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About the Calabi-Yau theorem{ Trimester on Kahler and Related Geometries (Nantes 2009) {Julien KellerL.A.T.P., Universite Marseille I { Universite de Provence{Lecture 2{Any comments, remarks and suggestions are welcome. These notes aresumming up the series of lectures given by the author during trimester onK ahler and Related Geometries (Nantes 2009). Please feel free to contactthe author at jkeller@cmi.univ-mrs.fr.1Numerical approximations of Calabi-Yau met-ricsWe have seen in the previous lecture a canonical way to nd a -balancedmetric and that -balanced metrics of order k converge when k tends toin nity towards the solution to the Calabi conjecture. Actually, a moreprecise statement is that one has convergence with quadratic speed in k.Thus we obtain an algorithm in order to compute a Calabi-Yau metric in anintegral K ahler class:1. Found a large number of points over the manifolds.2. Fix the volume form. Compute the volume at the points chosen previ-ously0 k3. Fix the space of holomorphic sections H (L ). Use the symmetries toreduce the dimension if possible0 k4. Fix a random hermitian matrix H 2Met(H (L ))0r5. Compute the iterates (Hilb FS) (H ) till one converges (usually 0r 10 is su cient). This requires to know the points over the manifoldand inverse a matrix to get an orthonormal basis.The algorithm for nding the -balanced metric has an exponential speed ofconvergence dependant on the smallest positive eigenvalue of the ...
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