A geometric characterization of toric varieties
-- Roberto Svaldi, March 4, 2016

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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. In general such bound does not exist. But
when -(K<sub>X</sub>+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. We prove the conjecture and show that if
the bound is achieved, or the number of components is close enough to
said sum, then X is a toric variety and D is close to being the toric
invariant divisor. This is joint work with M. Brown, J. McKernan,
R. Zong.