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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.

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