New Examples of Hyperkähler Manifolds by Dmitri Kaledin, Misha Verbitsky

By Dmitri Kaledin, Misha Verbitsky

This quantity introduces hyperkahler manifolds to those that haven't formerly studied them. The publication is split into elements on: hyperholomorphic sheaves and examples of hyperkahler manifolds; and hyperkahler constructions on overall areas of holomorphic cotangent bundles.

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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.

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