The unboundedness locus L(u) of a plurisubharmonic function u is the set of points x∈X such that u is unbounded in every neighbourhood of x. It always contains the polar locus of u. One knows that the polar locus of a non-trivial psh function is of zero Lebesgue-measure for instance, but is there anything similar known about L(u)?
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$L(u)$ can be the whole domain. In dimension $1$, take a dense countable set $\{ z_k\}$ and consider the (pluri) subharmonic function $\sum_k a_k\log|z-z_k|$, where $a_k>0$ tend to zero sufficiently fast. In other words, a (pluri) polar set where the (pluri) subharmonic function equals $-\infty$ can be dense.
answered Jul 9, 2014 at 20:41
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$\begingroup$ Thanks; so L(u) is as large as can be for a general plurisubharmonic function. I am particurlarly interested in understanding plurisubharmonic functions whose derivatives are locally in $L^2$. Is for this subclass or for other similar sublasses L(u) smaller? $\endgroup$– MateiJul 11, 2014 at 18:14
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$\begingroup$ I think $\sum_k a_k(-\log(-log|z-z_k|))$ does the same job and has derivatives in $L^2$. $\endgroup$– MateiJul 12, 2014 at 5:20
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