  We consider smooth, infinitely divisible random fields $(X(t),t\in M)$,
$M\subset {\mathbb{R}}^d$, with regularly varying Levy measure, and are
interested in the geometric characteristics of the excursion sets \[A_u=\{t\in
M:X(t)>u\}\] over high levels u. For a large class of such random fields, we
compute the $u\to\infty$ asymptotic joint distribution of the numbers of
critical points, of various types, of X in $A_u$, conditional on $A_u$ being
nonempty. This allows us, for example, to obtain the asymptotic conditional
distribution of the Euler characteristic of the excursion set. In a significant
departure from the Gaussian situation, the high level excursion sets for these
random fields can have quite a complicated geometry. Whereas in the Gaussian
case nonempty excursion sets are, with high probability, roughly ellipsoidal,
in the more general infinitely divisible setting almost any shape is possible.
