Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics

Abstract

Let $X^n\rightarrow \mathbb{P}^N$ be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of $(X,{\omega_{FS}}|_X)$ restricted to the Bergman metrics is completely determined by the $X$-hyperdiscriminant of format $(n-1)$ and the Chow form of $X$. As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in $G$ if and only if the hyperdiscriminant polytope dominates the Chow polytope for all maximal algebraic tori $H$ of $G$.

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      PAGES = {77--105},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14G40 (14C25 14D25)},
      MRNUMBER = {1420712},
      MRREVIEWER = {Klaus K{ü}nnemann},
      URL = {http://www.numdam.org/item?id=CM_1996__104_1_77_0},
      ZBLNUMBER = {0924.11055},
      }

Authors

Sean Timothy Paul

Department of Mathematics
University of Wisconsin Madison
480 Lincoln Drive
Madison, WI 53706-1388