By Saemann C.

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**Sample text**

42) on any two arbitrary patches Ua , Ub from the covering U of M . 34), elements ¯ −1 with ψ ∈ X. Thus, for every A0,1 ∈ H ˇ 0 (M, A) can be written as ψ ∂ψ ˇ 0 (M, A) of H we have corresponding elements ψ ∈ X. One of these ψ can now be used to define the transition functions of a topologically trivial rank n holomorphic vector bundle E over M by the formula fab = ψa−1 ψb on Ua ∩ Ub = ∅ . 43) 8 The corresponding picture in the non-Abelian situation has still not been constructed in a satisfactory manner.

These considerations motivate the following theorem: ˇ 1 (M, h) = 0, any small deformation of §3 Theorem. (Kodaira-Spencer-Nirenberg) If H ˇ 1 (M, h) = 0 and H ˇ 2 (M, h) = 0 then there exists a complex manifold M M is trivial. If H parameterizing a family of complex structures on M such that the tangent space to M ˇ 1 (M, h). is isomorphic to H ˇ ˇ 1 (M, h) gives the number of paThus, the dimension of the Cech cohomology group H ˇ 2 (M, h) gives the obstructions rameters of inequivalent complex structures on M , while H to the construction of deformations.

The K3 manifold’s name stems from the three mathematicians Kummer, K¨ ahler and Kodaira who named it in the 1950s shortly after the K2 mountain was climbed for the first time. §8 Rigid Calabi-Yau manifolds. There is a class of so-called rigid Calabi-Yau manifolds, which do not allow for deformations of the complex structure. 5, as it follows that the mirrors of these rigid Calabi-Yau manifolds have no K¨ ahler moduli, which is inconsistent with them being K¨ ahler manifolds. 2, §13. A ten-dimensional string theory is usually split into a four-dimensional theory and a six-dimensional N = 2 superconformal theory.