  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 M_{0,5}^bar 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 what happens when there
is an elliptic curve of low degree. (In other terminology, this concerns K3
surfaces that are unigonal, hyperelliptic, trigonal, or tetragonal.) It turns
out that the crucial issue is whether an elliptic fibration has a certain
special type of singular fiber.
