# zbMATH — the first resource for mathematics

The period map for cubic fourfolds. (English) Zbl 1177.32010
The primitive cohomology of nonsingular cubic fourfold $$Y \subset {\mathbb P}^{5}$$ is located in the middle dimension (four) and has as nonzero Hodge numbers $$h^{3,1} = h^{1,3} = 1$$ and $$h^{2,2}_{0} = 20.$$ The question that remained was the image of this period map. The main goal of this paper is to prove Hassett’s conjecture. But proof yield more, such as a new proof of Voisin’s injectivity theorem. The author shows that Vinberg’s Dynkin diagram of an arithmetic reflection group of hyperbolic type of rank 20 gives an insightful picture of the boundary strata and their incidence relations.
Theorem 4.1. The period map for cubic fourfolds with at most simple singularities, $$P : \dot{M} \to X,$$ is an open embedding with image $$\dot{X}.$$ It identifies the automorphic line bundle restricted to $$\dot{X}$$ with the line bundle $$O_{M^{o}}(2)$$ so that we obtain an isomorphism of $${\mathbb C}$$-algebras $\bigoplus_{k}H^{0}(\dot{{\mathbb D}}, A(k))^{\Gamma} \rightarrow {\mathbb C}[Sym^{3}V^{*}]^{SL(V)}$ which multiplies the degree by 2. The passage to $$Proj$$ makes the above embedding extend to an isomorphism of the GIT completion of $$\dot{M}$$ onto the Baily-Borel type compactification $$\dot{X}^{bb}$$ of $$\dot{X}.$$

##### MSC:
 32G20 Period matrices, variation of Hodge structure; degenerations 14J35 $$4$$-folds 32N15 Automorphic functions in symmetric domains
Full Text:
##### References:
 [1] Allcock, D.: The moduli space of cubic threefolds. J. Algebr. Geom. 12, 201–223 (2003) · Zbl 1080.14531 [2] Beauville, A., Donagi, R.: La variété des droites d’une hypersurface cubique de dimension 4. C.R. Acad. Sci. Paris Sér. I Math. 301, 703–706 (1985) · Zbl 0602.14041 [3] Griffiths, Ph.: On the periods of certain rational integrals. I, II. Ann. Math. (2) 90, 460–495 (1969); ibid. (2) 90, 496–541 (1969) · Zbl 0215.08103 [4] Hassett, B.: Special cubic fourfolds. Compos. Math. 120, 1–23 (2000) · Zbl 0956.14031 [5] Huybrechts, D.: Compact hyperkähler manifolds: basic results. Invent. Math. 152, 209–212 (2003) · Zbl 1029.53058 [6] Laza, R.: The moduli space of cubic fourfolds. J. Algebr. Geom. (to appear). Available at arXiv:0704.3256v1 [math.AG] [7] Laza, R.: The moduli space of cubic fourfolds via the period map. Ann. Math. (to appear). Available at arXiv:0705.0949v1 [math.AG] [8] Looijenga, E.: On the semi-universal deformation of a simple elliptic singularity I, II. Topology 16, 257–262 (1977); ibid. 17, 23–40 (1978) · Zbl 0373.32004 [9] Looijenga, E.: Compactifications defined by arrangements. II. Duke Math. J. 119, 527–588 (2003) · Zbl 1079.14045 [10] Looijenga, E., Swierstra, R.: On period maps that are open embeddings. J. Reine Angew. Math. 617, 169–192 (2008). Available at math.AG/0512489 · Zbl 1161.14007 [11] Scattone, F.: On the compactification of moduli spaces for algebraic K3 surfaces. Mem. Am. Math. Soc. 70(374), (1987), x+86 pp. · Zbl 0633.14019 [12] Vinberg, E.B.: The two most algebraic K3 surfaces. Math. Ann. 265, 1–21 (1985) · Zbl 0537.14025 [13] Voisin, C.: Théorème de Torelli pour les cubiques de $$\mathbb{P}$$5. Invent. Math. 86, 577–601 (1986). Erratum in Invent. Math. 172, 455–458 (2008) · Zbl 0622.14009 [14] Yokoyama, M.: Stability of cubic hypersurfaces of dimension 3 and 4. Preprint, Nagoya University · Zbl 1206.14028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.