Compactifications of smooth families and of moduli spaces of polarized manifolds

• 成果类型: Article
• 署名作者: Viehweg, Eckart
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• 发表日期: 2010
• 页码: 809-910
• 关键词: minimal models; general type; VARIETIES; projectivity

摘要

Let M-h be the moduli scheme of canonically polarized manifolds with Hilbert polynomial h. We construct for nu >= 2 with h(nu) > 0 a projective compactification (M) over bar (h) of the reduced moduli scheme (M-h)(red) such that the ample invertible sheaf lambda(nu), corresponding to det(f(*)omega(nu)(X0/Y0)) on the moduli stack, has a natural extension (lambda) over bar (nu) is an element of Pic((M) over bar (h))(Q). A similar result is shown for moduli of polarized minimal models of Kodaira dimension zero. In both cases natural means that the pullback of (lambda) over bar (nu) to a curve phi : C -> (M) over bar (h), induced by a family f(0) : X-0 -> C-0 = phi(-1)(M-h), is isomorphic to det(f(*)omega(nu)(X/C)) whenever f(0) extends to a semistable model f : X -> C. Besides of the weak semistable reduction of Abramovich-Karu and the extension theorem of Gabber there are new tools, hopefully of interest by themselves. In particular we will need a theorem on the flattening of multiplier sheaves in families, on their compatibility with pullbacks and on base change for their direct images, twisted by certain semiample sheaves.