We study pushforwards of log pluricanonical bundles on projective log canonical pairs
over the
complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell
in the log canonical setting. We show two effective global generation results. First,
when
surjects onto a projective variety, we show a quadratic bound for
generic generation for twists by big and nef line bundles. Second, when
is
fibered over a smooth projective variety, we show a linear bound for twists by ample
line bundles. These results additionally give effective nonvanishing statements. We
also prove an effective weak positivity statement for log pluricanonical bundles in this
setting, which may be of independent interest. In each context we indicate over which
loci positivity holds. Finally, using the description of such loci, we show an effective
vanishing theorem for pushforwards of certain log-sheaves under smooth
morphisms.
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