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The monodromy pairing for logarithmic $1$-motifs

Jonathan Wise

Vol. 4 (2022), No. 4, 587–633

We describe a 3-step filtration on all logarithmic abelian varieties with constant degeneration. The obstruction to descending this filtration, as a variegated extension, from logarithmic geometry to algebraic geometry is encoded in a bilinear pairing valued in the characteristic monoid of the base. This pairing is realized as the monodromy pairing in p-adic, -adic, and Betti cohomologies, and recovers the Picard–Lefschetz transformation in the case of Jacobians. The Hodge realization of the filtration is the monodromy weight filtration on the limit mixed Hodge structure.

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logarithmic geometry, abelian varieties, tropical geometry, monodromy pairing
Mathematical Subject Classification 2010
Primary: 14C22, 14D07, 14F42, 14H40, 14K05
Secondary: 14T05
Received: 24 March 2020
Revised: 8 June 2022
Accepted: 28 June 2022
Published: 15 January 2023
Jonathan Wise
Department of Mathematics
University of Colorado
Boulder, CO
United States