# zbMATH — the first resource for mathematics

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.

##### MSC:
 14J10 Families, moduli, classification: algebraic theory 14D20 Algebraic moduli problems, moduli of vector bundles 14D22 Fine and coarse moduli spaces
Full Text:
##### References:
 [1] Esnault, H, Viehweg, E.: Effective bounds for semi positive sheaves and for the height of points on curves over complex function fields. Compos. Math. (to appear) · Zbl 0742.14020 [2] Grothendieck, A.: Eléments de géométrie algébrique, III, 1. Publ. Math.,Inst. Hautes Etud. Sci.11, 1-167 (1961) [3] Kollár, J.: Toward moduli of singular varieties. Compos. Math.56, 369-398 (1985) · Zbl 0666.14003 [4] Kollár, J.: Projectivity of complete moduli. J.Differ. Geom. (to appear) · Zbl 0684.14002 [5] Matsusaka, T., Mumford, D.: Two fundamental theorems on deformations of polarized varieties. Am. J. Math.86, 668-684 (1964) · Zbl 0128.15505 [6] Mori, S.: Classification of higher-dimensional varieties.Algebraic Geometry. Bowdoin 1985. Proc. Symp. Pure Math.46, 269-331 (1987) [7] Mumford, D., Fogarty, J.: Geometric Invariant Theory, Second Edition. (Ergebnisse der Math., Vol. 34). Berlin-Heidelberg-New York: Springer 1982 · Zbl 0504.14008 [8] Pjatetskij-?apiro, I.I., ?afarevich, I.R.: A Torelli theorem for algebraic surfaces of type K 3. Math. USSR Izv.5, 547-588 (1971) · Zbl 0253.14006 [9] Popp, H.: Moduli theory and classification theory of algebraic varieties. (Lecture Notes in Math., Vol. 620. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0359.14005 [10] Viehweg, E.: Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces. Algebraic Varieties and Analytic Varieties. (Adv. Stud. Pure Math., Vol. 39, pp. 329-353. North-Holland 1983 · Zbl 0513.14019 [11] Viehweg, E.: Weak positivity and the additivity of the Kodaira dimension II: The local Torelli map. Clasification of Algebraic and Analytic Manifolds. (Progress in Math., Vol. 39, pp. 567-589). Boston: Birkhäuser 1983 · Zbl 0543.14006 [12] Viehweg, E.: Vanishing theorems. J. Reiner Angew. Math.335, 1-8 (1982) · Zbl 0485.32019 [13] Viehweg, E.: Weak positivity and the stability of certain Hilbert points. Invent. Math.96, 639-667 (1989) · Zbl 0695.14006 [14] Viehweg, E.: Weak positivity and the stability of certain Hilbert points, II. Invent. Math.101, 191-223 (1990) · Zbl 0721.14007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.