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Transactions of the American Mathematical Society

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Bott vanishing for algebraic surfaces


Author: Burt Totaro
Journal: Trans. Amer. Math. Soc. 373 (2020), 3609-3626
MSC (2010): Primary 14F17; Secondary 14J26, 14J28
DOI: https://doi.org/10.1090/tran/8045
Published electronically: January 28, 2020
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Abstract: Bott proved a strong vanishing theorem for sheaf cohomology on projective space. It holds for toric varieties, but not for most other varieties.

We prove Bott vanishing for the quintic del Pezzo surface, also known as the moduli space $ \overline {M_{0,5}}$ of 5-pointed stable curves of genus zero. This is the first non-toric Fano variety for which Bott vanishing has been shown, answering a question by Achinger, Witaszek, and Zdanowicz.

In another direction, we prove sharp results on which K3 surfaces satisfy Bott vanishing. For example, a K3 surface with Picard number 1 satisfies Bott vanishing if and only if the degree is 20 or at least 24. For K3 surfaces of any Picard number, we have complete results when the degree is big enough. We build on Beauville, Mori, and Mukai's work on moduli spaces of K3 surfaces, as well as recent advances by Arbarello-Bruno-Sernesi, Ciliberto-Dedieu-Sernesi, and Feyzbakhsh.

The most novel aspect of the paper is our analysis of Bott vanishing for K3 surfaces with an elliptic curve of low degree. (In other terminology, this concerns K3 surfaces that are monogonal, hyperelliptic, trigonal, or tetragonal.) It turns out that the crucial issue is whether an elliptic fibration has a certain special type of singular fiber.


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Additional Information

Burt Totaro
Affiliation: Department of Mathematics, University of California Los Angeles, Box 951555, Los Angeles, California 90095-1555
Email: totaro@math.ucla.edu

DOI: https://doi.org/10.1090/tran/8045
Received by editor(s): June 1, 2019
Received by editor(s) in revised form: September 8, 2019, and September 10, 2019
Published electronically: January 28, 2020
Additional Notes: This work was supported by National Science Foundation grant DMS-1701237, and by grant DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester.
Dedicated: For William Fulton on his eightieth birthday
Article copyright: © Copyright 2020 American Mathematical Society