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A review of geometrically defined functions on Newton–Okounkov bodies

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Abstract

This note is a presentation of two functions carrying geometric information recently defined on the Newton–Okounkov body by Boucksom and Chen (Compos Math 147:1205–1229, 2011) and Nystrom (Ann Sci ÉC Norm Supér (4) 47:1111–1161, 2014). None of the material presented here is original, and much can also be found in Boucksom’s (Séminaire Bourbaki, 2012) Bourbaki talk.

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Notes

  1. The ordering used is the lexicographic ordering.

  2. The question of priority between these two papers has been controversial.

  3. or K-semistable if the Donaldson–Futaki invariant of a test configuration is always positive but not necessarily strictly positive.

  4. Under this condition, which implies that \(V_k\) is of big type, we say that \(\oplus _k V_k\) contains an ample series.

  5. Changing the choice of non-vanishing local section will alter the map from \(H^0(kL)\) to \(\mathcal {O}_{X,x}\), but not the associated multi-valuation map.

  6. Leading with respect to the lexicographic order.

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Acknowledgements

The author wishes to thank the anonymous referee, whose careful reading has much improved the presentation of this note.

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  1. Institut Fourier, Université Grenoble Alpes, Grenoble, France

    Catriona Maclean

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  1. Catriona Maclean
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Maclean, C. A review of geometrically defined functions on Newton–Okounkov bodies. Rend. Circ. Mat. Palermo, II. Ser 66, 217–231 (2017). https://doi.org/10.1007/s12215-016-0282-6

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