Semialgebraic horizontal subvarieties of Calabi–Yau type

Abstract

We study horizontal subvarieties Z of a Griffiths period domain D. If Z is defined by algebraic equations, and if Z is also invariant under a large discrete subgroup in an appropriate sense, we prove that Z is a Hermitian symmetric domain D, embedded via a totally geodesic embedding in D. Next we discuss the case when Z is in addition of Calabi-Yau type. We classify the possible VHS of Calabi-Yau type parametrized by Hermitian symmetric domains D and show that they are essentially those found by Gross and Sheng-Zuo, up to taking factors of symmetric powers and certain shift operations. In the weight three case, we explicitly describe the embedding Z ֒→ D from the perspective of Griffiths transversality and relate this description to the Harish-Chandra realization of D and to the Korányi-Wolf tube domain description. There are further connections to homogeneous Legendrian varieties and the four Severi varieties of Zak. Convention:We abbreviate variation of Hodge structure by VHS. All VHS are polarizable/polarized, defined over Q unless otherwise specified, and satisfy the Griffiths transversality condition. A Hodge structure or VHS (of weight n) will be assumed to be effective (h p,q = 0 only for p, q ≥ 0, h n,0 = 0) unless otherwise noted. By a Tate twist, we can always arrange that a Hodge structure is effective. We denote by D a Griffiths period domain. Thus, D = G(R)/K, where G is an orthogonal or symplectic group defined over Q (the group preserving a pair (V, Q), where V is a Q-vector space and Q is a non-degenerate symmetric or alternating form defined over Q) and K is a compact subgroup of G(R), not in general maximal. Semi-algebraic implies Hermitian symmetricOur goal in this section is to prove Theorem 1.Definition 1.1. Let D = G(R)/K be a classifying space for Hodge structures with compact dualĎ = G(C)/P(C), where P(C) is an appropriate parabolic subgroup of G(C). A closed horizontal subvariety Z of D will be called semi-algebraic in D if Z is an open subset of its Zariski closureẐ ⊆Ď. Equivalently, there exists a closed subvarietyẐ of the projective varietyĎ such that Z is a connected component of Z ∩ D. Note that, if Z is semi-algebraic in D, then Z is a semi-algebraic set.Definition 1.2. Let D = G(R)/K be a classifying space for Hodge structures as above, and let Z be a closed horizontal subvariety of D. Let Γ = Γ Z be the stabilizer of Z in G(Z), i.e. Γ = {γ ∈ G(Z) : γ(Z) = Z}. Thus Γ acts properly discontinuously on Z. We call Γ\Z strongly quasi-projective if, for every subgroup Γ ′ of Γ of finite index, the analytic space Γ ′ \Z is quasi-projective, and thus the morphism Γ ′ \Z → Γ\Z is a morphism of quasi-projective varieties. In particular, if Γ\Z is strongly quasi-projective, then Γ\Z is quasi-projective.Remark 1.3.(i) If Γ acts on Z without fixed points and Γ\Z is quasi-projective, then by Riemann's existence theorem Γ\Z is automatically strongly quasiprojective.(ii) If D is Hermitian symmetric, so that the quotient of D by every arithmetic subgroup admits a Baily-Borel compactification, then ...

On some Hermitian variations of Hodge structure of Calabi-Yau type with real multiplication

“…In the two other tube domain cases mentioned above, III 3 and EVII, non-trivial weak real multiplication cannot arise. For the SU(3, 3) case, we showed in [FL11] that every totally real field E 0 can be realized as the endomorphism algebra of a Hermitian VHS of CY type defined over Q. This result is similar to that of van Geemen [vG08] for K3 type, but the representation theory is more involved.…”

“…Work of Gross [Gro94] and Sheng-Zuo [SZ10] shows that every Hermitian symmetric domain D carries a canonical R-variation of Hodge structure V of CY type (cf. also [FL11,§2] for more discussion). Furthermore, every other equivariant R-VHS (or Hermitian VHS) of CY type on D is obtained from V using certain standard constructions (see [FL11,Theorem 2.22]).…”

“…also [FL11,§2] for more discussion). Furthermore, every other equivariant R-VHS (or Hermitian VHS) of CY type on D is obtained from V using certain standard constructions (see [FL11,Theorem 2.22]). For example, each of the four rank 3 Hermitian symmetric tube domains D, namely III 3 , I 3,3 , II 6 , and EVII (corresponding to the real Lie groups Sp(6, R), SU(3, 3), SO * (12), and E 7,3 respectively), carries a weight 3 R-VHS of CY type, with the relevant Hodge number h 2,1 = 6, 9, 15, and 27 respectively, and every primitive irreducible weight 3 Hermitian VHS of CY type which is also of tube type is of this form.…”

“…The weight 2 case, or K3 type, was analyzed by Zarhin [Zar83] and van Geemen [vG08]. A basic invariant measuring the difference between the classification over Q and over R is the algebra E := End Hg (V s ) of Hodge endomorphisms of a general fiber V s of V. In the Calabi-Yau type case, E is either a totally real field or a CM field (see [Zar83] or [FL11,Prop. 3.1]).…”

“…If E = E 0 is a totally real field, we say that the Hermitian VHS has weak real multiplication by E 0 . In weight 3, we showed in [FL11,Theorem 3.18] that there are at most two primitive cases of Hermitian VHS of CY threefold type defined over Q with nontrivial weak real multiplication. These cases correspond to the domains I 3,3 and II 6…”