By Spencer Bloch, Igor V. Dolgachev, William Fulton

This quantity comprises the lawsuits of a joint USA-USSR symposium on algebraic geometry, held in Chicago, united states, in June-July 1989.

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G. for d = 2 as h0,0 h1,0 h2,0 1 h0,1 h1,1 h2,1 0 h0,2 h1,2 h2,2 = 1 0 1 . 57) 0 1 §4 Equivalent definitions of Calabi-Yau manifolds. Let us summarize all equivalent conditions on a compact complex manifold M of dimension n for being a Calabi-Yau manifold: ⊲ M is a K¨ ahler manifold with vanishing first Chern class. ⊲ M admits a Levi-Civita connection with SU(n) holonomy. ⊲ M admits a nowhere vanishing holomorphic (n, 0)-form Ωn,0 . ⊲ M admits a Ricci-flat K¨ ahler metric. ⊲ M has a trivial canonical bundle.

The mystery of the unnaturally big ratio of the Planck mass to the energy scale of electroweak symmetry breaking (∼ 300GeV), which comes with problematic radiative corrections of the Higgs mass. In the MSSM, these corrections are absent. ⊲ Dark matter paradox: the neutralino, one of the extra particles in the supersymmetric standard model, might help to explain the missing dark matter in the universe. This dark matter is not observed but needed for correctly explaining the dynamics in our galaxy and accounts for 25% of the total matter1 in our universe.

Also easily seen is the potential K = 2 ahler: since J is a fact that any orientable complex manifold M with dim M = 1 is K¨ real two-form, dJ has to vanish on M . These manifolds are called Riemann surfaces. Furthermore, any complex submanifold of a K¨ ahler manifold is K¨ ahler. n An important example is the complex projective space P , which is also a K¨ ahler i i manifold. 9) on the patch Ui , which globally defines a closed two-form J by J := i∂ ∂¯ ln Kj , as one easily checks. From this form, we obtain a metric by g(X, Y ) := J(X, IY ), the Fubini-Study metric of P n .

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