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32) ¯ on a complex manifold M together with the (r, s)th ∂-cohomology group H∂r,s ¯ (M ) = Z r,s cocycles ¯ (M ) . e. e. for s > 0 there is a form ¯ = ω. τ ∈ Ωr,s−1 (M ) such that ∂τ r,s The Hodge number h is the complex dimension of H∂r,s ¯ (M ). The corresponding Betti number of the de Rham cohomology of the underlying real manifold is given by bk = kp=0 hp,k−p and the Euler number of a d-dimensional real manifold is defined as χ = dp=0 (−1)p bp . The Poincar´e lemma can be directly translated to the complex situation and thus ¯ ¯ every ∂-closed form is locally ∂-exact.
2 Relative deformation theory §7 Normal bundle. Given a manifold X and a submanifold Y ⊂ X, we define the normal bundle N to Y in X via the short exact sequence 1 → T Y → T X|Y → N → 1 . 76) being slightly sloppy ˇ i (M, O(E)) and H ˇ n−i (M, O(E ∨ ⊗ Λn,0 )) are dual, where M is a compact complex The spaces H manifold of dimension n, E a holomorphic vector bundle and Λn,0 a (n, 0)-form. One can then pair elements of these spaces and integrate over M . 13 48 Complex Geometry Therefore, N = TTX|Y Y .
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.
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