Positivity of the CM line bundle for families of K-stable klt Fano varieties
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Positivity of the CM line bundle for families of K-stable klt Fano varieties Giulio Codogni1 · Zsolt Patakfalvi2
Received: 14 August 2018 / Accepted: 20 August 2020 © The Author(s) 2020
Abstract The Chow–Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K -poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semipositivity/positivity statements about the CM-line bundle for families with K -semi-stable/K -polystable fibers. We prove the necessary semi-positivity statements in the K -semi-stable situation, and the necessary positivity statements in the uniform K -stable situation, including in both cases variants assuming K -stability only for general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. We also present an application to the classification of Fano varieties. Additionally, our semi-positivity statements work in general for log-Fano pairs.
B Zsolt Patakfalvi
[email protected] Giulio Codogni [email protected]
1
Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della ricerca scientifica, 00133 Rome, Italy
2
EPFL, SB MATHGEOM CAG, MA B3 635 (Bâtiment MA), Station 8, 1015 Lausanne, Switzerland
123
G. Codogni, Z. Patakfalvi
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Technical statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Boundedness of the volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Byproduct statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Similar results in other contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Overview of K-stability for Fano varieties . . . . . . . . . . . . . . . . . . . . . 1.7 Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Semi-positivity statements. . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Nefness threshold, that is, Theorem 1.20. . . . . . . . . . . . . . . . . . . 1.7.3 Positivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Base-changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fiber product notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 General further notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Relative canonical divisor . . . . . . . . . . . . . . . . . . . . . . . . .
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