# 25 Beautiful Homes - February 2012 by Editor: Deborah Barker

By Editor: Deborah Barker

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Extra resources for 25 Beautiful Homes - February 2012

Example text

7. 6. Clearly, the vector θ ∈ Hr4 (M ) ∼ = S 2 (H 2 (M )) is 2 2 so(H (M ))-invariant. Moreover, the space of so(H (M ))-invariant vectors in S 2 (H 2 (M )) is one-dimensional. Finally, from an explicit computation of G it follows that G acts on H 4 (M ) as SO(H 2 (M )), and thus, the Lie algebra invariants coincide with invariants of G. We found that the space of G-invariants in Hr4 (M ) is one-dimensional and generated by θ. 12. 12 from dimension 4 to all dimensions. The space Hr2d (M )G of G-invariants in Hr2d (M ) is 1-dimensional for d even and zero-dimensional for d odd.

Since HI1,1 (M )∩H 2 (M, Z) has rank one, for all η ∈ HI1,1 (M )∩H 2 (M, Z), η = 0, we have | degI η| d. 1. 12. 6, θ ∈ HI2,2 (M ). Consider α2 ∈ HI2,2 (M ), where α is the generator of HI1,1 (M ) ∩ H 2 (M, Z). 5: Let J be an induced complex structure, J ◦ I = −J ◦ I, and degI , degJ the degree maps associated with I, J. Then degI α2 > 0, degJ α2 = 0, degI θ = degJ θ > 0. Proof: Since α is a K¨ahler class with respect to I, we have degI α2 > 0. Since the cohomology class θ is SU (2)-invariant, and SU (2) acts transitively on the set of induced complex structures, we have degI θ = degJ θ.

The Hermitian metric on B and the connection ∇ defined by this metric are called Yang-Mills if Λ(Θ) = constant · Id B , where Λ is a Hodge operator and Id B is the identity endomorphism which is a section of End(B). Further on, we consider only these Yang–Mills connections for which this constant is zero. A holomorphic bundle is called indecomposable if it cannot be decomposed onto a direct sum of two or more holomorphic bundles. The following fundamental theorem provides examples of Yang--Mills bundles.