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We introduce a log Picard variety over the complex number field by the method of log geometry in the sense of Fontaine-Illusie, and study its basic properties, especially, its relationship with the group of log version of m-torsors.
We develop the algebraic theory of log abelian varieties. This is Part II of our series of papers on log abelian varieties, and is an algebraic counterpart of the previous Part I (), where we developed the analytic theory of log abelian varieties.
We calculate l-adic nearby cycles in the étale cohomology for families with log smooth reduction using log étale cohomology. In particular, nearby cycles for log smooth families coincide with tame nearby cycles, as L. Illusie expected, and nearby cycles for semistable families depend only on the first infinitesimal neighborhood of the special fiber.
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