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Aug 20, 2021 at 16:51 comment added Dmitry Vaintrob @EvgenyShinder In fact, the simplest case of LCI subvariety is $X$ itself and I already don't know the answer for this case ($\omega$ is a rational top-degree form, $C$ is open and bounded).
Aug 20, 2021 at 9:50 comment added Evgeny Shinder Lci subvarieties are locally transverse intersections of Cartier divisors, hence one may try to understand the case of a Cartier divisor first.
Aug 20, 2021 at 4:22 history asked Dmitry Vaintrob CC BY-SA 4.0