Quotients of the complex ball by discrete groups

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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“…see Kirwan et al [43]). For us, it is important to note that the Picard number of M is 1 (see Mumford [51,pg.…”

Section: Consider a Minimal Blow-up Of B/γ Such That The Pull-back Ofmentioning

confidence: 93%

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

Casalaina-Martin¹,

Laza²

2009

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

118

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“…More complicated examples involving moduli spaces of K3 surfaces and of vector bundles over a Riemann surface have been worked out in [16] and [17] (see also [18]). …”

mentioning

confidence: 99%

Rational intersection cohomology of quotient varieties

Kirwan

1986

Invent Math

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Geometric invariant theory assigns a projective "quotient" variety to any linear action of a complex reductive Lie group G on a complex projective variety X. In this paper a procedure is described for computing the dimensions of the rational intersection homology groups of the quotient when X is nonsingular (see Theorem 3.1). The only condition which the action must satisfy is that the set of stable points for the action is not empty.The basic idea goes like this. Let the geometric invariant theory quotient of X by G be denoted by X//G to distinguish it from the ordinary topological quotient space X/G. In 1-12] a formula was given for the Betti numbers of the quotient in the special case when every semistable point is (properly) stable. In this case the quotient is everywhere locally isomorphic to the quotient of a nonsingular variety by a finite group action and hence its rational intersection cohomology groups are isomorphic to its ordinary rational cohomology groups. On the other hand given any linear action of G on X such that the set of stable points is nonempty, there is a systematic way to blow up X along a sequence of nonsingular subvarieties to obtain a variety X with a linear action of G lifting the action on X such that every semistable point of ~" is stable (see [13]). The quotient X//G may be thought of as a partial desingularisation of X//G (the more serious singularities have been resolved) and its Betti numbers can be computed by the method described in [12]. Finally in this paper it is shown how to relate the intersection Betti numbers of X//G to the Betti numbers of X//G (Theorem 3.1). Thus we end up with a procedure for computing the dimensions of the rational intersection cohomology groups of the quotient X//G in terms of the rational cohomology of certain nonsingular linear sections Z of X and the rational cohomology of the classifying spaces of certain reductive subgroups H of G, provided that we know how no(H ) acts on H*(Z) for appropriate Z and H acting on Z.The formulas for these intersection Betti numbers can become very complicated and involve many terms, especially as the rank of G increases. How-1 Current address: Mathematical Institute, Oxford, UK

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“…Though it is easy to determine the various chambers, it is more difficult to compute the quotients associated to them. In [KLW87], some of the quotients are computed for n = 5, 6, 7, 8. Using the techniques from the previous sections, we are able to compute these quotients for chambers contained in certain subcones of C G (X).…”

Section: Examplementioning

confidence: 99%

A correspondence of good G-sets under partial geometric quotients

Schmitt

2017

Beitr Algebra Geom

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Abstract. For a complex variety X with an action of a reductive group G and a geometric quotient π : X → X by a closed normal subgroup H ⊂ G, we show that open sets of X admitting good quotients by G = G/H correspond bijectively to open sets in X with good G-quotients. We use this to compute GIT-chambers and their associated quotients for the diagonal action of PGL2 on (P 1 ) n in certain subcones of the PGL2-effective cone via a torus action on affine space. This allows us to represent these quotients as toric varieties with fans determined by convex geometry.

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Quotients of the complex ball by discrete groups

Cited by 19 publications

References 6 publications

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

Rational intersection cohomology of quotient varieties

A correspondence of good G-sets under partial geometric quotients

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