It follows that1 ≤ L = M ≤ S on a Calabi-Yau threefold X (not nec-essarily simply connected) is a minimal surface of general type. This simple observa-tion yields a bridge between two important classes of algebraic varieties. Moreover,a smooth ample divisor S on a Calabi-Yau threefold is called a rigid ample surface if h ( X , O X ( S )) =
1. In this case, the geometric genus p g ( S ) : = h ( S , K S ) is zero and,by the Lefschetz theorem, the natural map π ( S ) → π ( X ) is an isomorphism. Thus,according to theorem 1.2, it may be interesting to ask the following question. Is there a simply connected smooth Calabi-Yau threefold X containing a rigidample surface S?
We remark that if we do not require the simple connectedness of X , such an example of ( X , S ) with the quaternion group of order 8 π ( X ) = H as its fundamental group wasconstructed by Beauville in [1]. R EFERENCE [1] A. Beauville. A Calabi-Yau threefold with non-abelian fundamental group. In
New trends in algebraicgeometry (Warwick, 1996) , volume 264 of
London Math. Soc. Lecture Note Ser. , pages 13–17. CambridgeUniv. Press, Cambridge, 1999.[2] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan. Existence of minimal models for varieties of loggeneral type.
J. Amer. Math. Soc. , 23(2):405–468, 2010.[3] E. Floris. Fundamental divisors on Fano varieties of index n − Geom. Dedicata , 162:1–7, 2013.[4] T. Fujita. Classification of polarized manifolds of sectional genus two. In
Algebraic geometry and com-mutative algebra, Vol. I , pages 73–98. Kinokuniya, Tokyo, 1988.[5] T. Fujita.
Classification theories of polarized varieties , volume 155 of
London Mathematical Society LectureNote Series . Cambridge University Press, Cambridge, 1990.[6] C. Jiang. On birational geometry of minimal threefolds with numerically trivial canonical divisors.
Math. Ann. , 365(1-2):49–76, 2016.[7] Y. Kawamata. Crepant blowing-up of 3-dimensional canonical singularities and its application todegenerations of surfaces.
Ann.of Math. (2) , 127(1):93–163, 1988.[8] Y. Kawamata. On effective non-vanishing and base-point-freeness.
Asian J. Math. , 4(1):173–181, 2000.Kodaira’s issue.[9] J. Koll´ar and S. Mori.
Birational geometry of algebraic varieties , volume 134 of
Cambridge Tracts in Mathe-matics . Cambridge University Press, Cambridge, 1998.[10] O. K ¨uchle. Some remarks and problems concerning the geography of Fano 4-folds of index and Picardnumber one.
Quaestiones Math. , 20(1):45–60, 1997.[11] J. Liu. Second chern class of Fano manifolds and anti-canonical geometry.
Math. Ann. , to appear, 2017.doi: 10.1007/s00208-018-1702-z.[12] W. Liu and S. Rollenske. Pluricanonical maps of stable log surfaces.
Adv. Math. , 258:69–126, 2014.[13] K. Oguiso. On polarized Calabi-Yau 3-folds.
J. Fac. Sci. Univ. Tokyo Sect. IA Math. , 38(2):395–429, 1991.[14] K. Oguiso and T. Peternell. On polarized canonical Calabi-Yau threefolds.
Math. Ann. , 301(2):237–248,1995.[15] W. Ou. On generic nefness of tangent sheaves. arXiv preprint arXiv:1703.03175 , 2017.
16] I. Reider. Vector bundles of rank 2 and linear systems on algebraic surfaces.
Ann. of Math. (2) ,127(2):309–316, 1988.J IE L IU , M ORNINGSIDE C ENTER OF M ATHEMATICS , A
CADEMY OF M ATHEMATICS AND S YSTEMS S CI - ENCE , C
HINESE A CADEMY OF S CIENCES , B
EIJING , 100190, C
HINA
E-mail address : [email protected]@amss.ac.cn