Abstract
The most important open questions in the theory of algebraic cycles are the Hodge Conjecture, and its companion problem, the Tate Conjecture. Both these questions attempt to give a description of those cohomology classes on a nonsingular proper variety which are represented by algebraic cycles, in terms of intrinsic structure which is present on the cohomology of such a variety (namely, a Hodge decomposition, or a Galois representation). For the Hodge conjecture, the case of divisors (algebraic cycles of codimension 1) was settled long ago by Lefschetz and Hodge, and is popularly known as the Lefschetz (1,1) theorem, though there is little general progress beyond that case. However, even this case of divisors is an open question for the Tate Conjecture, in general, even for divisors on algebraic surfaces. After giving an introduction to these problems, I will discuss the recent progress on the Tate Conjecture for K3 surfaces, around works of M. Lieblich, D. Maulik, F. Charles and K. Pera.
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Notes
Algebraic cycles of codimension i are elements of the free abelian group on integral subschemes of codimension i; cycles with rational coefficients are obtained by tensoring the group of cycles with \({\mathbb{Q}}\).
Tate’s lecture notes do make this point as well, and make the additional conjecture that (T1)–(T3) are “independent of ℓ” in a suitable sense: that the conjectures for one ℓ are equivalent to the conjectures for any ℓ.
It is known, however, that the “integral Hodge conjecture” is in general false in any codimension i with 2≤i<dimX, from examples of Atiyah–Hirzebruch and Kollár.
That any indefinite quadratic form over \({\mathbb{Q}}\) of dimension ≥5 has a nontrivial zero.
Near the end of this article, we comment a bit more on the above ingredients in Maulik’s work.
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Acknowledgements
I would like to thank Dipendra Prasad, Najmuddin Fakhruddin, Kapil Paranjape and Anand Sawant for their comments on versions of this document, which have helped improve it.
I acknowledge support from a J.C. Bose Fellowship of the DST, India, and a Humboldt Research Grant.
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Srinivas, V. The Tate conjecture for K3 surfaces—a survey of some recent progress. Acta Math Vietnam. 39, 69–85 (2014). https://doi.org/10.1007/s40306-014-0048-1
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DOI: https://doi.org/10.1007/s40306-014-0048-1