### Article

## Standard conjectures in model theory, and categoricity of comparison isomorphisms

We formulate two conjectures about etale cohomology and fundamental groups motivated by categoricity conjectures in model theory. One conjecture says that there is a unique Z-form of the etale cohomology of complex algebraic varieties, up to Aut(C)-action on the source category; put differently, each comparison isomorphism between Betti and etale cohomology comes from a choice of a topology on C. Another conjecture says that each functor to groupoids from the category of complex algebraic varieties which is similar to the topological fundamental groupoid functor, in fact factors through it, up to a field automorphism of the complex numbers acting on the category of complex algebraic varieties. We also try to present some evidence towards these conjectures, and show that some special cases seem related to Grothendieck standard conjectures and conjectures about motivic Galois group.

The crystal structures of rare-earth diaryl- or dialkylphosphate derivatives are poorly explored. Crystals of bis[bis(2,6-diisopropylphenyl) phosphato-kappa O] chloridotetrakis( methanol-kappa O) neodymium methanol disolvate, [Nd(C24H34O4P)Cl( CH4O)(4)]center dot 2CH(3)OH, (1), and of the lutetium, [Lu(C24H34O4P) Cl(CH4O)(4)]center dot-2CH(3)OH, (2), and yttrium, [Y(C24H34O4P) Cl(CH4O)(4)]center dot 2CH(3)OH,(3), analogues have been obtained by reactions between lithium bis(2,6-diisopropylphenyl)-phosphate and LnCl(3)(H2O)(6) (in a 2:1 ratio) in methanol. Compounds (1)-(3) crystallize in the C2/c space group. Their crystal structures are isomorphous. The molecule possesses C-2 symmetry with a twofold crystallographic axis passing through the Ln and Cl atoms. The bis(2,6-diisopropylphenyl) phosphate ligands all display a kappa O-1-monodentate coordination mode. The coordination polyhedron for the metal atom [coordination number (CN) = 7] is a distorted pentagonal bipyramid. Each [Ln{O2P(O-2,6-(Pr2C6H3)-Pr-i)(2)}(2)Cl(CH3OH)(4)] molecular unit exhibits two intramolecular O-H center dot center dot center dot O hydrogen bonds, forming six-membered rings, and two intramolecular O-H center dot center dot center dot Cl interactions, forming four-membered rings. Intermolecular O-H . . . O hydrogen bonds connect each unit via four noncoordinating methanol molecules with four other units, forming a two-dimensional hydrogen-bond network. Crystals of bis[bis(2,6-diisopropylphenyl) phosphato-kappa O] tetrakis(methanol-kappa O)(nitrato-kappa(2) O, O ') neodymium methanol disolvate, [Nd(C24H34O4P)(NO3)(CH4O)(4)]center dot 2CH(3)OH, (4), have been obtained in an analogous manner from NdCl3(H2O)(6). Compound (4) also crystalizes in the C2/c space group. Its crystal structure is similar to those of (1)-(3). The kappa(2) O, O '-bidentate nitrate anion is disordered over a twofold axis, being located nearly on it. Half of the molecule is crystallographically unique (CNNd = 8). Unlike (1)-(3), complex (4) exhibits disorder of all three methanol molecules, one isopropyl group of the phosphate ligand and the NO3- ligand. The structure of (4) displays intra- and intermolecular O-H center dot center dot center dot O hydrogen bonds similar to those in (1)-(3). Compounds (1)-(4) represent the first reported mononuclear bis[bis(diaryl/dialkyl) phosphate] rare-earth complexes.

The article suggests a genre model for novel-myth which can be discerned in «The God of Small Things» by Arundhati Roy. The author argues against devising the model solely on the basis of parallels between the content of the novel and mythological images and motifs. Instead such mythomodeling categories as time, recollection and reader are investigated. It is demonstrated how the discrepancy between the time of story and discourse (G. Genette) defines the mythical nature of characters and their relations. The interplay of pro- and retrospections resulting from the second of these categories allows to define the novel as the ritual of recollection (M. Eliade). As a result, the reader in the novel becomes the participant of this ritual and is isomorphic in this role to the characters.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.