This reading seminar focuses on the topics of the recent paper [AJL21] on weighted cscK metrics and semisimple principal fibration construction, by Apostolov-Jubert-Lahdili. Since this paper extends various important recent results on the existence of canonical Kähler metrics, it can serve as a pretext to study them as well, depending on the wishes of the participants.
Semisimple principal fibration construction: from the data of
The emblematic example of such a construction is Calabi's ansatz for extremal metrics on Hirzebruch surfaces [Cal82]. Various generalizations of the Calabi ansatz have been used over the years, a notable illustration being the series [ACG***] comprising examples of Kähler manifolds without extremal Kähler metrics, which are not destabilized by standard (equivariant) test configurations, which motivated the introduction of uniform K-stability.
One of the main results of [AJL21] is the following variational caracterisation of the existence of extremal Kähler metrics on semisimple principal fiber bundles.
Theorem [Extremal metrics on semisimple principal fibrations] : The following are equivalent:
This theorem allows to prove the existence of new canonical Kähler metrics, especially if one uses in addition the resolution of the uniform YTD conjecture by Simon Jubert when the fiber is toric [J21], or the resolution of the YTD conjecture by Han and Li for weighted solitons [HL20]. The proof (in the coercivity implies existence direction) relies on an adaptation of Chen's continuity method to the construction, as well as on the revolutionary results of Chen-Cheng adapted to the extremal setting by He [CC18, He19]. This provides a striking illustration of the relevance of weighted cscK metrics, introduced by Lahdili and studied in a series of papers [Lah**]. Another main result of their paper takes place entirely on the side of weighted cscK metrics, forgetting about fibrations. It only provides a direction in the above theorem, but for general weights.
Theorem [Existence implies weighted coercivity] : Let \((X,\omega_X)\) be a weighted cscK manifold. Then the weighted Mabuchi functional is \(T^{\mathbb{C}}\)-coercive on the space of \(T\)-invariant Kähler metrics in \([\omega_X]\).
The proof builds on the general framework developed by Darvas and Rubinstein [DR17], and requires adapting various key results from the classical case to the weighted case (e.g. Chen Tian's formula for weighted K-energy, the extension of various functionals to finite energy metrics, Berman-Darvas-Lu regularity result [BDL20], etc). A remarkable aspect of the proof, giving a global coherence to the paper is the use of the above semisimple principal fibration construction to extend the functionals to finite energy metrics. The idea has been used previously by Han and Li in the context of weighted solitons, inspired by the much older ideas of Donaldson [Don05] and equivariant cohomology. As one naturally expects, the above theorem has consequences on the K-stability side:
Theorem [Existence implies weighted K-stability] : Let \((X,\omega_X)\) be a weighted cscK manifold, then \((X,[\omega_X])\) is uniformly weighted K-stable.
This last theorem is based on the general framework developed by Boucksom-Hisamoto-Jonsson [BHJ19], Sjostrom Dyrefelt [Sjo20], etc.: one extracts information about non-archimedean functionals from their archimedean versions by studying the slopes along geodesic rays.