Higher Residue Pairing for $p$-adic Isocrystals and the $p$-adic Riemann--Hilbert Correspondence
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2014
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| author | Rahmati, Mohammad Reza |
| author_facet | Rahmati, Mohammad Reza |
| contents | We construct a canonical sesquilinear pairing on the relative crystalline cohomology of a smooth proper family of varieties over a complete discretely valued $p$-adic field. Motivated by the role of Saito's higher residue pairing in the theory of primitive forms and complex variations of Hodge structure, we develop a $p$-adic analogue based on the twisted relative de~Rham--Witt complex. We show that this twisted complex defines a filtered $F$-isocrystal whose cohomology carries a natural flat, Frobenius-compatible, and non-degenerate bilinear form. Its specialization at the uniformizer recovers the classical Grothendieck residue on the special fiber, providing a direct bridge between crystalline geometry and residue theory.
Using the $p$-adic Riemann--Hilbert correspondence of Faltings and Liu--Zhu, we further identify the resulting pairing with the unique flat extension of this residue form to the corresponding $p$-adic local system. The construction is functorial in the family and compatible with base change and $p$-adic comparison isomorphisms. This yields a genuine $p$-adic analogue of Saito's higher residue pairing and supplies foundational ingredients for a prospective theory of $p$-adic primitive forms, $p$-adic TERP structures, and $p$-adic Frobenius manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1405_6243 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Higher Residue Pairing for $p$-adic Isocrystals and the $p$-adic Riemann--Hilbert Correspondence Rahmati, Mohammad Reza Algebraic Geometry We construct a canonical sesquilinear pairing on the relative crystalline cohomology of a smooth proper family of varieties over a complete discretely valued $p$-adic field. Motivated by the role of Saito's higher residue pairing in the theory of primitive forms and complex variations of Hodge structure, we develop a $p$-adic analogue based on the twisted relative de~Rham--Witt complex. We show that this twisted complex defines a filtered $F$-isocrystal whose cohomology carries a natural flat, Frobenius-compatible, and non-degenerate bilinear form. Its specialization at the uniformizer recovers the classical Grothendieck residue on the special fiber, providing a direct bridge between crystalline geometry and residue theory. Using the $p$-adic Riemann--Hilbert correspondence of Faltings and Liu--Zhu, we further identify the resulting pairing with the unique flat extension of this residue form to the corresponding $p$-adic local system. The construction is functorial in the family and compatible with base change and $p$-adic comparison isomorphisms. This yields a genuine $p$-adic analogue of Saito's higher residue pairing and supplies foundational ingredients for a prospective theory of $p$-adic primitive forms, $p$-adic TERP structures, and $p$-adic Frobenius manifolds. |
| title | Higher Residue Pairing for $p$-adic Isocrystals and the $p$-adic Riemann--Hilbert Correspondence |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/1405.6243 |