The determination of the nonlinear term by the bifurcating curve of a solution of a nonlinear boundary valne problem

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Using the Atiyah-Singer index and the Atiyah-Bott-Lefschetz-Singer fixed point formula, we obtained an explicit calculation formula for the lifted Futaki invariant. (The Futaki invariant is a lie algebra homomorphism and the lifted Futaki invariant is a lifting of the Futaki invariant to a Lie group homomorphism.) Using the calculation formula above, we obtained the following results.(1) The lifted Futaki invariant of a complex 2-dimensional kaehler surface with positive first Chern class and with a generic complex structure vanishes if and only if the surface admits an Einstein-Kaehler metric.(2) The lifted Futaki invariant for a certain general automorphism of a complete intersection vanishes. (Note that every complete intersection is expected to admit an Einstein-Kaehler metric.)We moreover defined the determinant of elliptic operators, obtained an explicit calculation formula for the determinant and proved the special case of the Witten holonomy formula as an application of the calculation formula.Furthermoro, using the determinant of elliptic operators, we defined a Lie group homomorphism which is a lifting of the Futaki-Morita invariant polynomial (which is a generalization of the Futaki invariant) and obtained the explicit calculation formula for the Lie group homomorphism.