On the integral Hodge conjecture for real varieties, I

We formulate the "real integral Hodge conjecture", a version of the integral Hodge conjecture for real varieties, and raise the question of its validity for cycles of dimension 1 on uniruled and Calabi-Yau threefolds and on rationally connected varieties. We relate it to the problem of determining the image of the Borel-Haefliger cycle class map for 1-cycles, with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the Chow group of 1-cycles of real threefolds. New results about these problems are obtained along the way.Comment: 67 pages; v2: minor modifications; v3: Section 1.1.3 slightly expanded, final versio

On the cycle class map for zero-cycles over local fields

We study the Chow group of zero-cycles of smooth projective varieties over local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus, over local fields of residue characteristic different from l. The same statement holds for semistable K3 surfaces defined over C((t)), but does not hold in general for surfaces over strictly local fields.Comment: 37 pages (with an appendix by Spencer Bloch); bibliography updated, final versio

On the fibration method for zero-cycles and rational points

Conjectures on the existence of zero-cycles on arbitrary smooth projective varieties over number fields were proposed by Colliot-Th\'el\`ene, Sansuc, Kato and Saito in the 1980's. We prove that these conjectures are compatible with fibrations, for fibrations into rationally connected varieties over a curve. In particular, they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers. We prove the analogous result for rational points, conditionally on a conjecture on locally split values of polynomials which a recent work of Matthiesen establishes in the case of linear polynomials over the rationals.Comment: 54 pages; v3: minor updates, added Remark 9.12(ii), v4: improved exposition, final versio

A restriction isomorphism for cycles of relative dimension zero

We study the restriction map to the closed fiber of a regular projective scheme over an excellent henselian discrete valuation ring, for a cohomological version of the Chow group of relative zero-cycles. Our main result extends the work of Saito--Sato to general perfect residue fields.Comment: 34 pages; final versio

Index of varieties over Henselian fields and Euler characteristic of coherent sheaves

Let X be a smooth proper variety over the quotient field of a Henselian discrete valuation ring with algebraically closed residue field of characteristic p. We show that for any coherent sheaf E on X, the index of X divides the Euler-Poincar\'e characteristic \chi(X,E) if p=0 or p>dim(X)+1. If 0<p\leq dim(X)+1, the prime-to-p part of the index of X divides \chi(X,E). Combining this with the Hattori-Stong theorem yields an analogous result concerning the divisibility of the cobordism class of X by the index of X. As a corollary, rationally connected varieties over the maximal unramified extension of a p-adic field possess a zero-cycle of p-power degree (a zero-cycle of degree 1 if p>dim(X)+1). When p=0, such statements also have implications for the possible multiplicities of singular fibers in degenerations of complex projective varieties.Comment: 20 pages; final versio

Z\'ero-cycles sur les fibrations au-dessus d'une courbe de genre quelconque

Let X be a smooth and proper variety over a number field k. Conjectures on the image of the Chow group of zero-cycles of X in the product of the corresponding groups over all completions of k were put forward by Colliot-Th\'el\`ene, Kato and Saito. We prove these conjectures for the total space of fibrations, over curves with finite Tate-Shafarevich group, into rationally connected varieties which satisfy weak approximation, under an abelianness assumption on the singular fibers. ---- Soit X une vari\'et\'e propre et lisse sur un corps de nombres k. Des conjectures sur l'image du groupe de Chow des z\'ero-cycles de X dans le produit des m\^emes groupes sur tous les compl\'et\'es de k ont \'et\'e propos\'ees par Colliot-Th\'el\`ene, Kato et Saito. Nous d\'emontrons ces conjectures pour l'espace total de fibrations en vari\'et\'es rationnellement connexes v\'erifiant l'approximation faible, au-dessus de courbes dont le groupe de Tate-Shafarevich est fini, sous une hypoth\`ese d'ab\'elianit\'e sur les fibres singuli\`eres.Comment: 46 pages; Th\'eor\`eme 4.8 slightly improved; final versio

Park City lecture notes: around the inverse Galois problem

The inverse Galois problem asks whether any finite group can be realised as the Galois group of a Galois extension of the rationals. This problem and its refinements have stimulated a large amount of research in number theory and algebraic geometry in the past century, ranging from Noether's problem (letting X denote the quotient of the affine space by a finite group acting linearly, when is X rational?) to the rigidity method (if X is not rational, does it at least contain interesting rational curves?) and to the arithmetic of unirational varieties (if all else fails, does X at least contain interesting rational points?). The goal of the present notes, which formed the basis for three lectures given at the Park City Mathematics Institute in August 2022, is to provide an introduction to these topics.Comment: 31 pages; some clarifications in Section 2.