# Plurisubharmonic functions on Kähler manifolds, intuition?

As the question suggests, what is the intuition for working with plurisubharmonic functions on Kähler manifolds?

In fact plurisubharmonic functions are invariant under holomorphic transformation, so we can define purisubharmonic functions on any Kähler manifolds and in general on any complex manifolds.

The main intuition is in positivity theory. In fact, Lelong showed that if $$u$$ is a plurisubharmonic function then $$dd^cu$$ is a closed positive current with coefficient as Borel measure.

Pierre Lelong, Fonctions plurisousharmoniques et formes différentielles positives. Gordon and Breach, 1965

One of the applications of Plurisubharmonic functions is in the study of degenerate complex monge-ampere operators for finding canonical metric (Kähler-Einstein metric) on singular varieties. In fact the wedge product of two currents is not well defined in general. Bedford and Taylor defined $$(dd^cu)^n$$ for $$u\in \operatorname{PSH}(\Omega)\cap L_\text{loc}^\infty(\Omega)$$. This is highly nontrivial, since one cannot in general multiply distributions.

Another motivation for the study of plurisubharmonic functions is the following informal principle, stated in many places: plurisubharmonic functions play the role of convex functions in complex geometry.

Here is one formula to motivate this principle. If f is a smooth function on a Kähler manifold, then the Levi form of f evaluated on the pair of vectors (u,Ju) is equal to the Hessian of f (thought of as a quadratic form) evaluated on u plus the Hessian of f evaluated on Ju. Here u is a tangent vector and J the complex structure.

Hence convexity in the Riemannian sense implies plurisubharmonicity.

Another instance of this principle: the sublevel sets of a smooth psh function can be made convex locally (in a chart) by a holomorphic change of coordinates.

These ideas are well explained in Chapter 2 of Cieliebak and Eliashberg's book "From Stein to Weinstein and back". See also "Function Theory on Manifolds Which Possess a Pole" by Greene and Wu for a different viewpoint.