Moduli of varieties of general type

By J. Kollár. 2010

143 citations recorded by the source corpus.

(1.1) Identify a class of objects whose moduli theory is nice. In some cases the answer is obvious, for instance, we should study the moduli theory of smooth projective curves. In other cases, it took some time to understand what the correct objects should be: Abelian varieties should be replaced by polarized Abelian varieties, K3 surfaces by marked K3 surfaces and vector bundles by stable vector bundles. We see later that smooth, projective varieties of general type should be replaced by their canonical models. We discuss this in Section 2. In all these cases, we get non-proper moduli spaces. This is inconvenient for many reasons, for instance it is hard to count objects with various properties. To remedy this, one should look for a compactification whose points have clear geometric meaning.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/00b6ecb53d6ec984200f4ae5ab737b827182f28f

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Citations in the corpus

ACC for log canonical thresholds (2012, background): “3) is to prove the boundedness of the moduli functor for canonically polarised varieties; see [26].”
Explicit Gromov-Hausdorff compactifications of moduli spaces of K\"ahler-Einstein Fano manifolds (2017, background): “These abstract Fano K-moduli spaces KM generalize analogous compactifications of moduli of varieties with positive canonical bundle (KSBA compactifications [36]), and thus they are important spaces to study also for purely algebro-geometric reasons.”
Semi-positivity in positive characteristics (2012, background): “PPLICATIONS 4.A. Projectivity of proper moduli spaces Recently there have been great advances in constructing moduli spaces of varieties (or pairs) of (log-)general type in characteristic zero, c.f., [Kol10]. However, the method is not new. It has been worked out in [Kol90] and is as follows. First,”
Semipositivity theorems for moduli problems (2012, methodology): “The main purpose of this paper is to give a proof of the following “folklore statement” (see, for example, [Ale96], [Kar00], [Kov09], [AH11], [Kol13a], and [Fn11b]) based on [FF14] (see also [FFS14]).”
Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, background): “Let now M denote the moduli space of all n-dimensional stable varieties with a ﬁxed Hilbert polynomial [ Kol13, Kol ].”