On compactifying moduli and degenerations of K-trivial varieties

Abstract: Some background review: the KSBA moduli of varieties of ample canonical classes is interpreted via K-stability resp., KE metrics (O’10, resp., Berman-Guenancia’13). A recent trend since 2012 is to establish its Fano analogue, and study their K-stability itself, which still continues to be developed by more and more contributors wonderfully. Luckily, in both cases, K-polystable / KE varieties (should) form projective (compact) moduli schemes. However, nevertheless of general K-moduli expectation, such existence of projective moduli of K-polystable/cscK (polarized) varieties is NOT true “at the boundary”, even for classical K-trivial / Calabi-Yau cases. Indeed, as a general theory, no “canonical” algebro-geometric compactification theory of moduli of polarized CY vars seems established. E.g. An idea pursued and fairly developed is to attach ample extra divisors on the CY vars (to pass to “K:ample”-like situations) and take their “log KSBA” compactifications, but different choice of the extra divisors can lead to different log KSBA compactifications. In our talk, based on our several recent papers (partially j.w.w. Yoshiki Oshima), we discuss the possibilities of still getting “canonical (geometric) compactifications” of the moduli of polarized K-trivial / CY varieties and corresponding "canonical limits", especially giving more explicit conjectures in hyperKahler / K3 case, with certain confirmations. This involves not only classical AG but also DG of collapsing CY metrics, symmetric space theory (Lie, Cartan, .. Satake..), non-archimedean/tropical geometry, and some mirror symmetric phenomena. Examples and pictures will be used for the illustration.

algebraic geometry

Audience: researchers in the topic

ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

The seminar takes place on Tuesdays and Thursdays via Zoom. Zoom passwords are given via mailing list on Fridays. To join the mailing list go to the website.

If you use a calendar system, you can see the individual seminars at bit.ly/zag-seminar-calendar

Times vary to accommodate speakers time zones but times will be announced in GMT time.