×

zbMATH — the first resource for mathematics

Standard conjecture of Künneth type with torsion coefficients. (English) Zbl 1386.14049
Let \(X\) be a complex smooth projective variety of dimension \(n\). Let \(cl_{X,\mathbb{Q}}^j:\text{CH}^j(X)\rightarrow H^{2j}(X,\mathbb{Q})\) denote the cycle class map from the Chow group of codimension-\(j\) cycles on \(X\). For each integer \(i\), one has the rational Künneth projector \(\pi_{X,\mathbb{Q}}^i\in H^{2n}(X\times X,\mathbb{Q})\), which acts as 1 on \(H^i(X,\mathbb{Q})\) and as 0 on \(H^j(X,\mathbb{Q})\) for all \(j\neq i\). Grothendieck’s standard conjecture of Künneth type asserts that the Künneth projectors \(\pi_{X,\mathbb{Q}}^i\) are algebraic, namely \(\pi_{X,\mathbb{Q}}^i\in \text{Im}(\text{cl}_{X\times X,\mathbb{Q}}^n)\otimes\mathbb{Q}\) for all \(i\). The present paper is concerned with the question, raised by A. Venkatesh, which asks whether the mod-\(p\)-analogue of the standard conjecture holds true or not. More precisely the problem is whether or not the mod-\(p\)-Künneth projectors \(\pi_{X,\mathbb{F}_p}^i\), which is defined similarly by replacing the coefficient \(\mathbb{Q}\) by \(\mathbb{F}_p\), belong to \(\text{Im}(\text{cl}_{X\times X,\mathbb{F}_p}^n)\) for all \(i\). One of the main result of this article is to answer the question in the negative by showing that, if \(H^*(X,\mathbb{Z})\) has nontrivial \(p\)-torsion, then Venkatesh’s question has a negative answer for \(X\). Furthermore he shows that for any integers \(i>0\) and \(n\geq i+1\), there exists a projective smooth variety \(X\) of dimension \(n\) such that \(\pi_{X,\mathbb{F}_p}^i\) is not in the image of the mod-\(p\) cycle class map. On the other hand, he looks into the question in the case of abelian varieties, and shows that Venkatesh’s question has an affirmative answer for all primes \(p\), if \(X\) is the Jacobian of a curve.
MSC:
14C25 Algebraic cycles
14H40 Jacobians, Prym varieties
55S05 Primary cohomology operations in algebraic topology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 10.1007/978-3-662-06280-7
[2] 10.1016/j.aim.2006.01.005 · Zbl 1109.14012
[3] 10.1112/plms/s3-25.1.75 · Zbl 0244.14017
[4] 10.1016/0040-9383(62)90094-0 · Zbl 0108.36401
[5] 10.1007/BF01472135 · Zbl 0566.14003
[6] ; Beauville, J. Ramanujan Math. Soc., 25, 253, (2010)
[7] 10.1007/s11511-016-0136-2 · Zbl 1349.14031
[8] 10.1007/978-3-662-06307-1
[9] ; Cartan, Homological algebra. Princeton Mathematical Series, 19, (1956)
[10] 10.1112/S0010437X12000607 · Zbl 1312.14012
[11] 10.2140/ant.2015.9.1035 · Zbl 1321.11050
[12] 10.1007/BF01455794 · Zbl 0552.14011
[13] ; Katz, Groupes de monodromie en géométrie algébrique, II : Exposés X-XXII. Lecture Notes in Mathematics, 340, 254, (1973)
[14] 10.1007/BF01405203 · Zbl 0275.14011
[15] 10.1007/BF02803582 · Zbl 0445.22018
[16] ; Kleiman, Dix exposés sur la cohomologie des schémas. Advanced Studies in Pure Mathematics, 3, 359, (1968)
[17] 10.1007/BF01244303 · Zbl 0806.14001
[18] 10.1007/978-3-642-51445-6
[19] 10.1215/S0012-7094-99-09620-5 · Zbl 0976.14009
[20] 10.1515/crelle-2016-0048 · Zbl 1440.14137
[21] 10.2307/1970825 · Zbl 0249.18022
[22] 10.1515/9781400882441 · Zbl 0724.11031
[23] 10.1090/pspum/055.1/1265529
[24] 10.4007/annals.2015.182.3.3 · Zbl 1345.14031
[25] ; Serre, Symposium internacional de topología algebraica, 24, (1958)
[26] 10.1016/j.aim.2004.10.022 · Zbl 1088.14002
[27] 10.1112/S0010437X08003527 · Zbl 1153.14028
[28] 10.1070/IM2011v075n05ABEH002563 · Zbl 1234.14009
[29] 10.1070/IM2015v079n01ABEH002738 · Zbl 1318.14012
[30] 10.1090/S0894-0347-97-00232-4 · Zbl 0989.14001
[31] 10.4007/annals.2016.183.1.4 · Zbl 1358.11063
[32] 10.1007/978-0-387-73892-5 · Zbl 1131.32001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.