Perfect-pairing conjecture for logarithmic K-theory

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Let XX be as above, let d=dim⁡(X)d=\dim(X), and take integers m,m′,r,r′m,m',r,r' with m,m′≥0m,m'\geq 0, m+m′=2dm+m'=2d, and m−2r=m′−2r′≥0m-2r=m'-2r'\geq 0. The construction in the source gives a pairing

K(X,m,r)×K(X,m′,r′)⟶Q.K(X,m,r)\times K(X,m',r')\longrightarrow\mathbb{Q}.

Perfect-pairing conjecture. This pairing is perfect, and both vector spaces are finite-dimensional over Q\mathbb{Q}. This is proposed as a logarithmic intersection-theoretic counterpart of Poincaré duality; no resolution is stated.

References

Primary source

Kazuya Kato, Chikara Nakayama and Sampei Usui, “Logarithmic Tate conjectures over finite fields”, arXiv:2502.16974 (2025).

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