To understand in detail duality between heterotic string and F theory compactifications, it is important to understand the construction of holomorphic G bundles over elliptic CalabiYau manifolds, for various groups G. In this paper, we develop techniques to describe these bundles, and make several detailed comparisons between the heterotic string and F theory.

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Rhenium and osmium concentrations and osmium isotopic ratios of group IIA, IIIA, IVA, and IVB iron meteorites were determined by negative thermal ionization mass spectrometry with modified digestion and equilibration techniques. Precise isochrons are defined for all four groups. An absolute age of 4558 million years is assumed for group IIIA irons, leading to closure ages of 4537 ± 8, 4456 ± 25, and 4527 ± 29 million years for the group IIA, IVA, and IVB irons, respectively. The initial osmium-187/osmium-188 ratios of the IIA, IIIA, and IVA groups, as a function of crystallization age, suggest that the rhenium/osmium ratio of the parental materials to these asteroidal cores was similar to that of ordinary chondrites. Data for IVB irons, in contrast, indicate a different mode of formation and derivation from a distinctly nonchondritic osmium reservoir.

We study string compactifications with sixteen supersymmetries. The moduli space for these compactifications becomes quite intricate in lower dimensions, partly because there are many different irreducible components. We focus primarily, but not exclusively, on compactifications to seven or more dimensions. These vacua can be realized in a number ways: the perturbative constructions we study include toroidal compactifications of the heterotic/type I strings, asymmetric orbifolds, and orientifolds. In addition, we describe less conventional M and F theory compactifications on smooth spaces. The last class of vacua considered are compactifications on singular spaces with non-trivial discrete fluxes.We find a number of new components in the string moduli space. Contained in some of these components are M theory compactifications with novel kinds of "frozen" singularities. We are naturally led to conjecture the existence of new dualities relating spaces with different singular geometries and fluxes. As our study of these vacua unfolds, we also learn about additional topics including: F theory on spaces without section, automorphisms of del Pezzo surfaces, and novel physics (and puzzles) from equivariant K-theory. Lastly, we comment on how the data we gain about the M theory three-form might be interpreted.

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The algebraic topology of smooth algebraic varieties Publications mathématiques de l'I.H.É.S., tome 48 (1978), p. 137-204 © Publications mathématiques de l'I.H.É.S., 1978, tous droits réservés. L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ '42 JOHNW.MORGAN 143 145 19 Proposition (1.9) ([4], (i .2.8)).-Let (V, W, F) be a mixed Hodge structure. Then VC^^^A^ gives afunctorial bigrading compatible with tensor products and duals: P,q 1) W,(Vc)= © A^, P+q^i 2) F^'(Vc) =©A^, and P^J 3) the composition A^^W^(Vc)->Gr^ (V^)= © ^^ sends A^ isomorphically onto ^^. p'+q'=p+q Proof.-Functoriality is clear from the definition of the A^' 3 's. From 3) parts i) and 2) follow immediately. To prove 3) we first note that the map A^-^ © ^p'^' P' Q.' has its image contained in ^p fq. The next proposition is then used to show that AF^-^^^ is an isomorphism. Proposition (1.10).-Let (V, W, F) be a mixed Hodge structure. Then W^(Vc)=R^+L^. Proof.-Clearly R^+L^^W^,, and Gr^(R^+L^)-> Gr^(Vc) is onto for n ^p + q. Since W^ = o for N sufficiently small it follows that: R^+L^=W,^(Vc). Note.-It is not always true that A^=A^. However, modulo W^+^_i(Vc) this is true.