Lectures on Fibre Bundles and Differential Geometry by J.L. Koszul, S. Ramanan

By J.L. Koszul, S. Ramanan

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However there is an obvious strategy for removing this restriction. We work with suitable generic perturbations of the Calabi-Yau structure, involving triples ω, ρ, ρ with ω ∧ ρ = 0. It is very reasonable to expect that for generic perturbations of this kind all solutions are regular. But as we explained in Section 3 we have then to give up the assumption that σ is closed, so we get nonzero Fredholm indices for adapted bundles. But this just means that, in the finite-dimensional analogue, we need to compute twisted cohomology using a 1-form with zeros of different indices so we can have a nontrivial chain complex.

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