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On real Riemannian manifold , the heat kernel of the laplacian have an asymptotic expansion . But on complex manifold , i haven't seen a result like this , i.e. the heat kernel of the Kodaira Laplacian have an asymptotic expansion as the real case . Maybe I know so little , so I want to ask that Is there an asymptotic expansion of the heat kernel of the Kodaira laplacian ?

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Have you looked at "Holomorphic Morse Inequalities and Bergman Kernels" by Ma and Marinescu? I just saw the introduction, but it looks like it might have something similar to what you want. –  Kimball Jun 14 '11 at 10:37
    
Thanks very much! –  HKSHLZW Jun 14 '11 at 14:30

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