Complex Hyperbolic Structures for Moduli Spacesof Calabi-Yau Varieties
Chenglong Yu
YMSC, Tsinghua University(joint with Zhiwei Zheng)
Tsinghua-Tokyo workshop on Calabi-Yau
Jan 16th

Chenglong Yu

YMSC, Tsinghua University

Moduli spaces and locally symmetric spaces
Many locally symmetric spaces have modular interpretations. Let
M be moduli of certain algebraic varieties, D Hermitian symmetricdomain and Γ ⊂ Aut(D) discrete lattice acting on D. The period
map induces
M ←→ Γ\DFamous examples include:1

M : moduli of polarized abelian varieties, D : Siegel upper halfspace2

M : moduli of polarized K 3 surfaces, D : Type-IV domain

Chenglong Yu

YMSC, Tsinghua UniversityComplex hyperbolic balls
Let h be a Hermitian form on vector space V with signature (1, n).
The complex hyperbolic ball of dimension n isBn = {v ∈ P(V ) | h(v , v ) > 0}
Examples of moduli spaces with ball structure:1(Deligne-Mostow) Moduli of (n + 3)-points on P1

2(Allcock-Carlson-Toledo) Moduli of cubic surfaces and cubicthreefolds

3(Kondō) Moduli of genus 3 curves and genus 4 curves.

Chenglong Yu

YMSC, Tsinghua University

Cyclic cover of projective line
Euler, Riemann, Schwarz, Picard, · · · , Shimura, Deligne-Mostow,
Thurston: consider cyclic covers of P1
Cµ : y d = (x − x1 )a1 · · · (x − xn+3 )an+3
Pwith 0 < µi = adi < 1 and i µi ∈ Z. The cyclic group Z/dZ acts
on Cµ and decomposes H 1 (Cµ ) by characters.
PWhenµi = 2, Hχ1 (Cµ ) has hermitian form with sign (1, n).
Moduli of such Cµ is M = PSL(2, C)\ (P1 )n+3 − diagonals
Monodromy representation ρ : π1 (M) → PU(1, n),Γµ = Im(ρ)
M∼= Γ\BnChenglong Yu

BBYMSC, Tsinghua University

Connection with hypergeometric functions when n = 1
Euler introduced hypergeometric functions
 X∞
a b(a)k (b)k x k;x =,2 F1c(c)k k!

|x| < 1k=0

where (a)k = a(a − 1) · · · (a − k + 1).
It satisfies a 2nd -order differential equation
d 2ydy+ (c − (a + b + 1)x)
− aby = 0dx 2dx
It has an integral representation

Z 1a bΓ(c);x =z a−1 (1 − z)c−a−1 (1 − zx)−b dz
2 F1c
Γ(a)Γ(c − a) 0Z ∞Γ(c)z b−c (z − 1)c−a−1 (z − x)−b dz=
Γ(a)Γ(c − a) 1x(1 − x)Chenglong Yu

YMSC, Tsinghua University

Monodromy representation
The differential equation has three regular singular points{0, 1, ∞} on the Riemann sphere P1 .
The analytic continuation of two linearly independent solutions
F1 , F2 gives rise to a representationρ : π1 (P1 − {0, 1, ∞}) → PGL(2, C).
Schwarz found that the two solutions are algebraic functions iff
Γ = Im(ρ) is finite.
Schwarz also showed that the inverse of FF21 being single-valued
is equivalent to Γ being discrete.

Chenglong Yu

YMSC, Tsinghua University

Discreteness, arithmeticity and Commensurability
Discreteness. When Γ is discrete, it is a triangle group,generated by even number of reflections about sides of aspherical, euclidean or hyperbolic triangle. (Schwarz, Mostow,Knapp,· · · )Arithmeticity. Γ is arithmetic if the discreteness is provided byZ being discrete in R. Takeuchi (1977) listed all the arithmetic
triangle groups (85 examples).
Commensurability. Γ1 , Γ2 are commensurable in G , iff there isg ∈ G , such that g Γ1 g −1 ∩ Γ2 has finite indices in g Γ1 g −1 , Γ2 .
This is commensurability of triangles.
Chenglong Yu

YMSC, Tsinghua University

Why ball quotients?
Discreteness. Deligne-Mostow and Thurston’s list: 94 + 10
tuples of µ for n ≥ 2.
Arithmeticity. Nonarithmetic lattices in PU(1, 2) and one
example in PU(1, 3).
For other simple groups, there are either infinitely many
commensurability classes of nonarithmetic lattices, (O(1, n) by
Gromov-Piatetski-Shapiro), or only arithmetic lattices(Margulis superrigidity, Corlette, Gromov-Schoen).
Other constructions in PU(1, 2) by Barthel-Hirzebruch-Höfer
via Bogomolov-Miyaoka-Yau inequality, and Yau’s criterion forball quotients.
Chenglong Yu

YMSC, Tsinghua University

Cyclic cover of P1 × P1Kondō considers nonhyperelliptic curves of genus 4, D ⊂ P1 × P1 .
Taking triple covers of P1 × P1 branching along D, gives surfaces
S → P1 × P1 .
S is K 3.
Hχ2 (S) has hermitian form with sign (1, 9).
Moduli of genus 4 curves M4 is birational to Γ\B9 .
Nikulin theory on K 3 lattice relates this ball quotient toDeligne-Mostow ball quotient with µ1 = · · · = µ12 = 61 up to
finite cover.
Moduli spaces of genus three curves, cubic surfaces and cubic
threefolds are ball quotiens in a similar way.
Chenglong Yu

YMSC, Tsinghua University

Cyclic cover of P3
Sheng-Xu-Zuo studied cyclic cover Y → P3 branching along 6hyperplanes.
Y is a Calabi-Yau orbifold admitting crepant resolution.
Z/3Z operation decomposes H 3 (Y ) as follows.
H 3,0 H 2,1 H 1,2 H 0,331300Hχ (Y )3Hχ̄ (Y )00
31
Period domain is B3 .
This is also related to Deligne-Mostow’s example
C : y 3 = (x − x1 ) · · · (x − x6 ).
Sheng-Xu proved global Torelli theorem for this family.
Sheng-Xu-Zuo classified such examples for cyclic covers of Pn
branching along hyperplanes.
Chenglong Yu

YMSC, Tsinghua UniversityClassification for P1 × P1 × P1
3:1

Let Y −−→ P1 × P1 × P1 be cyclic cover branching along simple
normal crossing divisor D = D1 + · · · + Dr ∈ |O(3, 3, 3)|. As Di
vary in |Li |, we obtain a family of Calabi-Yau orbifolds.
Theorem (Y.-Zheng)
The period map of this family factors through complex hyperbolicball if and only if L1 · · · Lr are
1

L1 = (3, 3, 0) and L2 = (0, 0, 3); (Voisin, Borcea and Rohde)2

L1 = (3, 2, 0) and L2 = (0, 1, 3);

3

L1 = (2, 2, 0), L2 = (1, 0, 2) and L3 = (0, 1, 1);

4L1 = (2, 1, 0), L2 = (1, 0, 2) and L3 = (0, 2, 1).

or their refinements. Moreover, the ball quotients in the 4 maximal
cases have dimensions 9, 9, 7, 6 respectively.
Chenglong Yu

YMSC, Tsinghua UniversityRefinement and half-twist
d:1

Let Y −−→ X be cyclic cover branching along simple normal
dcrossing divisor D = D1 + · · · + Dr ∈ | − d−1
KX |. Then Ddcorresponds a partition of − d−1 KX .
Refinements of D preserves the ball-type property.
When d = 3, consider
X ′ = X × P1 , Di′ = Di × P1 , Dr′ +1 = X × 3pts
The corresponding Y ′ is called half-twist of Y .
Half-twists generate the ball-type examples.
When X = (P1 )n , all ball-type examples are generated byrefinements and half-twists from the previous list together with
one more example for n = 4.
Chenglong Yu

YMSC, Tsinghua UniversityCrepant resolution and completeness
The family of Calabi-Yau orbifolds Y admits crepant resolutions Yeby Sheng-Xu-Zuo.
Theorem (Y-Zheng)
If the family of Calabi-Yau manifolds Ye is of ball type and complete,then the divisor D is a refinement of the following 5 cases:
1

L1 = (3, 1, 0), L2 = (0, 2, 1), L3 = (0, 0, 2);2

L1 = (3, 0, 0), L2 = (0, 2, 1), L3 = (0, 1, 2);

3

L1 = (2, 1, 0), L2 = (1, 0, 2), L3 = (0, 2, 1);

4

L1 = (2, 1, 0), L2 = (1, 0, 1), L3 = (0, 1, 1), L4 = (0, 1, 1);

5

L1 = (2, 1, 0), L2 = (1, 1, 0), L3 = (0, 1, 1), L4 = (0, 0, 2).

The dimensions of the balls for the five cases are 5, 4, 6, 5, 4.
Higher dimensional families are generated by half-twists.
Chenglong Yu

YMSC, Tsinghua University

Ingredients in the Classification
Local Torelli for equisingular deformation of cyclic covers.
Stability and moduli dimension in GIT give the Hodge number.
The classification method works for toric base or homogeneousvariety.
Refinements relation comes from a generalization of
Clemens-Schmid long exact sequence by Kerr-Laza.
The monodromy group is arithmetic subgroup in PU(1, n) byBorel extension.
Half twist is Y ′ = (Y × E )/(Z/3Z), where E is the elliptic
curve with j(E ) = 0.

Chenglong Yu

YMSC, Tsinghua University

Relation to Deligne-Mostow
Most of the examples are Deligne-Mostow ball quotients up tofinite index. Consider S → P1 × P1 and D1 ∈ |O(3, 1)| andD2 ∈ |O(0, 2)|. The branching divisor D is as follows.
AG H

KCBFEIJD

Chenglong Yu

YMSC, Tsinghua University

The fibration S → P1 is isotrivial elliptic fibration with 6
singular fibers in both directions.
The singular fibers in first projection gives rise toDeligne-Mostow tuple µ = ( 13 , 13 , 31 , 13 , 31 , 13 ) by Kodaira.
The second projection gives rise to Deligne-Mostow tupleν = ( 23 , 23 , 61 , 16 , 61 , 16 ).
Corollary: the two Deligne-Mostow lattices are the same up tofinite index (commensurable).
In dimension 3, most examples are isotrivial fibrations of K 3
surfaces. Singular fibers give the Deligne-Mostow data.

Chenglong Yu

YMSC, Tsinghua University

Commensurability relationsTheorem (Deligne-Mostow, Sauter (1980s))
Commensurability pairs Γµ ∼ Γν in PU(1, 2) with explicit indices.
12µ = (a, a, b, b, 1 − 2a − 2b),
ν = (1 − b, 1 − a, a + b − 12 , a + b − 12 , 1 − a − b).
µ = ( 12 − a, 12 − a, 12 − a, 16 + a, 2( 16 + a)),ν = ( 16 , 16 , 61 , 56 − a, 23 + a).
The pairs were found by Mostow with computer investigation.
Kappes-Möller (2012),McMullen (2013) proved that thosepairs provide all commensurability classes for non-arithmeticDeligne-Mostow lattices.
Commensurability invariants are adjoint trace fields and ratios
of Chern numbers similar as Hirzebruch proportionality.

Chenglong Yu

YMSC, Tsinghua UniversityTheorem (Y.-Zheng)
Commensurabilities with explicit indices. (Not necessarily discrete)1

n = 2, reprove Deligne-Mostow, Sauter.

2n = 3, two infinite series.
µ=( 16 , 61 , 16 , 61 , 1 − a, 13 + a)

ν = (a, a, a, 23 − a, 32 − a, 23 − a)andµ = ( 12 − a, 21 − a, 12 − a, 12 − a, 2a, 2a)ν=3Chenglong Yu( 21 − a, 21 − a, 21 − a, a, a, 12 + a)n ≥ 3, finite pairs.
YMSC, Tsinghua University

Cyclic covers of Calabi-Yau type
The proof is based on moduli spaces M of Calabi-Yau type
cyclic covers over (P1 )m .
Y : y d = (f1 )a1 · · · (fk )ak
The Calabi-Yau condition hχm,0 (Y ) = 1, and ball-typeconditions
Xdeg fi = (3, · · · , 3)X
µi deg fi = (1, · · · , 1)
Asymmetric solutions give rise to commensurability pairs.
Chenglong Yu

YMSC, Tsinghua UniversityReprove Deligne-Mostow, Sauterdeg f1 = (2, 1),deg f2 = (1, 0),deg f3 = deg f4 = (0, 1),

2a1 + a2 = a1 + a3 + a4 = d
Then Y admits two fibrations with five singular fibres.

Chenglong Yu

YMSC, Tsinghua University

Commensurability invariants
The arithmetic Deligne-Mostow lattices Γµ are related to
K = (Q[ζd ] ∩ Q)-algebraic groups PU(hµ ).
Γµ ∼ Γν if and only if PU(hµ ) ∼= PU(hν ) as K -algebraic
groups.
In Deligne-Mostow theory with n ≥ 2, this is equivalent toQ[ζd ] being the same and hµ conformal to hν , which can bedetermined by lattice invariants.
When n = 1, this is no longer true. When dµ = 4, dν = 6,
there are PU(hµ ) ∼= PU(hν ) as Q-algebraic groups.
This approach does not give commensurability indices.

Chenglong Yu

YMSC, Tsinghua University

Thank you!
Happy birthday, Professor Yau!

Chenglong Yu

YMSC, Tsinghua University