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Weak positivity and the stability of certain Hilbert points. III. (English) Zbl 0746.14014
In parts I and II of this paper [Invent. Math. 96, No. 3, 639-667 (1989; Zbl 0695.14006) and 101, No. 1, 191-223 (1990; Zbl 0721.14007)] the author used the concept of “weak positivity” to construct quasi- projective moduli spaces for canonically polarised compact complex manifolds. The object of the present paper is to extend this to compact complex manifolds with arbitrary polarisation. In fact, the author constructs a quasi-projective coarse moduli scheme \(M\) parametrising pairs, consisting of a compact complex manifold \(F\) with numerically effective canonical sheaf and an ample invertible sheaf \(H\) with given Hilbert polynomial, up to isomorphisms preserving the ample sheaf. This is not quite the desired result; for this one would need to take the quotient of \(M\) by \(\text{Pic}^ \tau\) and, at the time of writing, the author was not able to construct this quotient as a quasi-projective scheme. [This problem has now been solved; see the author, Math. Ann. 289, No. 2, 297-314 (1991; Zbl 0729.14010).]
As in the previous parts, the same result holds for normal Gorenstein varieties with at most rational singularities, provided that the moduli functor is bounded and separated.

14J10 Families, moduli, classification: algebraic theory
14D20 Algebraic moduli problems, moduli of vector bundles
14D22 Fine and coarse moduli spaces
Full Text: DOI EuDML
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