Algebraic Geometry Talk Announcement Time/Date: 4pm, 17 September, 2021 Contraction theorem for Kahler manifolds Omprokash Das TIFR. 170921 Abstract Let X be a smooth variety and V a subvariety of X. Under what conditions on X and V we can contract V from X, i.e there is a morphism f:X\to Y such that \dim f(V)<\dim V and X\V is isomorphic to Y\f(V)? This is an important question in birational geometry. In 1920 Castelnuovo provided an answer to this question when \dim X=2, he showed that if X is a smooth projective surface and C is a (1)curve, then C can be contracted to a point. In higher dimensions, for a smooth complex projective variety X, there is a result called the `BasePoint Free Theoremâ€™ which allows us to contraction certain type of subvarieties from X. The Basepoint free theorem is one of the fundamental theorems in the Minimal Model Program (MMP). In this talk I will discuss how to adapt the statement of the Basepoint free conjecture to compact Kahler manifolds, what are the challenges and the known cases in lower dimensions.
