\(p\)-adic deformation of algebraic cycle classes. (English) Zbl 1301.19005
This paper deals with Fontaine-Messing’s \(p\)-adic variational Hodge conjecture [J.-M. Fontaine and W. Messing, Contemp. Math. 67, 179–207 (1987; Zbl 0632.14016)] which is a \(p\)-adic analogue of Grothendieck’s variational Hodge conjecture in characteristic zero.
Let \(X\) be a smooth projective variety over the Witt ring \(W\) of a perfect field \(k\) of characteristic \(p > 0\) (with field of fractions \(K\)) and let \(\xi_1\) be an element of the Grothendieck group \(K_0(X_1)_{\mathbb Q}\) of the special fibre \(X_1= X \otimes_W k\) such that the preimage of the Chern character element \({\mathrm{ch}}(\xi_1)\) under Berthelot’s crystalline-de Rham comparison isomorphism \(H^{\scriptscriptstyle \bullet}_{\mathrm{dR}}(X/W) \;\; \tilde{\rightarrow} \;\; H^{\scriptscriptstyle \bullet}_{\mathrm{cris}}(X_1/W)\) lies in \(\bigoplus_r \, F^rH^{2r}_{\mathrm{dR}}(X_K/K)\) where \(F^rH^{2r}_{\mathrm{dR}}(X_K/K)\) denotes the \(r^{\mathrm{th}}\) step in the Hodge filtration on the de Rham cohomology \(H^{2r}_{\mathrm{dR}}(X_K/K)\). Then Fontaine-Messing’s conjecture predicts that there exists an element \(\xi \in K_0(X)_{\mathbb Q}\) such that \({\mathrm{ch}}(\xi |_{X_1}) = {\mathrm{ch}}(\xi_1)\) in \(\bigoplus_r \, H^{2r}_{\mathrm{cris}}(X_1/W)_K\). The line-bundle version of this conjecture has been proved by P. Berthelot and A. Ogus [Notes on crystalline cohomology. Princeton, New Jersey: Princeton University Press (1978; Zbl 0383.14010)].
The main result of the paper under review is that there exists a \(\hat{\xi}\) in the projective limit \(({\mathrm{lim}} \, K_0(X_n))_{{\mathbb Q}}\) such that \(\hat{\xi} |_{X_1} = \xi_1\) in \(K_0(X_1)_{\mathbb Q}\) provided we have \(p > {\mathrm{dim}}(X_1) + 6\). This settles (what the authors call) the deformation part of Fontaine-Messing’s conjecture. What’s left to prove Fontaine-Messing’s conjecture is to lift the pro-element \(\hat{\xi} \in ({\mathrm{lim}} \, K_0(X_n))_{\mathbb Q}\) (or a variant of it) to an element \(\xi \in K_0(X)_{\mathbb Q}\). Unlike for Pic, there is unfortunately no general approach to this algebraisation part of Fontaine-Messing’s conjecture available.
For the proof of their main result, the authors create a continuous Chow theory \({\mathrm{CH}}_{\mathrm{cont}}(X_{\scriptscriptstyle \bullet})\) and first prove the analogue of their main result with \({\mathrm{lim}} \, K_0(X_n)\) replaced with \({\mathrm{CH}}_{\mathrm{cont}}(X_{\scriptscriptstyle \bullet})\). To this end, they glue the Suslin-Voevodsky motivic complex on \(X_1\) [A. Suslin and V. Voevodsky, NATO ASI Ser., Ser. C, Math. Phys. Sci. 548, 117–189 (2000; Zbl 1005.19001)] with the Fontaine-Messing-Kato syntomic complex on \(X_{\scriptscriptstyle \bullet}\) [K. Kato, Adv. Stud. Pure Math. 10, 207–251 (1987; Zbl 0645.14009)] to obtain a motivic pro-complex \({\mathbb Z}_{X_{\scriptscriptstyle \bullet}}(r)\) of the \(p\)-adic formal scheme \(X_{\scriptscriptstyle \bullet}\) associated to \(X\) on the Nisnevich site of \(X_1\). They then define \({\mathrm{CH}}^r_{\mathrm{cont}}(X_{\scriptscriptstyle \bullet}) = H^{2r}_{\mathrm{cont}}(X_1, {\mathbb Z}_{X_{\scriptscriptstyle \bullet}}(r))\) and they construct a certain obstruction map and relate it to the Hodge theoretic properties of the cycle class in crystalline cohomology. To finish the proof of their main result, the authors finally construct a Chern character \({\mathrm{ch}}: K_0^{\mathrm{cont}}(X_{\scriptscriptstyle \bullet}) _{\mathbb Q} \rightarrow \bigoplus_{r \leq {\mathrm{dim}}(X_1)} \, {\mathrm{CH}}^r_{\mathrm{cont}}(X_{\scriptscriptstyle \bullet})_{\mathbb Q}\) and, using deep results from topological cyclic homology theory due to T. Geisser and L. Hesselholt [Trans. Am. Math. Soc. 358, No. 1, 131–145 (2006; Zbl 1087.19003); J. Am. Math. Soc. 19, No. 1, 1–36 (2006; Zbl 1087.19002); Ann. Sci. Éc. Norm. Supér. (4) 37, No. 1, 1–43 (2004; Zbl 1062.19003)], they show that \(\mathrm{ch}\) is an isomorphism for \(p > {\mathrm{dim}}(X_1)+6\).
The paper ends with three appendices about homological algebra, homotopical algebra and a crystalline construction of the motivic complex.
Let \(X\) be a smooth projective variety over the Witt ring \(W\) of a perfect field \(k\) of characteristic \(p > 0\) (with field of fractions \(K\)) and let \(\xi_1\) be an element of the Grothendieck group \(K_0(X_1)_{\mathbb Q}\) of the special fibre \(X_1= X \otimes_W k\) such that the preimage of the Chern character element \({\mathrm{ch}}(\xi_1)\) under Berthelot’s crystalline-de Rham comparison isomorphism \(H^{\scriptscriptstyle \bullet}_{\mathrm{dR}}(X/W) \;\; \tilde{\rightarrow} \;\; H^{\scriptscriptstyle \bullet}_{\mathrm{cris}}(X_1/W)\) lies in \(\bigoplus_r \, F^rH^{2r}_{\mathrm{dR}}(X_K/K)\) where \(F^rH^{2r}_{\mathrm{dR}}(X_K/K)\) denotes the \(r^{\mathrm{th}}\) step in the Hodge filtration on the de Rham cohomology \(H^{2r}_{\mathrm{dR}}(X_K/K)\). Then Fontaine-Messing’s conjecture predicts that there exists an element \(\xi \in K_0(X)_{\mathbb Q}\) such that \({\mathrm{ch}}(\xi |_{X_1}) = {\mathrm{ch}}(\xi_1)\) in \(\bigoplus_r \, H^{2r}_{\mathrm{cris}}(X_1/W)_K\). The line-bundle version of this conjecture has been proved by P. Berthelot and A. Ogus [Notes on crystalline cohomology. Princeton, New Jersey: Princeton University Press (1978; Zbl 0383.14010)].
The main result of the paper under review is that there exists a \(\hat{\xi}\) in the projective limit \(({\mathrm{lim}} \, K_0(X_n))_{{\mathbb Q}}\) such that \(\hat{\xi} |_{X_1} = \xi_1\) in \(K_0(X_1)_{\mathbb Q}\) provided we have \(p > {\mathrm{dim}}(X_1) + 6\). This settles (what the authors call) the deformation part of Fontaine-Messing’s conjecture. What’s left to prove Fontaine-Messing’s conjecture is to lift the pro-element \(\hat{\xi} \in ({\mathrm{lim}} \, K_0(X_n))_{\mathbb Q}\) (or a variant of it) to an element \(\xi \in K_0(X)_{\mathbb Q}\). Unlike for Pic, there is unfortunately no general approach to this algebraisation part of Fontaine-Messing’s conjecture available.
For the proof of their main result, the authors create a continuous Chow theory \({\mathrm{CH}}_{\mathrm{cont}}(X_{\scriptscriptstyle \bullet})\) and first prove the analogue of their main result with \({\mathrm{lim}} \, K_0(X_n)\) replaced with \({\mathrm{CH}}_{\mathrm{cont}}(X_{\scriptscriptstyle \bullet})\). To this end, they glue the Suslin-Voevodsky motivic complex on \(X_1\) [A. Suslin and V. Voevodsky, NATO ASI Ser., Ser. C, Math. Phys. Sci. 548, 117–189 (2000; Zbl 1005.19001)] with the Fontaine-Messing-Kato syntomic complex on \(X_{\scriptscriptstyle \bullet}\) [K. Kato, Adv. Stud. Pure Math. 10, 207–251 (1987; Zbl 0645.14009)] to obtain a motivic pro-complex \({\mathbb Z}_{X_{\scriptscriptstyle \bullet}}(r)\) of the \(p\)-adic formal scheme \(X_{\scriptscriptstyle \bullet}\) associated to \(X\) on the Nisnevich site of \(X_1\). They then define \({\mathrm{CH}}^r_{\mathrm{cont}}(X_{\scriptscriptstyle \bullet}) = H^{2r}_{\mathrm{cont}}(X_1, {\mathbb Z}_{X_{\scriptscriptstyle \bullet}}(r))\) and they construct a certain obstruction map and relate it to the Hodge theoretic properties of the cycle class in crystalline cohomology. To finish the proof of their main result, the authors finally construct a Chern character \({\mathrm{ch}}: K_0^{\mathrm{cont}}(X_{\scriptscriptstyle \bullet}) _{\mathbb Q} \rightarrow \bigoplus_{r \leq {\mathrm{dim}}(X_1)} \, {\mathrm{CH}}^r_{\mathrm{cont}}(X_{\scriptscriptstyle \bullet})_{\mathbb Q}\) and, using deep results from topological cyclic homology theory due to T. Geisser and L. Hesselholt [Trans. Am. Math. Soc. 358, No. 1, 131–145 (2006; Zbl 1087.19003); J. Am. Math. Soc. 19, No. 1, 1–36 (2006; Zbl 1087.19002); Ann. Sci. Éc. Norm. Supér. (4) 37, No. 1, 1–43 (2004; Zbl 1062.19003)], they show that \(\mathrm{ch}\) is an isomorphism for \(p > {\mathrm{dim}}(X_1)+6\).
The paper ends with three appendices about homological algebra, homotopical algebra and a crystalline construction of the motivic complex.
Reviewer: Bernhard Köck (Southampton)
MSC:
| 19E15 | Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) |
| 14C25 | Algebraic cycles |
| 14D07 | Variation of Hodge structures (algebro-geometric aspects) |
| 14D15 | Formal methods and deformations in algebraic geometry |
| 14F30 | \(p\)-adic cohomology, crystalline cohomology |
| 14F40 | de Rham cohomology and algebraic geometry |
Keywords:
\(p\)-adic deformation; algebraic cycles; crystalline Chern character; de Rham cohomology; Hodge filtration; de Rham-Witt complex; Grothendieck’s variational Hodge conjecture; Fontaine-Messing’s \(p\)-adic variational Hodge conjecture; motivic pro-complex; Suslin-Voevodsky motivic complex; Fontaine-Messing-Kato syntomic complex; fundamental triangle; crystalline Hodge obstruction; topological cyclic homological theory; Milnor \(K\)-theoryCitations:
Zbl 0632.14016; Zbl 0383.14010; Zbl 1005.19001; Zbl 0645.14009; Zbl 1087.19003; Zbl 1087.19002; Zbl 1062.19003References:
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