Given a pair (X, D), where X is a proper variety and D a
divisor with mild singularities, it is natural to ask how to bound the
number of components of D or rather the sum of their (positive) coefficients.
In general such bound does not exist. But when -(K_X+D) is positive, i.e.
ample (or nef), then a conjecture of Shokurov says this bound should
coincide with the sum of the dimension of X and its Picard number.
When the bound is achieved, Shokurov conjectured that then X would
be a toric variety and D a choice of toric invariant divisor. We prove
a strengthened version of this conjecture and show that it suffices
that the number of components of D is close enough to said sum
for X to be a toric variety and D to be very close to a toric invariant
divisor. Moreover, if time permits I will explain how to use the same kind
of techniques to prove rationality of certain varieties.
This is joint work with M. Brown, J. McKernan, R. Zong.