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Positivity of the Chow-Mumford line bundle for families of K-stable Q-Fano varieties

Recent Progress in Moduli Theory May 06, 2019 - May 10, 2019

May 07, 2019 (11:00 AM PDT - 12:00 PM PDT)
Speaker(s): Zsolt Patakfalvi (École Polytechnique Fédérale de Lausanne (EPFL))
Location: MSRI: Simons Auditorium
  • Fano

  • CM line bundle

  • K-stable

  • Moduli space

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification



The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of polarized varieties, in particular on the base of families of Q-Fano varieties (that is, Fano varieties with klt singularities). It is conjectured that the CM line bundle yields a polarization on the conjectured moduli space of K-polystable Q-Fano varieties. This boils down to showing semi-positivity and positivity statements about the CM-line bundle for families with K-semi-stable and K-polystable Q-Fano fibers, respectively. I present a joint work with Giulio Codogni where 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 very 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. I also present applications to fibered Fano varieties.

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