Tropical Compactifications
Jenia Tevelev
Friday 1/28/5

Many important varieties of algebraic geometry
(e.g. various moduli spaces like M_g) are not compact.
For many practical purposes (e.g. to understand
their intersection theory) it is necessary to find
reasonably good compactifications.
One simple example is a complement in P^n
to a hyperplane arrangement. It has rich geometry
and topology related to hypergeometric functions, etc.
But its nice compactification is actually not P^n
but rather a certain blow-up of it.
Another example is to take an action of
a reductive group H on a variety X.
The set of orbits does not have any geometry
but it's always possible to shrink X to an open subset
such that the set of orbits in this subset
is a smooth (but not compact) variety.
Many moduli spaces appear this way.

There are many methods to produce a compactification.
For example, in the equivariant setting one can take
Mumford's GIT quotients, or Chow/Hilbert quotients.
A very general method to recompactify everything
is provided by Mori theory (logcanonical models).
Unfortunately, this approach has several disadvantages:
logcanonical models are difficult to describe,
they are not functorial, and their existence is still conjectural
(it relies on the main conjecture of Mori theory --
the finite generation of the logcanonical ring,
or the existence of flips).

We study a new class of compactifications of very affine varieties
(closed subvarieties of an algebraic torus) defined
by imposing the polyhedral structure on their tropicalization
(a non-archimedean analogue of a complex amoeba of complex analysis).
These compactifications always have divisorial
boundary with combinatorial normal crossings.
Examples include a log canonical model of a complement
to a hyperplane arrangement (including Grothendieck-Knutsen space
M_{0,n} of stable rational curves) and tropical recompactifications
of Kapranov's Chow quotients of Grassmannians.