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This course is an introduction to complex analytic techniques in
algebraic geometry. Topics that will likely be covered include:
(1) Plurisubharmonic functions and closed positive currents, Lelong numbers;
(2) H¨ormander’s L2 estimates for @;
(3) Ohsawa-Takegoshi extension theorem (as a consequence of H¨ormander);
(4) Demailly’s regularization theorem for currents;
(5) Multiplier ideal sheaves and Nadel vanishing theorem;
(6) Applications to algebraic geometry;
(7) Invariance of plurigenera;
(8) Fujita’s approximation theorem;
(9) Boucksom-Demailly-Paun-Peternell’s characterization of uniruled manifolds;
(10) Openness conjecture and strong openness conjecture.
References
[1] Demailly, J.-P. Analytic Methods in Algebraic Geometry, Higher Education Press, Surveys of Modern
Mathematics, Vol. 1, 2010.
[2] Lazarsfeld, R. Positivity in algebraic geometry. I, II. Springer-Verlag, Berlin, 2004.
[3] Guan, Qi’an; Zhou, Xiangyu. A proof of Demailly’s strong openness conjecture. Ann. of Math. (2) 182
(2015), no. 2, 605–616.
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