  These are the lecture notes for the Morningside Center of Mathematics
Geometry Summer School on August 15-20, 2022. These lectures sketch the results
by Yau, Demailly-Paun, the author, and Datar-Pingali about generalized
Monge-Amp\`ere equations and subvarieties and aim to use these results to study
the Hodge conjecture.

Differential Geometry;Algebraic Geometry;Complex Variables