A well-known theorem by Y.-T. Siu states that plurigenera are invariant under smooth projective deformation. More generally, we can ask whether or not a similar invariance is also true for smooth deformation of Kähler manifolds. However, such a general question is still an open problem. 
In the lectures, we discuss the generalized problem for Kähler manifolds from variational points of view. Actually, given a pluricanonical section in the central fiber, by fixing its perturbed section in the nearby fiber, we consider a suitable functional for variation of by diffeomorphisms such that its critical point allows us to obtain a holomorphic section of the nearby fiber.

Some basic knowledge of Kähler geometry

[1] J.-P. Demailly: Structure theorems for projective and Kähler varieties, Lectures given at the PCMI Graduate Summer School held at Park City in July 2008. 
[2] J. Morrow and K. Kodaira: Complex Manifolds, AMS Chelsea Publishing Vol.355 (1971). 
[3] Y.-T. Siu: Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type,
in Complex Geometry, dedicated to H. Grauert, Springer (2002), 223-277. 
[4] Y..-T. Siu: Some Recent Transcendental techniques in algebraic and complex geometry, arXiv: math/0212402, ICM2002, Vol. 1, 439-448.