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The period map for cubic threefolds

Published online by Cambridge University Press:  17 July 2007

Eduard Looijenga  and
Rogier Swierstra
Eduard Looijenga
Affiliation:
Mathematisch Instituut, Universiteit Utrecht, PO Box 80.010, NL-3508 TA Utrecht, The Netherlands looijeng@math.uu.nl
Rogier Swierstra
Affiliation:
Mathematisch Instituut, Universiteit Utrecht, PO Box 80.010, NL-3508 TA Utrecht, The Netherlands swierstra@math.uu.nl
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Abstract

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Allcock, Carlson and Toledo defined a period map for cubic threefolds which takes values in a ball quotient of dimension 10. A theorem of Voisin implies that this is an open embedding. We determine its image and show that on the algebraic level this amounts to identification of the algebra of $\operatorname{SL}(5,\mathbb{C})$-invariant polynomials on the representation space $\operatorname{Sym}^3(\mathbb{C}^5)^*$ with an explicitly described algebra of meromorphic automorphic forms on the complex 10-ball.

Keywords

cubic threefoldcubic fourfoldball quotientperiod map32G20 (primary)14J3032N15 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 4 , July 2007 , pp. 1037 - 1049
Copyright
Foundation Compositio Mathematica 2007