ORPHAN CALABI-YAU OPERATOR WITH ARITHMETIC MONODROMY GROUP TYMOTEUSZ CHMIEL Abstract. We present an example of a Picard-Fuchs operator of a one-parameter family of Calabi-Yau

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ORPHAN CALABI-YAU OPERATOR WITH ARITHMETIC MONODROMY GROUP

TYMOTEUSZ CHMIEL

Abstract. We present an example of a Picard-Fuchs operator of a one-parameter family of Calabi-Yau

threefolds which does not have a point of maximal unipotent monodromy, yet its monodromy group is of

ﬁnite index in Sp(4,Z). In particular, it contains inﬁnitely many maximally unipotent elements. We also

state some related results for the remaining 17 double octic orphan operators.

Introduction

The monodromy group is one of the most important invariants of a diﬀerential equation of Fuchsian

type. If Pis a Picard-Fuchs operator of a one-parameter family of Calabi-Yau threefolds, the monodromy

group Mon(P) is naturally realized as a subgroup of Sp(4,Z). This poses a question whether Mon(P) is

Zariski-dense in Sp(4,Z). If it is, one can further ask whether it is arithmetic, i.e. of ﬁnite index, or thin.

Indexes of several monodromy groups have been computed for operators with a singular point whose local

monodromy has maximal unipotency index (see [3, 16, 17, 11]). Existence of a point of maximal unipotent

monodromy, or a MUM point, is also crucial in the context of mirror symmetry.

In this paper we focus on operators which do not have a MUM singularity. They are called orphan

operators. Since every ﬁnite index subgroup of Sp(4,Z) contains maximally unipotent matrices, one could

expect that monodromy groups of orphan operators are small in this sense.

We study orphan Picard-Fuchs operators of one-parameter families of double octic Calabi-Yau threefolds

deﬁned over Q(see [7]). We construct rational bases using an approach which is easily applicable in general.

Using rational realizations we study properties of the corresponding monodromy groups as linear groups

generated by local monodromy operators. In particular, we obtain symplectic realizations for (subgroups of)

double octic orphan monodromy groups.

The main result of this paper is the following theorem (Thm. 1, Thm. 3 and Thm. 4):

Theorem.

(1) Monodromy groups of all double octic orphan operators are Zariski-dense in Sp(4,Z).

(2) Monodromy groups of double octic orphan operators contain maximally unipotent elements, with

the possible exception of operator 35.

(3) Monodromy group of the double octic orphan operator 250 is of ﬁnite index in Sp(4,Z).

These surprising results show that the na¨ıve expectation that the monodromy group of a Picard-Fuchs

operator without a MUM point are small is false.

In section 1 we present basic facts concerning Picard-Fuchs operators. Section 2 explains our method

of ﬁnding rational bases for the monodromy action. Section 3 contains results obtained using this method

when applied to orphan families of double octic Calabi-Yau threefolds. In section 4 we construct symplectic

bases for double octic orphan monodromy groups and show that they are Zariski-dense in Sp(4,Z). We also

prove that the monodromy group of the operator 250 is arithmetic.

1. Monodromy group of a Picard-Fuchs operator

ACalabi-Yau threefold is a smooth complex projective variety Xof dimension 3 such that ωX≃ OXand

H1(X, OX) = 0. By the Bogomolov-Tian-Todorov theorem, the Hodge number h2,1(X) is the dimension of

the smooth deformation space of X. When h2,1(X) = 1, the universal deformation space is one-dimensional,

2010 Mathematics Subject Classiﬁcation. Primary: 14D05; Secondary: 11F06, 14J32.

Key words and phrases. Calabi-Yau manifolds, monodromy, Picard-Fuchs operator, arithmetic group.

The ﬁrst author was supported by the National Science Center grant no. 2023/49/N/ST1/04089. Data sharing not applicable

to this article as no datasets were generated or analysed during the current study.

arXiv:2210.02367v3 [math.AG] 26 Jun 2024

i.e. there exists a (germ of a) smooth curve S, a distinguished point t0∈ S and a family of smooth Calabi-Yau

threefolds (Xt)t∈S =X → S such that X≃Xt0. We say that Xdeforms in a one-parameter family.

Fix a family of complex volume elements ωt∈H3,0(Xt) depending holomorphically on tand a locally

constant family of 3-cycles ∆t∈H3(Xt,Z). Period function of the family Xis y(t) := R∆tωt.It is deﬁned

in some neighbourhood Uof t0. By the Hodge decomposition, dim H3(Xs,C) = b3(Xs) = 2(h3,0(Xs) +

h2,1(Xs)) = 4 for all s∈U. Thus the elements

∇4

∂

∂t ωtt=s,∇3

∂

∂t ωtt=s,∇2

∂

∂t ωtt=s,∇∂

∂t ωt,t=s, ωs∈H3(Xs,C)

are linearly dependent over C.

It follows that the period function y(t) satisﬁes a diﬀerential equation Py= 0 for some order four

diﬀerential operator Pwith coeﬃcients in O(U). This is the Picard-Fuchs operator of the one-parameter

family X → S. By Sol(P, t0) we denote the space of solutions of P= 0 in a neighbourhood of t0∈ S. There

is an isomorphism Sol(P, t0)≃H3(Xt0,C).

For [γ]∈π1(S, t0) a solution f∈ Sol(P, t0) can be continued analytically along γ. This deﬁnes the

monodromy representation Mon :π1(S, t0)→Aut (Sol(P, t0)). For a ﬁxed basis Bof Sol(P, s0) we have

the corresponding matrix representation MonB:π1(S, t0)→GL(4,C). We deﬁne Mon(P) := im Mon and

MonB(P) := im MonB.

Let S=P1\Σ for some ﬁnite set Σ. The fundamental group π1(P1\Σ) is generated by loops γσencircling

singular points σ∈Σ. Consequently, the local monodromy operators Mσ:= Mon(γσ), σ∈Σ, generate the

monodromy group. Operators Mσare quasi-unipotent, i.e. (Mk

σ−Id)4= 0 for some k∈N≥1(see [12]).

Picard-Fuchs operators are Fuchsian: they have only regular singular points. Type of a singularity is

determined by the Jordan form of its local monodromy. We have the following possibilities:

Type of singularity Jordan form of local monodromy

MUM 





1100

0110

0011

0001







nK





1 1 0 0

0 1 0 0

0 0 ζn1

0 0 0 ζn







nC





1 0 0 0

0ζn1 0

0 0 ζn0

0 0 0 ζ2







F





1 0 0 0

0ζn10 0

0 0 ζn20

0 0 0 ζn1ζn2







Here ζmdenotes some primitive root of unity of order m.

Singular points of type MUM are called points of maximal unipotent monodromy. Singular points of type

nKare named due to their connection with K3 surfaces. Singular points of type 1

nCare also called conifold

points and are connected to (singular models of) rigid Calabi-Yau threefolds. Singular points of type Fare

called ﬁnite singularities and have local monodromy of ﬁnite order. For each point σof type 1

nCthere exists

the solution fc, unique up to scaling, such that im(Mn

σ−Id) = C·fc. It is called the conifold period.

An operator without a MUM singularity is called an orphan operator.

2. Rational basis for the monodromy action

An important disclaimer for all of the results presented in this paper is the following:

all considered monodromy groups are determined numerically.

This approach is common when studying Picard-Fuchs operators beyond the hypergeometric case (see [8, 10,

11, 4, 5]). The relative error of our numerical approximations is smaller than 10−100. Numerical identiﬁcation

of the coeﬃcients of the monodromy matrices yields the best results when MonB(P)⊂GL(4,Q).

Let Pbe a Calabi-Yau operator and put SQ:= H3(Xt0,Q)→H3(Xt0,Q)⊗C≃H3(Xt0,C)≃ Sol(P, t0).

It is a monodromy-invariant rational subspace and with respect to any basis of SQthe monodromy matrices

are rational.

We start with two general lemmas.

Lemma 1. Assume that the monodromy group M on(P)acts irreducibly on Sol(P, t0). Then for any solution

0̸=f∈ SQthere exist monodromy operators M1, . . . , M4∈Mon(P)such that

B:= nM1(f), M2(f), M3(f), M4(f)o

is a a basis of SQ.

Proof. Since SQis monodromy-invariant, B ⊂ SQ. Assume that for all M1, . . . , M4∈Mon(P) the elements

of Bspan inside Sol(P, t0) a subspace of dimension ≤3. Let Sbe the C-linear span of M(f), M∈Mon(P).

The subspace Sis monodromy-invariant and dim S≤3. But then S̸=Sol(P, t0), which contradicts the fact

that the monodromy group Mon(P) acts irreducibly. □

Fix a non-zero solution f∈ SQand let B=B(f) be as in the lemma. For α∈C∗we have B(αf) = α·B(f),

MonB(f)=MonB(αf)and MonB(f)(P)⊂GL(4,Q) if and only if MonB(αf)(P)⊂GL(4,Q). Thus we can

use Lemma 1 when f̸∈ SQbut [f]∈P(SQ)⊂P(Sol(P, t0)).

Lemma 2. Let A:= Pn

i=1 qiMi,qi∈Q, Mi∈Mon(P). Assume that rank(A) = 1, resp. corank(A) = 1,

and take f∈ Sol(P, t0)such that im(A) = C·f, resp. ker(A) = C·f. Then αf ∈ SQfor some α∈C∗.

Proof. Aacts as an endomorphism of SQ. Let g∈ SQbe such that im A|SQ=Q·g, resp. ker A|SQ=Q·g.

Then g∈C·f.□

If Mσis a local monodromy operator around σ∈Σ and A=Pn

i=0 aiMi

σ,ai∈Q, the rank and the

co-rank of Acan be computed by assuming that Mσis given in its Jordan form. In the construction of the

Doran-Morgan basis in [8] one takes A:= (M0−Id)3, where 0 is a MUM point. Then rank(A) = 1 and the

generator of im(A) is the unique holomorphic solution in a neighbourhood of 0. It is equivalent to taking

A:= M0−Id, since then corank(A) = 1 and the kernel of Ais spanned by the same solution.

Orphan operators do not have a point of maximal unipotent monodromy. Instead we use a singular point

s∈Σ of type 1

nC. Put A:= Mn

s−Id. The Jordan form of Mn

sis







1000

0110

0010

0001







and rank(A) = 1. The subspace im(A) is spanned by the conifold period fc. By Lemma 2 we may assume

fc∈ SQ. Let M1,· · · , M4be as in the conclusion of Lemma 1 applied to f=fcand put fi:= Mi(f). Fix a

non-singular base point t0∈Σ and consider the matrix

R:= f(j−1)

i(t0)i,j=1...,4

The Frobenius method implies that the map

Sol(P, t0)∋g7→ 





g(t0)

g′(t0)

g′′(t0)

g(3)(t0)







∈C4

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ORPHANCALABI-YAUOPERATORWITHARITHMETICMONODROMYGROUPTYMOTEUSZCHMIELAbstract.WepresentanexampleofaPicard-Fuchsoperatorofaone-parameterfamilyofCalabi-Yauthreefoldswhichdoesnothaveapointofmaximalunipotentmonodromy,yetitsmonodromygroupisoffiniteindexinSp(4,Z).Inparticular,itcontainsinfinitelymanymaximal...

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