Birational Geometry of Algebraic Varieties

By Shigeetj Iitaka. 2010

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K-STABILITY OF FANO MANIFOLDS WITH NOT SMALL ALPHA INVARIANTS (2016, methodology): “For the theory of minimal model program, we refer the readers to the book [KM98]; for the theory of toric geometry, we refer the readers to the book [Ful93].”
A geometric characterization of toric varieties (2016, unknown): “We will follow the terminology from [22].”
F-singularities via alterations (2011, background): “45 in [KM98], this valuation ring can be realized as OZ,E for some prime divisor E on θ : Z −→X as above, so that ν = ordEi for some i.”
Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, methodology): “Following the by now common terminology of Mori theory and the minimal model program (cf e.g. [ KM98 ]), a pair ( X, D ) is by deﬁnition a complex normal projective variety X carrying a Weil Q -divisor D (not necessarily eﬀective).”
Categorical geometric skew Howe duality (2009, background): “es π : Y →X one has π∗(OY ) = OX. Given a resolution π : Y →X of singularities, to show that π∗(OY ) = OX it suﬃces to show that X is normal, Cohen-Macaulay, and that R0π∗ωY ∼=ωX (see Theorem 5.10 of [KM]). But if one can show that R0π ∗ωY and ωX are isomorphic up to tensoring by a line bundle then ”