Plurisubharmonic function: Difference between revisions

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*If <math>f_1</math> and <math>f_2</math> are plurisubharmonic functions, then the function <math>f=\max(f_1,f_2)</math> is plurisubharmonic.

*If <math>f_1,f_2,\dots</math> is a decreasing sequence of plurisubharmonic functions then its pointwise limit is plurisubharmonic.

*Every continuous plurisubharmonic function can be obtained as the limit of a monotonically decreasing sequence of smooth plurisubharmonic functions. Moreover, this sequence can be chosen uniformly convergent.<ref>R. E. Greene and H. Wu, ''<math>C^\infty</math>-approximations of convex, subharmonic, and plurisubharmonic functions'', Ann. Scient. Ec. Norm. Sup. 12 (1979), 47&ndash;84.</ref>

*The inequality in the usual [[semi-continuity]] condition holds as equality, i.e. if <math>f</math> is plurisubharmonic then <math>\limsup_{x\to x_0}f(x) =f(x_0)</math>.

* Plurisubharmonic functions are [[Subharmonic function|subharmonic]], for any [[Kähler manifold|Kähler metric]].