In algebraic geometry, the abundance conjecture is a conjecture in
birational geometry
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...
, more precisely in the
minimal model program
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its orig ...
,
stating that for every
projective variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
with
Kawamata log terminal singularities over a field
if the
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, it ...
is
nef, then
is
semi-ample.
Important cases of the abundance conjecture have been proven by
Caucher Birkar
Caucher Birkar ( ku, کۆچەر بیرکار, lit=migrant mathematician, translit=Koçer Bîrkar; born Fereydoun Derakhshani ( fa, فریدون درخشانی); July 1978) is an Iranian peoples, Iranian Kurds, Kurdish mathematician and a profes ...
.
References
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Algebraic geometry
Birational geometry
Unsolved problems in geometry
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