The formal expansion kernel conjecture for logarithmic crystalline cohomology

Let XX be a smooth toric variety of dimension nn, let YY be a smooth hypersurface, and consider logarithmic crystalline cohomology Hcrisn(X0,Y0)H^n_{\operatorname{cris}}(X_0,Y_0) together with the formal expansion map

P ⁣:HDRn(X,Y)RHDRn(R[[t1,,tn]]/R).P\colon H^n_{\operatorname{DR}}(X,Y)\otimes R_\infty\to H^n_{\operatorname{DR}}(R_\infty[[t_1,\ldots,t_n]]/R_\infty).

Let Un1U_{\leq n-1} denote the slope-at-most-n1n-1 subcrystal. The formal expansion kernel conjecture. Un1U_{\leq n-1} is the kernel of the formal expansion map PP. This conjecture identifies the slope filtration piece with the classes vanishing under local formal expansion, linking the Hodge-theoretic slope decomposition to Katz's local expansion method. The supplied text does not state whether the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

An Huang, Bong Lian, Shing-Tung Yau and Chenglong Yu, “Hasse-Witt matrices, unit roots and period integrals”, arXiv:1801.01189 (2018).

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