Suita conjecture
In mathematics, the Suita conjecture is a conjecture related to the theory of the Riemann surface, the boundary behavior of conformal maps, the theory of Bergman kernel, and the theory of the L2 extension. The conjecture states the following:

It was first proved by  for the bounded plane domain and then completely in a more generalized version by . Also, another proof of the Suita conjecture and 
some examples of its generalization to several complex variables (the multi (high) - dimensional Suita conjecture) were given in  and . The multi (high) - dimensional Suita conjecture fails in non-pseudoconvex domains. This conjecture was proved through the optimal estimation of the Ohsawa–Takegoshi L2 extension theorem. Notes

References 

Several complex variables
Algebraic geometry