ScalingStacks

Theorem 1.2 . [0596]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context Β· Original author HTML

Theorem 1.2.

Let XX be an nn-dimensional proper algebraic variety over KK and 𝔛\mathfrak{X} a strongly nondegenerate polystable formal model of XanX^{\textup{an}} over K∘K^{\circ} with associated skeleton Ξ”\Delta. Let Ο„\tau be an nn-dimensional open face of Ξ”\Delta with associated point SS in the special fibre of 𝔛\mathfrak{X}. Let hh be a convex function on Ο„\tau and denote by π’ͺΒ―h∘p𝔛\overline{\mathcal{O}}^{h\circ p_{\mathfrak{X}}} the trivial line bundle on the strictly KK-analytic space pπ”›βˆ’1​(Ο„)p_{\mathfrak{X}}^{-1}(\tau) endowed with the metric given by βˆ₯1βˆ₯=eβˆ’h∘p𝔛\|1\|=e^{-h\circ p_{\mathfrak{X}}}. Then

c1(π’ͺΒ―h∘p𝔛)n=[K~(S):K~]β‹…n!β‹…MA(h)c_{1}\left(\overline{\mathcal{O}}^{h\circ p_{\mathfrak{X}}}\right)^{n}=[\tilde{K}(S):\tilde{K}]\cdot n!\cdot\MA(h)

on pπ”›βˆ’1​(Ο„)p_{\mathfrak{X}}^{-1}(\tau).

Original mathematics by the credited authors. Source-backed reader collection; mathematical self-containment is not assessed.