Abstract. A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of Allcock-Carlson-Toledo and Looijenga-Swierstra is also considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves.

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We describe the GIT compactification of the moduli space of cubic fourfolds (cubic hypersurfaces in the five dimensional projective space), with a special emphasis on the role played by singularities. Our main result is that a cubic fourfold with only isolated simple (A-D-E) singularities is GIT stable. Conversely, with some minor exceptions, the stability for cubic fourfolds is characterized by this condition.License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf Case α: X is singular along a line and a quartic elliptic curve. Case β: X has two E 8 singularities (of the same modulus).Case γ: X is singular along a conic and has an isolated singularity of E 7 . Case δ: X has three E 6 singularities (of the same modulus). Case : X is singular along a rational normal curve of degree 4; X is stable. Case φ: X is singular along a sextic elliptic curve; X is stable.(See table 3.) Furthermore, the boundary components β and are 3-dimensional and they meet along a surface σ. The surface σ meets the 2-dimensional boundary components γ and φ along a curve τ . Finally, the curve τ meets the 1-dimensional components α and δ in a point ζ (see figures 1 and 2).The description of the GIT compactification M might seem complicated, but we note that there is quite a bit of structure. To start, we note that, as discussed in section 8, the GIT computation for cubic fourfolds is closely related to that for cubic threefolds (Allcock [1] and Yokoyama [31]) and that for plane sextics (Shah [25]). It follows that one can essentially reconstruct the cubic fourfold case from these two lower dimensional cases. At a deeper level, in all three cases mentioned here, the structure of the moduli space is dictated by the Hodge theoretical properties of the varieties under consideration. We only lightly touch on this in section §8.3. Nonetheless, it is quite apparent from our computations that the relationship between the GIT construction and Hodge theoretical construction of the moduli space of cubic fourfolds is very similar to that for low degree K3 surfaces (see [25] , [26] and [17, §8]).A few words about the organization of this paper. A standard GIT analysis consists of three steps. The first one is a purely combinatorial one, and consists of identifying certain maximal subsets of monomials. We discuss this step for cubic fourfolds in section 2. The next step (corresponding to sections 3 and 4 in our text) attaches some geometric meaning to the combinatorial results obtained in the previous step. The results in this step typically describe the stability of hypersurfaces in terms of a "bad flag" (see Theorem 3.2 and the discussion from [21, §4.2]). Unfortunately, this geometric interpretation is rather coarse, so one needs to refine these results. Typically, by using some classification of singularities, one can interpret the existence of bad flags in terms of singularities. In our situation, we divide the analysis in two cases: isolated (section 5) or non-isolated (section 6) singul...

We characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space. It follows then that the geometric invariant theory (GIT) compactification of the moduli space of cubic fourfolds is isomorphic to the Looijenga compactification associated to this arrangement. This paper builds on and is a natural continuation of our previous work on the GIT compactification of the moduli space of cubic fourfolds.

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 ...