تقرير
Weighted K-stability for a class of non-compact toric fibrations
| العنوان: | Weighted K-stability for a class of non-compact toric fibrations |
|---|---|
| المؤلفون: | Cifarelli, Charles |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Differential Geometry, Mathematics - Algebraic Geometry, Mathematics - Symplectic Geometry |
| الوصف: | We study the weighted constant scalar curvature, a modified scalar curvature introduced by Lahdili depending on weight functions $(v, w)$, on certain non-compact semisimple toric fibrations, a generalization of the Calabi Ansatz defined by Apostolov--Calderbank--Gauduchon--T{\o}nnesen-Friedman. We show that the natural analog of the weighted Futaki invariant of Lahdili can under reasonable assumptions be interpreted on an unbounded polyhedron $P \subset \mathbb{R}^n$ associated to $M$. In particular, we fix a certain class $\mathcal{W}$ of weights $(v, w)$, and prove that if $M$ admits a weighted cscK metric, then $P$ is K-stable, and we give examples of weights on $\mathbb{C}^2$ for which the weighted Futaki invariant vanishes but do not admit $(v, w)$-cscK metrics. Following Jubert, we introduce a weighted Mabuchi energy $\mathcal{M}_{v,w}$ and show that the existence of a $(v, w)$-cscK metric implies that it $\mathcal{M}_{v,w}$ proper, and prove a uniqueness result using the method of Guan. We show that weighted K-stability of the abstract fiber $\mathbb{C}$ is sufficient for the existence of weighted cscK metrics on the total space of line bundles $L \rightarrow B$ over a compact K\"ahler base, extending a result of Lahdili in the $\mathbb{P}^1$-bundles case. The right choice of weights corresponds to the (shrinking) K\"ahler-Ricci soliton equation, and we give an interpretation of the asyptotic geometry in this case. Comment: Added Corollary 1.3 and its proof, and other minor changes, as per the suggestion of the referee. Final version, to appear in J. Geom. Anal |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2303.03263 |
| رقم الأكسشن: | edsarx.2303.03263 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |