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**Quasi-projectivity of moduli spaces of polarized varieties.**
*(English)*
Zbl 1068.32011

It has been known for some time that there are reasonable moduli spaces of nonuniruled polarized projective manifolds. Certainly, they are algebraic spaces: locally a finite quotient of an algebraic variety. The main result of this article asserts that these moduli spaces are quasi-projective.

In fact, all that is used is the existence of a moduli space for a class of polarized projective manifolds as a proper quotient of an open subspace of a Hilbert scheme. The tools are from pluripotential theory and especially the use of singular Hermitian metrics. It would be a very satisfactory result. – Unfortunately, it appears that this main result is false. There are three objections to the proof itself due to János Kollár and Eckart Viehweg and carefully reported by Vasile Brînzănescu in MR2081436 (2005h:14089). It seems that only one of these objections can be fixed and, in fact, there is a counterexample [J. Kollár, Ann. Math. (2) 164, No. 3, 1077–1096 (2006; Zbl 1140.14011)].

In fact, all that is used is the existence of a moduli space for a class of polarized projective manifolds as a proper quotient of an open subspace of a Hilbert scheme. The tools are from pluripotential theory and especially the use of singular Hermitian metrics. It would be a very satisfactory result. – Unfortunately, it appears that this main result is false. There are three objections to the proof itself due to János Kollár and Eckart Viehweg and carefully reported by Vasile Brînzănescu in MR2081436 (2005h:14089). It seems that only one of these objections can be fixed and, in fact, there is a counterexample [J. Kollár, Ann. Math. (2) 164, No. 3, 1077–1096 (2006; Zbl 1140.14011)].

Reviewer: Michael G. Eastwood (Adelaide)

### MSC:

32G13 | Complex-analytic moduli problems |

14D20 | Algebraic moduli problems, moduli of vector bundles |

32U05 | Plurisubharmonic functions and generalizations |

32U25 | Lelong numbers |

32W05 | \(\overline\partial\) and \(\overline\partial\)-Neumann operators |

14J15 | Moduli, classification: analytic theory; relations with modular forms |