Maximal Families of Calabi–Yau Manifolds with Minimal Length Yukawa Coupling
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Abstract
For each natural odd number n≥3, we exhibit a maximal family of ndimensional Calabi–Yau manifolds whose Yukawa coupling length is 1. As a consequence, Shafarevich’s conjecture holds true for these families. Moreover, it follows from Deligne and Mostow (Publ. Math. IHÉS, 63:5–89, 1986) and Mostow (Publ. Math. IHÉS, 63:91–106, 1986; J. Am. Math. Soc., 1(3):555–586, 1988) that, for n=3, it can be partially compactified to a Shimura family of ball type, and for n=5,9, there is a sub \({\mathbb{Q}}\)PVHS of the family uniformizing a Zariski open subset of an arithmetic ball quotient.
Keywords
Calabi–Yau Yukawa Coupling Hodge TheoryMathematics Subject Classification (2010)
14D07 14J321 Introduction
Theorem 1.1
 (i)
The family f _{ n } admits a simultaneous resolution \(\tilde{f}_{n}: \tilde{\mathcal{X}}_{n}\to\mathfrak{M}_{n,n+3}\) which is a maximal family of ndimensional projective CY manifolds.
 (ii)
\(\varsigma(\tilde{f}_{n})=1\). Consequently, Shafarevich’s conjecture holds for \(\tilde{f}_{n}\).
 (iii)
The family \(\tilde{f}_{3}\) admits a partial compactification to a Shimura family over an arithmetic quotient of \({\mathbb{B}}^{3}\).
 (iv)
The families \(\tilde{f}_{5},\tilde{f}_{9}\) have a sub \({\mathbb{Q}}\)PVHS which uniformizes a Zariski open subset of an arithmetic ball quotient.
2 The Cyclic Cover and Its Crepant Resolution
The meaning of letters in the tuple (n,m,r) will be fixed throughout the paper: n is a natural odd number ≥3, m=n+3 and \(r=\frac{m}{2}\).
Remark 2.1
The technique of this section can be applied equally to a tuple (n,m,r) where n is a natural number, r a positive factor of m and \(m=n+1+\frac{m}{r}\), and yields the same result as the special case.
2.1 The Cyclic Cover of \({\mathbb{P}}^{n}\)
2.2 The Crepant Resolution
In this paragraph we aim to obtain a good smooth model of the cyclic cover X. First recall the order function on a smooth variety. Let M be a smooth variety over an algebraically closed field and \(\mathcal{I}\) be an ideal sheaf on M. For any point x∈M, the order of \(\mathcal{I}\) at x is defined as \({\rm Ord}_{x}\mathcal{I}:=\max\{r\mid \mathcal{I}_{x}\subset m_{x}^{r}{\mathcal{O}}_{M,x}\}\), where m _{ x } is the maximal ideal of the local ring \({\mathcal{O}}_{M,x}\). For a smooth and irreducible closed subvariety Z of M, the order of \(\mathcal{I}\) along Z is defined as \({\rm Ord}_{Z}\mathcal{I}:= {\rm Ord}_{p}\mathcal{I}\), where p∈Z is the generic point of Z. We have the following wellknown formula for the canonical bundle under a blowup.
Lemma 2.2
Proof
The proof is a direct application of the adjunction formula. □
In order to fix notation, we recall the following definition.
Definition 2.3
 (1)
either x∉E _{ i }, or
 (2)
E _{ i }=(z _{ c(i)}=0) in a neighborhood of x for some c(i), and
 (3)
c(i)≠c(i′) if i≠i′.
 (4)
\(Z=(z_{j_{1}}=\cdots=z_{j_{s}}=0)\) for some j _{1},…,j _{ s }, again in some open neighborhood of x.
 (1)
either X,E _{1},…,E _{ s },F _{1},…,F _{ t } meet transversally at x, or
 (2)there exists a coordinate system (y _{1},…,y _{ m },x _{1},…,x _{ n }) at x admissible to E _{1},…,E _{ s },F _{1},…,F _{ t } (refer to Definition 2.3 for the notation) such that
 the defining equation of X isin a nonempty open neighborhood of x, where the integers satisfy$$y_{1}^{a_{1}}\cdots y_{p}^{a_{p}}x_{1} \cdots x_{q}=0 $$and$$1\leq p\leq m,\quad 1\leq q\leq n,\quad a_{1}\geq1,\quad\ldots,\quad a_{p} \geq1, $$

for each 1≤i≤p, y _{ i }=0 is a defining equation for some \(E_{c_{i}}\in\mathfrak{E}\) in a nonempty open neighborhood of x, and

for each 1≤j≤q, x _{ j }=0 is a defining equation for some \(F_{d_{j}}\in\mathfrak{F}\) in a nonempty open neighborhood of x.

 (3)
\(\forall E\in\mathfrak{E}\cup\mathfrak{F}\), ∀ closed points x _{1},x _{2}∈E∩X, e(E,x _{1})=e(E,x _{2}).
 (4)
\(\forall E\in\mathfrak{E}\), \(\forall F\in\mathfrak{F}\), if there exists a closed point x∈X such that e(E,x)>0 and e(F,x)>0, then E∩F⊂X.
Proposition 2.4
 (1)
\(\mathfrak{E}_{1}\cup\mathfrak{F}_{1}\) is a set of smooth divisors meeting transversally on M _{1}, and X _{1} is a hypersurface of M _{1} binomial with respect to \((\mathfrak{E}_{1}, \mathfrak{F}_{1})\).
 (2)
Each irreducible component of the singular locus of X, say \(\operatorname{Sing}(X)\), has the form \(E_{i_{1}}\cap F_{j_{1}}\cap F_{j_{2}}\subset X\) or \(E_{i_{1}}\cap E_{i_{2}}\cap F_{j_{1}}\cap F_{j_{2}}\subset X\), for 1≤i _{1}≠i _{2}≤s, 1≤j _{1}≠j _{2}≤t.
 (3)
For the induced morphism \(X_{1}\xrightarrow{\pi_{1}}X\) we have \(\pi_{1}^{1}(E_{1}\cap F_{1}\cap \operatorname{Sing}(X))\rightarrow E_{1}\cap F_{1}\cap \operatorname{Sing}(X)\) is a \({\mathbb{P}}^{1}\)bundle, and \(\pi _{1}^{1}(XE_{1}\cap F_{1}\cap \operatorname{Sing}(X))\rightarrow XE_{1}\cap F_{1}\cap \operatorname{Sing}(X)\) is an isomorphism.
 (4)Let \(\mathfrak{T}=\{E_{i_{1}}', \ldots, E_{i_{k}}', F_{j_{1}}',\ldots, F_{j_{l}}'\}\subset\mathfrak{E}_{1} \cup \mathfrak{F}_{1}\) be a subset of \(\mathfrak{E}_{1} \cup\mathfrak {F}_{1}\) satisfying \(V=E_{i_{1}}'\cap\cdots\cap E_{i_{k}}'\cap F_{j_{1}}'\cap\cdots\cap F_{j_{l}}'\subset X_{1}\), then V=∅ if \(\{E_{1}', F_{1}'\}\subset\mathfrak{T}\). Suppose V≠∅, then$$ \nonumber \pi(V)=\left \{ \begin{array}{l@{\quad}l} E_{i_{1}}\cap\cdots\cap E_{i_{k}}\cap F_{j_{1}}\cap\cdots\cap F_{j_{l}}, & \textit{if } E_{s+1}'\notin\mathfrak{T}; \\ [4pt] E_{1}\cap F_{1}\cap E_{i_{2}}\cap\cdots\cap E_{i_{k}}\cap F_{j_{1}}\cap\cdots\cap F_{j_{l}}, & \textit{if } E_{i_{1}}'=E_{s+1}'. \end{array} \right . $$
 (5)
Notations as in (4). If \(\{E_{1}', F_{1}'\}\cap \mathfrak{T}\neq\emptyset\), then the induced morphism \(V\xrightarrow {\tilde{\pi}} \pi(V)\) is an isomorphism. If \(\{E_{1}', F_{1}'\} \cap\mathfrak{T}= \emptyset\), then the induced morphism \(V\xrightarrow{\tilde{\pi}} \pi(V)\) satisfies that \(\tilde{\pi }^{1}(E_{1}\cap F_{1}\cap\pi(V))\rightarrow E_{1}\cap F_{1}\cap\pi (V)\) is a \({\mathbb{P}}^{1}\)bundle, and \(\tilde{\pi}^{1}(\pi (V)E_{1}\cap F_{1})\rightarrow\pi(V)E_{1}\cap F_{1}\) is an isomorphism.
Proof
The verification is straightforward in local coordinates. □
 (0)
If the maximal value max_{ x∈X } f(x)=(0,0,0), then X is already a smooth variety meeting transversally with the divisors in \(\mathfrak{E}\cup\mathfrak{F}\).
 (1)If max_{ x∈X } f(x)>(0,0,0), take any closed point x∈X such that f(x) attains the maximal value of f. It is not difficult to see that we can choose \(E_{i}\in\mathfrak{E}\), \(F_{j}\in \mathfrak{F}\) such that e(E _{ i },x)>0 and e(F _{ j },x)>0. Then blow up M along Z=E _{ i }∩F _{ j } (we have Z=E _{ i }∩F _{ j }⊂X by the definition of binomial hypersurfaces). Let E be the exceptional divisor. Let M _{1}=Bl _{ Z } M, and \(X_{1},E_{1}',\ldots ,E_{s}', F_{1}',\ldots, F_{t}'\) be the strict transforms of X,E _{1},…,E _{ s },F _{1},…,F _{ t }, respectively. Let \(\mathfrak{E}_{1}=\{E, E_{1}',\ldots, E_{s}'\}\), \(\mathfrak {F}_{1}=\{F_{1}',\ldots, F_{t}'\}\), then according to Proposition 2.4, X _{1} is a hypersurface of M _{1} binomial with respect to \((\mathfrak {E}_{1},\mathfrak{F}_{1})\). So we can define a function \(f_{1}:X_{1}\rightarrow\mathbb{N}\times\mathbb{N}\times\mathbb {N}\) in the same way as above. Let π _{1}:X _{1}→X be the blowup morphism. Then it is direct to verify:Note also that since Z has codimension 2 everywhere in M, and X has order 1 at the generic point of each irreducible component of Z, we have \(K_{X_{1}}\simeq\pi_{1}^{*}K_{X}\), according to Lemma 2.2.

for any point x∈X _{1}, f _{1}(x)≤f(π _{1}(x)), and

the maximal value drops strictly, i.e. \(\max_{x\in X_{1}}f_{1}(x)< \max_{x\in X}f(x)\).

 (2)If \(\max_{x\in X_{1}}f_{1}(x)>(0,0,0)\), then continue to blow up M _{1} and get…$$M_{2},\quad X_{2},\quad \mathfrak{E}_{2},\quad \mathfrak{F}_{2},\quad f_{2}. $$
Theorem 2.5

π is crepant, i.e. \(K_{X_{N}}\simeq\pi^{*}K_{X}\);

π is a strong resolution of X, i.e. if \(U=X\operatorname{Sing}(X)\) is the regular part of X, then π induces an isomorphism \(\pi ^{1}(U)\xrightarrow{\sim} U\);

π is a projective morphism, moreover, it is a composition of blowups along smooth centers.
Proof
Most of the theorem follows from the above discussions. We just explain why π is a strong resolution. Note that in each blowup step the smooth center Z _{ i } is contained completely in X _{ i }. So in the regular part of X _{ i }, we just blow up a Cartier divisor. Therefore, the regular part of X _{ i } remains unchanged. □
Now we give an application of Theorem 2.5. Suppose Q is a smooth variety, and \(D=\sum_{j=1}^{t}F_{j}\) is a simple normal crossing divisor on Q defined by the section \(s_{D}\in\varGamma(Q,\mathcal{O}_{Q}(D))\). Let a be a positive integer and \(L\in{\rm Pic}(Q)\) such that aL=D. Then in the total space of L: \(M={\rm Tot}(L)\xrightarrow{p} Q\), we have the tautological section s _{0}∈Γ(M,p ^{∗} L). The hypersurface X of M defined by the equation \(s_{0}^{a}=p^{*}s_{D}\) is called the afold cyclic cover of Q branched along D. It is easy to verify that X is a hypersurface of M binomial with respect to \((\mathfrak{E}, \mathfrak{F})=(\{E_{0}=(s_{0}=0)\},\{p^{*}F_{1},\ldots, p^{*}F_{t}\})\). So we can apply Theorem 2.5 to get a crepant resolution of X. That is, we have the following corollary.
Corollary 2.6
Suppose Q is a smooth variety and \(D=\sum_{j=1}^{t}F_{j}\) is a simple normal crossing divisor on Q, for any a≥1, if there exists L∈Pic(Q) such that aL=D, then the afold cyclic cover of Q branched along D admits a crepant resolution, which can be obtained by applying the crepant resolution algorithm (∗).
For the cyclic cover X constructed in Sect. 2.1, we can simply apply the above result to obtain a crepant resolution \(\sigma: \tilde{X} \to X\).
2.3 The Middle Cohomology Does Not Change Under Resolution
Lemma 2.7
Proposition 2.8
We need some lemmas.
Lemma 2.9
Let \(f: \tilde{X}\rightarrow X\) be a proper modification with discriminant D. Put E=f ^{−1}(D). For p,q≥0, if H ^{ p,q }(X)=H ^{ p,q }(E)=0, then \(H^{p,q}(\tilde{X})=0\).
Proof
This follows directly from [10, CorollaryDefinition 5.37]. □
Lemma 2.10
Let \(\pi: \tilde{V}\rightarrow V\) be a surjective morphism between projective varieties, Z⊂V a closed subvariety such that \(\pi^{1}(Z)\xrightarrow{\pi}Z \) is a \({\mathbb{P}}^{1}\)bundle and \(\pi^{1}(VZ)\xrightarrow{\pi}VZ\) is an isomorphism. For p,q≥0 and p≠q, if H ^{ p,q }(Z)=0, then the natural homomorphism \(H^{p,q}(V)\xrightarrow{\pi^{*}} H^{p,q}(\tilde{V})\) is surjective.
Proof
This follows from Lemma 2.9 and the Leray–Hirsch Theorem for the \({\mathbb{P}}^{1}\)bundle \(\pi^{1}(Z)\xrightarrow{\pi}Z \). □
We come to the proof of Proposition 2.8.
Proof
Corollary 2.11
Let \(\pi: X\rightarrow{\mathbb{P}}^{n}\) be the rfold cyclic cover of \({\mathbb{P}}^{n}\) branched along m hyperplanes in general position in Sect. 2.1. Let \(\tilde{X}\xrightarrow{\sigma} X\) be the crepant resolution constructed after Corollary 2.6. The obtained \(\tilde{X}\) is a smooth projective CY manifold.
Proof
3 The Hodge Structure of the Cyclic Cover
Let \(\pi: X\to{\mathbb{P}}^{n}\) be the rfold cyclic cover branched along \(H=\sum_{i=1}^{m}H_{i}\) where {H _{1},…,H _{ m }} is a hyperplane arrangement of \({\mathbb{P}}^{n}\) in general position. In this section we investigate the Hodge structure \(H^{n}(X,{\mathbb{Q}})\). We first record its Hodge numbers.
Lemma 3.1
Proof
By Lemma 2.7, we can derive the Hodge numbers of X from those of Y, which is a smooth complete intersection. By the work of Terasoma [12], one can represent the cohomology classes of Y by a certain Jacobian ring, together with an explicit description of the action of G _{1}. After the computation has been implemented, we found that [2, Lemma 8.2] actually contains our result (set \(\mu=(\frac{1}{r},\ldots,\frac{1}{r})\) in the cited lemma). The detail is therefore omitted. □
3.1 The Cyclic Cover of \({\mathbb{P}}^{1}\) Branched Along m Distinct Points
Lemma 3.2
Proof
See for example [6, Lemma 4.2]. □
The next lemma is purposed for a later use.
Lemma 3.3
Proof
Now we perform a similar construction to the one taken in [3, Sect. 2.3]. Let \(\gamma: ({\mathbb{P}}^{1})^{n}\to\mathrm {Sym}^{n}({\mathbb{P}}^{1})={\mathbb{P}}^{n}\) be the Galois cover with Galois group S _{ n }, the permutation group of n letters, and the identification attaches to a divisor of degree n the ray of its equation in \(H^{0}({\mathbb{P}}^{1},{\mathcal{O}}(n))\).
Lemma 3.4
Put \(H_{i}=\gamma(\{p_{i}\}\times({\mathbb{P}}^{1})^{n1})\). Then (H _{1},…,H _{ m }) is a hyperplane arrangement in \({\mathbb{P}}^{n}\) in general position.
Proof
The divisors of degree n in \({\mathbb{P}}^{1}\) containing a given point form a hyperplane and, as a divisor of degree n cannot contain n+1 distinct points, no n+1 hyperplanes in the arrangement do meet. □
In this case, we have more: for any natural number n (not necessarily odd), we will show that any (ordered) m hyperplane arrangement in \({\mathbb{P}}^{n}\) is projectively equivalent to a(n) (ordered) one arising from the above way. In fact, we will prove a stronger statement. Let \(\mathfrak{M}_{1,n+3}\) be the moduli space of ordered n+3 distinct points in \({\mathbb{P}}^{1}\) and similarly \(\mathfrak {M}_{n,n+3}\) ordered n+3 hyperplane arrangements in \({\mathbb{P}}^{n}\) in general position.
Lemma 3.5
Proof
Claim 3.6
Proof
3.2 The Abel–Jacobi Map and the Hodge Structure of the Cyclic Cover
Proposition 3.7
Proof
4 Maximal Families of CY Manifolds with Length 1 Yukawa Coupling
Our aim in this section is to exhibit families of CY manifolds with claimed properties, and make some complements to these families at the end.
Theorem 4.1
Proof
It follows from Propositions 2.8 and 3.7. □
Remark 4.2
Corollary 4.3
 (i)
The family \(\tilde{f}_{n}\) is maximal.
 (ii)
\(\varsigma(\tilde{f}_{n})=1\). Consequently, Shafarevich’s conjecture holds true for \(\tilde{f}_{n}\).
 (iii)
A suitable partial compactification of the family \(\tilde{f}_{3}\) is a Shimura family of U(1,3)type.
 (iv)
For n=5,9, the sub \({\mathbb{Q}}\)PVHS \({\mathbb{H}}_{\mathrm{unif},{\mathbb{Q}}}\subset{\mathbb{H}}_{n}\) gives a uniformization to a Zariski open subset of an arithmetic ball quotient.
Proof
We conclude the paper with the following remark.
Remark 4.4
(i) For n=3, the Hodge numbers of \(\tilde{X}\) read h ^{1,1}=51,h ^{2,1}=3. Rohde has constructed in his doctor thesis (see [11]) a maximal family of CY 3folds with the same Hodge numbers which is also a Shimura family. Note that the parameter of his family comes also from \(\mathfrak{M}_{1,6}\). Are these two families birationally equivalent?
(ii) One constructs more maximal families of CY manifolds from the moduli space of hyperplane arrangements in a projective space. Do our families exhaust all possibilities with length 1 Yukawa coupling?
(iii) We have shown that \({\mathbb{H}}_{\mathrm{unif},{\mathbb{Q}}}\subset {\mathbb{H}}_{n}\) exists only for n=3,5,9, which are the unique three cases appeared in Mostow’s list with equal μ _{ i }s. Is there some deeper reason than a mere coincidence?
Notes
Acknowledgements
This work is supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG, and partially supported by the University of Science and Technology of China.
We would like to thank Guitang Lan for helpful discussions, particularly in Lemma 3.5. Our special thanks go to Igor Dolgachev who has drawn our attention to the n=3 case of the paper.
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