For any linear algebraic group G, we define a ring CH * BG, the ring of characteristic classes with values in the Chow ring (that is, the ring of algebraic cycles modulo rational equivalence) for principal G-bundles over smooth algebraic varieties. We show that this coincides with the Chow ring of any quotient variety (V − S)/G in a suitable range of dimensions, where V is a representation of G and S is a closed subset such that G acts freely outside S. As a result, computing the Chow ring of BG amounts to the computation of Chow groups for a natural class of algebraic varieties with a lot of torsion in their cohomology. Almost nothing is known about this in general. For G an algebraic group over the complex numbers, there is an obvious ring homomorphism CH * BG → H * (BG, Z). Less obviously, using the results of [42], this homomorphism factors through the quotient of the complex cobordism ring M U * BG by the ideal generated by the elements of negative degree in the coefficient ring M U * = Z[x 1 , x 2 , . . . ], that is, through the ring M U * BG⊗ MU * Z. (For clarity, let us mention that the classifying space of a complex algebraic group is homotopy equivalent to that of its maximal compact subgroup. Moreover, every compact Lie group arises as the maximal compact subgroup of a unique complex reductive group.)The most interesting result of this paper is that in all the examples where we can compute the Chow ring of BG, it maps isomorphically to the topologically defined ring M U * BG ⊗ MU * Z. Namely, this is true for finite abelian groups, the symmetric groups, tori, GL(n, C), Sp(2n, C), O(n), SO(2n + 1), and SO(4). (The computation of the Chow ring for SO(2n + 1) is the result of discussion between me and Rahul Pandharipande. Pandharipande then proceeded to compute the Chow ring of SO (4), which is noticeably more difficult [31].) We also get various additional information about the Chow rings of the symmetric groups, the group G 2 , and so on.Unfortunately, the map CH * BG → M U * BG ⊗ MU * Z is probably not always an isomorphism, since this would imply in particular that M U * BG is concentrated in even degrees. By Ravenel, Wilson, and Yagita, M U * BG is concentrated in even degrees if all the Morava K-theories of BG are concentrated in even degrees [33]. The latter statement, for all compact Lie groups G, was a plausible conjecture of Hopkins, Kuhn, and Ravenel [19], generalizing the theorem of Atiyah-HirzebruchSegal on the topological K-theory of BG [1], [5]. But it has now been disproved by Kriz [22], using the group G of strictly upper triangular 4 × 4 matrices over Z/3.Nonetheless, there are some reasonable conjectures to make. First, if G is a complex algebraic group such that the complex cobordism ring of G is concentrated in even degrees, say after tensoring with Z (p) (Z localized at p) for a fixed prime number p, then the homomorphismshould become an isomorphism after tensoring with Z (p) . Second, we can hope that the Chow ring of BG for any complex algebraic group G has the good properties whi...

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A fundamental problem of algebraic geometry is to determine which varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties from both sides. In particular, we want to know which smooth hypersurfaces in projective space are rational. An easy case is that smooth complex hypersurfaces of degree at least n + 2 in P n+1 are not covered by rational curves and hence are not rational.By far the most general result on rationality of hypersurfaces is Kollár's theorem that for d at least 2⌈(n + 3)/3⌉, a very general complex hypersurface of degree d in P n+1 is not ruled and therefore not rational [13, Theorem 5.14]. Very little is known about rationality in lower degrees, except for cubic 3-folds and quintic 4-folds [4], [19, Chapter 3].A rational variety is also stably rational, meaning that some product of the variety with projective space is rational. Many techniques for proving non-rationality give no information about stable rationality. Voisin made a breakthrough in 2013 by showing that a very general quartic double solid (a double cover of P 3 ramified over a quartic surface) is not stably rational [23]. These Fano 3-folds were known to be non-rational over the complex numbers, but stable rationality was an open question. Voisin's method was to show that the Chow group of zero-cycles is not universally trivial (that is, the Chow group becomes nontrivial over some extension of the base field), by degenerating the variety to a nodal 3-fold which has a resolution of singularities X with nonzero torsion in H 3 (X, Z).Colliot-Thélène and Pirutka simplified and generalized Voisin's degeneration method. They deduced that very general quartic 3-folds are not stably rational [6]. This was striking, in that non-rationality of smooth quartic 3-folds was the original triumph of Iskovskikh-Manin's work on birational rigidity, while stable rationality of these varieties was unknown [11]. Beauville applied the method to prove that very general sextic double solids, quartic double 4-folds, and quartic double 5-folds are not stably rational [1,2].In this paper, we show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all d ≥ 2⌈(n + 2)/3⌉ and n ≥ 3, a very general complex hypersurface of degree d in P n+1 is not stably rational (Theorem 2.1). The theorem covers all the degrees in which Kollár proved non-rationality. In fact, we get a bit more, since Kollár assumed d ≥ 2⌈(n + 3)/3⌉. For example, very general quartic 4-folds are not stably rational, whereas it was not even known whether these varieties are rational.The method applies to some smooth hypersurfaces over Q in each even degree. Section 3 gives some examples over Q which are not stably rational over C.The idea is that the most powerful results are obtained by degenerating a smooth complex variety to a singular variety in positive characteristic, rather than to a sin-

Atiyah and Hirzebruch gave the first counterexamples to the Hodge conjecture with integer coefficients. In particular, there is a smooth complex projective variety X of dimension 7 and a torsion element of H 4 (X, Z) which is not the class of a codimension-2 algebraic cycle [4]. In this paper, we provide a more systematic explanation for their examples: for every smooth complex algebraic variety X, we show that the cycle map, from the ring of cycles modulo algebraic equivalence on X to the integer cohomology of X, lifts canonically to a more refined topological invariant of X, the ring M U * X ⊗ MU * Z, where M U * X is the complex cobordism ring of X. Here M U * X is a module over the graded ringand we map M U * to Z by sending all the generators x i to 0. The ring M U * X ⊗ MU * Z is the same as the integer cohomology ring if the integer cohomology is torsion-free, but in general the map M U * X ⊗ MU * Z → H * (X, Z) need not be either injective or surjective, although the kernel and cokernel are torsion. This more refined cycle map gives a new way to prove that the Griffiths group (the kernel of the map from cycles modulo algebraic equivalence to integer cohomology) can be nonzero, without any use of Hodge theory.The resulting examples answer some questions on algebraic cycles by Colliot-Thélène and Schoen.Our examples are all quotients of complete intersections by finite groups, as are Atiyah-Hirzebruch's examples. First, we find smooth complex projective varieties X of dimension 7, definable over Q, such that the map CH 2 X/2 → H 4 (X, Z/2) is not injective. Here CH i X is the group of codimension i algebraic cycles on X modulo rational equivalence. Kollár and van Geemen [5], p. 135, gave the first examples of smooth complex projective varieties with CH 2 X/n → H 4 (X, Z/n) not injective for some n, answering a question by Colliot-Thélène [9], p. 14. Over non-algebraically closed fields k there are other examples of smooth projective varietiesdue to Colliot-Thélène and Sansuc as reinterpreted by Salberger (see [10] and [9], Remark 7.6.1), Parimala and Suresh [29], and Bloch and Esnault [8]. Of these examples, only Bloch and Esnault's elements of CH 2 (X k )/n are shown to remain nonzero in CH 2 (X k )/n, as happens in our example.Also, we find codimension-3 cycles, on certain smooth complex projective varieties X of dimension 15, which are torsion in the Chow group CH 3 X, which map to 0 in H 6 (X, Z) and even in Deligne cohomology (i.e., the intermediate Jacobian), but which are not algebraically equivalent to 0. The variety X and the cycles we consider can be defined over Q. By contrast, for all X over C, the map from the torsion subgroup of CH i X to Deligne cohomology was known to be injective for i ≤ 2 by Merkur'ev-Suslin [22], p. 338, and for i = dim X by Roitman [34], and Schoen [35], p. 13, asked whether the map was injective in general. Similarly, it is conjectured that codimension-2 cycles which map to 0 in Deligne cohomology are algebraically equivalent to 0, and our construction shows that this i...

Abstract. We prove the equivalence of two fundamental properties of algebraic stacks: being a quotient stack in a strong sense, and the resolution property, which says that every coherent sheaf is a quotient of some vector bundle. Moreover, we prove these properties in the important special case of orbifolds whose associated algebraic space is a scheme.