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On the Kodaira dimension of the moduli space of abelian varieties. (English) Zbl 0508.14038

Author’s introduction: Let \(H_g\) be the Siegel upper half space of genus \(g\), \(A_g=H_g/\mathrm{Sp}(2g, \mathbb Z)\). The purpose of this paper is to prove that for \(g\ge 9\), the moduli space \(A_g\) of principally polarized abelian varieties of genus \(g\) over \(\mathbb C\) is of general type.
By the method of the toroidal compactifications, we can construct a projective variety \(\bar A_g\) such that \(\bar A_g - A_g\) has normal crossing and \(\bar A_g\) has only finite quotient singularities. Resolving these singularities, we get a projective non-singular model \(\tilde{A_g}\). We shall study the extension problems of pluri-canonical differential forms to \(\bar A_g\) and \(\tilde{A_g}\) and prove that there are sufficiently many pluri-canonical forms which extend to \(\tilde{A_g}\) such that the plurigenera \(P_k\) of \(\tilde{A_g}\) grows with the same order as \(k^{g(g+1)/2}\) for \(g\ge 9\), therefore \(A_g\) is of general type.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
11F55 Other groups and their modular and automorphic forms (several variables)
14K05 Algebraic theory of abelian varieties
14J10 Families, moduli, classification: algebraic theory
32N05 General theory of automorphic functions of several complex variables
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References:

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