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On the body of ample angles of asymptotically log Fano varieties


In dimension two, we reduce the classification problem for asymptotically log Fano pairs to the problem of determining generality conditions on certain blow-ups. In any dimension, we prove the rationality of the body of ample angles of an asymptotically log Fano pair, i.e., these convex bodies are always rational polytopes.

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Correspondence to Yanir A. Rubinstein.

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This note grew from interactions between J.M.G. and Y.A.R, Y.A.R.’s lecture, and the ensuing conversations between P.C. and Y.A.R. at the conference on “Birational Geometry, Kähler–Einstein Metrics and Degenerations" that took place in November 2019. Thanks go to J. Park and POSTECH for the excellent conference and hospitality. Thanks to I.A. Cheltsov for co-organizing the conference as well as many helpful discussions. The research of P.C. was supported by an EPSRC fellowship.

The research of Y.A.R. was supported by NSF grants DMS-1515703,1906370 and the Rosi & Max Varon Visiting Professorship (Fall 2019 and Spring 2020) at the Weizmann Institute of Science to which he is grateful for the excellent research conditions.

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Cascini, P., Martinez-Garcia, J. & Rubinstein, Y.A. On the body of ample angles of asymptotically log Fano varieties. Rend. Circ. Mat. Palermo, II. Ser (2022).

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  • Asymptotically log Fano varieties
  • Asymptotically log Del Pezzo surfaces
  • Body of ample angles

MSC subject codes:

  • 14J45
  • 14J26 (primary)
  • 14J10
  • 14E05 (secondary)