DIFFERENTIAL GEOMETRY OF THE SIEGEL MODULAR THREEFOLD & ALGEBRO-ARITHMETIC DATA OF K3 SURFACES - MICHAEL T. SCHULTZ (UTAH STATE UNIVERSITY) JOINT ...

DIFFERENTIAL GEOMETRY OF THE SIEGEL
MODULAR THREEFOLD & ALGEBRO-ARITHMETIC
DATA OF K3 SURFACES

MICHAEL T. SCHULTZ (UTAH STATE UNIVERSITY)
JOINT MATHEMATICS MEETING 2021
AMS SPECIAL SESSION ON ALGEBRA AND ARITHMETIC GEOMETRY

DG OF THE SIEGEL MODULAR THREEFOLD & AG OF K3 SURFACES

OVERVIEW
1. Period domains for double sextic K3 surfaces

2. Humbert Surfaces and Picard-Rank 18 K3s

3. The Differential Geometric Perspective

Results: Using Mehran’s generalization of the Shioda-Inose structure for K3 surfaces admitting
rational double cover of a Kummer surface, together with high Picard-rank K3 surfaces and the
holomorphic conformal geometry of the Siegel Modular Threefold, we can detect subvarieties
of arithmetic signi cance, i.e., certain Humbert Surfaces.
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PERIOD DOMAINS OF
 DOUBLE SEXTIC K3
 SURFACES
 SECTION I

A con guration of six lines ℓ1, …, ℓ6 in ℙ2 .
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PERIOD DOMAINS OF DOUBLE SEXTIC K3 SURFACES

FUNDAMENTAL CONCEPTS
▸ Recall that an algebraic K3 surface is a smooth, compact, algebraic surface X satisfying
 1
 KX = 0, h (X, X) = 0.
 2
▸ A double sextic K3 is the smooth minimal resolution of a double cover of ℙ branched over six lines ℓ1, …, ℓ6,
 given here in af ne form:
 ℓi(x, t) = ai1 + ai2 x + ai3t, i = 1,…,6

▸ Hence, the Picard-rank ρ ≥ 16 = 15 + 1, the blowup of 15 intersection points pij = ℓi ∩ ℓj and the hyperplane class.

▸ These K3s have been studied extensively: from the geometric, modular, and string theoretic perspective in
 (Clingher, Malmendier, Shaska 2018), from the period domain perspective (Matsumoto, Sasaki, Yoshida 1992),
 and from the perspective of mirror symmetry (Hosono, Lian, Tagaki, Yau 2018, 2019).

▸ When the family ℓ = {ℓ1, …, ℓ6} is tangent to a smooth conic F, the double cover C of F branched at the six
 points of tangency is generically a genus-2 curve, and the K3 surface is then the Kummer surface
 X = Kum(Jac(C)). This implies ρ ≥ 17.
 
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PERIOD DOMAINS OF DOUBLE SEXTIC K3 SURFACES

FUNDAMENTAL CONCEPTS, CONT.
▸ When Y is a K3, A a principally polarized abelian surface, and τ : Y ⤏ Kum(A) is a degree two rational map, there is a tight
 connection between A and the Kummer surface Kum(A) via a Shioda-Inose structure providing Hodge isometries between
 period lattices:
 τ*
 TY(2) TKum(A) ⟺ TY ≅ TA
 20−ρ−1
▸ For X = X(ℓ) a double sextic K3, and a basis Σ 1, …, Σ20−ρ ∈ T X, the period point of X in ℙ is determined by the period
 integrals
 dx ∧ dt
 ∬Σ
 ωj =
 6
 j ∏i=1 ℓi(x, t)
▸ As the con guration ℓ changes, we obtain the period mapping Φ : ℳ → ℙ20−ρ−1, a multivalued holomorphic map de ned on
 the moduli space ℳ of six line con gurations in ℙ2, whose image Φ(ℳ) is a quasi-projective variety of dimension
 dim(Φ(ℳ)) = rank(TX ) − 2.

▸ For X = Kum(Jac(C)), employing a related construction a la (Mehran, 2006), we obtain a rational double cover
 Y = Ya,b,c ⤏ Kum(Jac(C)) as an elliptic K3, realized as a Twisted Legendre Pencil (Hoyt, 1989). Periods are described by
 classical special functions like 2F1, 3F2, Appell′s F2, and GKZ functions; these are closely related to periods of abelian surfaces.
 2
 y = x(x − 1)(x − t)(t − a)(t − b)(t − c)
 
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PERIOD DOMAINS OF DOUBLE SEXTIC K3 SURFACES

AN EXAMPLE
▸ Let A = ℰ × ℰ′ be the product of isogenous elliptic curves. Then X = Kum(A) is of Picard-rank ρ = 19, and there is a degree
 two map τ : Ỹλ ⤏ X with Ỹλ = Yλ,0,∞ the Twisted Legendre Pencil
 2
 y = x(x − 1)(x − t)t(t − λ)
▸ Hence for a suitable basis of TỸλ, the period of Ỹλ realizes the classical Gauss integral representation of 3F2:

 ( )
 dx ∧ dt 1 1 1 1
 ∬
 ω= = 3F2 , , ; 1,1
 x(x − 1)(x − t)t(t − λ) 2 2 2 λ
▸ Consequently, for λ near 0, the period map satis es the third order Picard-Fuchs ODE:
 3 2
 d ω d ω dω
 8λ (λ − 1) 3 + 12λ (3λ − 2) 2 + (26λ − 8)
 2
 +ω=0
 dλ dλ dλ
 1 2
▸ As λ ∈ ℙ − {0,1,∞} varies, the image of the period map lies on non-degenerate quadric in ℙ . This implies that the Picard-

 (2 2 )
 1 1
 Fuchs system decomposes as the outer tensor product of second order ODEs for 2F1 , ; 1 , whose arguments encode the
 isogeny relation ℰ ∼ ℰ′. This is relevant to the Mirror-Moonshine phenomenon for Mn-polarized K3s (Lian, Yau 1996), with
 ⊕2 ⊕2
 Mn = H ⊕ E8(−1) ⊕ ⟨−2n⟩ .
 
 
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PERIOD DOMAINS OF DOUBLE SEXTIC K3 SURFACES

AND NOW, A WORD FROM OUR SPONSOR
▸ The Picard-Fuchs system of a multi-parameter family of K3 surfaces corresponding Kummer surfaces is
 a system of coupled linear PDEs with regular singular points annihilating the period integrals. This is
 an invariant of the variation of Hodge structure.

▸ Via (Mehran, 2006), we may study rational double covers of Kummer surfaces, whose Picard Fuchs
 systems are closely related to the period integrals of abelian surfaces.
 N
▸ When the period domain is realized as a hyperquadric in ℙ , we say the Picard-Fuchs system satis es
 the quadric condition (Sasaki,Yoshida 1988), meaning the differential system decomposes as a
 bilinear form of lower rank systems. For the examples presented here, this decomposition always has
 arithmetic signi cance.

▸ The quadric condition is completely determined by a relatively simple differential geometric invariant
 of the period domain.
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HUMBERT SURFACES
 & PICARD-RANK 18
 K3 SURFACES
 SECTION II

A Kummer Surface, formed from a nodal quartic in ℙ3.

HUMBERT SURFACES & PICARD-RANK 18 K3 SURFACES

DEGENERATING FROM ρ = 17 TO ρ = 18
▸ When the Twisted Legendre Pencil Ya,b,c of ρ = 17 degenerates c → ∞ , the period integral degenerates to Appell’s

 (2 2 2 )
 1 1 1
 classical multivariate hypergeometric function F2 , , ; 1,1 z1, z2 , with z1, z2 rational functions of a, b.

▸ This yields a 2-parameter family of Picard-Rank 18 K3s Ỹa,b = Ya,b,∞.

▸ The original rank 5=22-17 Picard-Fuchs system then degenerates to a rank 4=22-18 system annihilating F2 , which

 (2 2 ) (2 2 )
 1 1 1 1
 decomposes as the outer tensor product 2F1 , ; 1 λ1 ⊠ 2F1 , ; 1 λ2 . This is precisely the quadric

 (2 2 2 )
 1 1 1
 condition for F2 , , ; 1,1 .

 2
▸ Here λ ,
 1 2λ are the elliptic moduli in the Legendre form of an elliptic curve ℰ i : y = x(x − 1)(x − λi).

▸ Hence, we obtain a degree two rational map τ : Ỹa,b ⤏ Kum(ℰ1 × ℰ2). This degeneration of Picard-Rank agrees
 exactly with results established in (Clingher, Doran, Malmendier 2018).

▸ Are there other arithmetically meaningful degenerations to ρ = 18?

HUMBERT SURFACES & PICARD-RANK 18 K3 SURFACES

 HUMBERT SURFACES
 ▸ In the moduli space 2 of principally polarized abelian surfaces (or the Siegel Modular
 Threefold ℍ2 /Γ(2)), the Humbert Surfaces HΔ are described by the projection of the divisor
 2
 az1 + bz2 + cz3 + d(z2
 − z1z3) + e = 0
 2
 a, …, e ∈ ℤ, & Δ = b − 4ac − 4de
 2
 ▸ For Δ = δ , these loci describe principally polarized abelian surfaces (Birkenhake, Wilhelm
 2003)
 (δ,δ)
 A = Jac(C) ℰ1 × ℰ2 .

 2
 ▸ Genus-2 curves C with the Rosenhain form y = x(x − 1)(x − λ1)(x − λ2)(x − λ1λ2) have Jacobians
 that lie in a connected component of H4.

 ▸ Then Kum(A) is of Picard-Rank ρ = 18, with period lattice TKum(A) = H(2) ⊕ ⟨4⟩ ⊕ ⟨−4⟩ .

 ▸ We should be able to degenerate the Picard-Fuchs system down to this loci as well.
 

THIS IS RATHER
COMPLICATED!

SECTION III
 THE DIFFERENTIAL
The differential geometry of hyperquadrics
 GEOMETRIC PERSPECTIVE
 in ℙN makes a surprise appearance!

THE DIFFERENTIAL GEOMETRIC PERSPECTIVE

HOLOMORPHIC CONFORMAL STRUCTURES
▸ Here, all Picard-Fuchs are a system of linear PDEs in n variables of rank n + 2 (for n = 2,3) of the form
 2 2
 ∂ω ∂ω k ∂ω 0
 = gij + Aij + Aij ω = 0
 ∂x ∂x
 i j ∂x ∂x
 1 n ∂x k
 i j
▸ The principal part g = gij dx ⊗ dx determines a holomorphic conformal structure (HCS) on the period
 domain (Sasaki, Yoshida 1988). These structures and orbifold uniformizing differential equations (Yoshida,
 1987) are multi-parameter relatives of holomorphic anomaly equations for elliptic modular surfaces
 (Malmendier, MTS 2020).

▸ The classi cation of complex surfaces admitting HCS was completed by (S. Kobayashi, Ochiai 1982, R.
 Kobayashi, Naruki 1998), and then for projective threefolds by (Jahnke, Radloff 2004).

▸ The quadric condition is completely determined by a rank-3 symmetric, covariant tensor: the Wilcyznski-
 Fubini-Pick form ϕ = hijk dx i ⊙ dx j ⊙ dx k.

▸ When the quadric condition holds, and n ≥ 3, the entire system is determined by the conformal metric: under
 θ i θ
 a normalization det(e gij) = 1, we have for Γjk, Sij the Christoffel symbols and Schouten tensor of e gij:
 i i i 0
 Ajk = Γjk − gjkΓ1n, Aik = − Sik + gikS1n
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THE DIFFERENTIAL GEOMETRIC PERSPECTIVE

DETECTING CERTAIN HUMBERT SURFACES VIA CONFORMAL GEOMETRY
▸ The Humbert surface H4 satis es the quadric condition: this is a manifestation of the classical
 Jacobi Reduction, which relates the Gauss Hypergeometric function with Lauricella’s multivariate
 hypergeometric function FD.

▸ At the level of Kummer surfaces, this is dif cult to see using purely algebro-geometric techniques
 from the rst part of the talk.

▸ However, the Picard-Fuchs system corresponding to H4 is determined by the pullback of the
 conformal metric (Hara, Sasaki, Yoshida 1989)
 g = (λ1 − λ2)λ3(λ3 − 1) dλ1 ⊙ dλ2 + (λ2 − λ3)λ1(λ1 − 1) dλ2 ⊙ dλ3 + (λ3 − λ1)λ2(λ2 − 1) dλ1 ⊙ dλ3
 along the inclusion map H4 ↪ ℍ2 /Γ(2), (λ1, λ2, λ1λ2) → (λ1, λ2, λ3).

▸ The decomposition of this rank-4 system into the outer tensor product of ODEs will reveal the
 transformation corresponding to the (2,2)-isogeny Jac(C) → ℰ1 × ℰ2.
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OUTLOOK

FUTURE DIRECTIONS
▸ Motivated by purely algebro-geometric and arithmetic questions, the conformal
 differential geometry of period domains - in particular, the Siegel Modular
 Threefold - allows for a unique and potentially easier way to study certain Humbert
 surfaces.

▸ Currently working on explicit computations for H4 (connecting to the classical work
 of Jacobi), H9 (classical work of Hermite; Shaska 2001, 2002), and H16 (current work
 of Braeger, Clingher, Malmendier, Spatig).

▸ We expect that similar results will hold more generally for Humbert surfaces HΔ
 2
 with Δ = δ .

I would like to acknowledge the help and encouragement of my advisor,
 Dr. Andreas Malmendier, both with this research and preparing for this talk.

Additionally, I am grateful for the support of a GRCO grant from the school of
 graduate studies at Utah State University.

THANK YOU!