Weight functions on non-archimedean analytic spaces and the Kontsevich-Soibelman skeleton

By M. Mustaţă, Johannes Nicaise. 2012

106 citations recorded by the source corpus.

We associate a weight function to pairs consisting of a smooth and proper variety X over a complete discretely valued field and a differential form on X of maximal degree. This weight function is a real-valued function on the non-archimedean analytification of X. It is piecewise affine on the skeleton of any regular model with strict normal crossings of X, and strictly ascending as one moves away from the skeleton. We apply these properties to the study of the Kontsevich-Soibelman skeleton of such a pair, and we prove that this skeleton is connected when X has geometric genus one. This result can be viewed as an analog of the Shokurov-Kollar connectedness theorem in birational geometry.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/46fe4da635cf18e1c7db437a63a73285b6a63a28

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In Geometry and Topology

1. Mattias Jonsson
2. S. Boucksom
3. Chi Li
4. C. Favre
5. Chenyang Xu
6. Johannes Nicaise
7. Robert J. Berman
8. M. Mustaţă

In Mathematical Physics

1. Mattias Jonsson
2. S. Boucksom
3. C. Favre
4. Chi Li
5. Chenyang Xu
6. Johannes Nicaise
7. J. Kollár
8. Giulia Saccà

In Statistical and Nonlinear Physics

1. Chenyang Xu
2. S. Boucksom
3. Mattias Jonsson
4. Giulia Saccà
5. Claire Voisin
6. J. Kollár
7. R. Laza
8. C. Favre

Citations in the corpus

• Remarks on degenerations of hyper-K\"ahler manifolds (2017, methodology): “Relevant for us is the fact that associated to a K-trivial degeneration X/∆ there is an intrinsic (depending only on X ∗/∆∗) topological space, that we call (following [59]) the essential skeleton, Sk(X ) associated to the degeneration.”
• Singular semipositive metrics in non-Archimedean geometry (2011, background): “regarding Appendix B. Finally, we would like to thank the anonymous referee for a careful reading and many useful remarks. 2A simpler proof of the relevant vanishing theorem appeared very recently in [MN12], which was posted after the rst version of the present paper. 6 SEBASTIEN BOUCKSOM, CHARLES F”
• The essential skeleton of a degeneration of algebraic varieties (2013, methodology): “n Y such that KY +∆ is Q-linearly equivalent to f∗(K X +(Xs) red) and f∗∆ = (Xs) red. THE ESSENTIAL SKELETON OF A DEGENERATION OF ALGEBRAIC VARIETIES 5 (2.2.5)We will use the following notations from [MN13]. If X is a normal model of Xover C, xis a point of Xcη and Dis a divisor on X that is support”
• Tropical and non-Archimedean limits of degenerating families of volume forms (2016, background): “ shows that the lsc metric A X:= ˚ Klog X=S + A X (5.4) on Kan X is independent of X. This is a special case of Temkin’s canonical metrization of the canonical bundle [Tem14].8 The weight function of [MN15] associated to a pluricanonical form !2H0(X;mK X) is the function A X 1 m logj!jon Xan. 5.7. T”