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

利用Atiyah-Singer指数和Atiyah-Bott-Lefschetz-Singer不动点公式，我们得到了提升的Futaki不变量的显式计算公式。(Futaki不变量是李代数同态，而提升的Futaki不变量是Futaki不变量到李群同态的提升。)利用上面的计算公式，我们得到了如下结果：(1)具有正第一类且具有一般复结构的复二维Kaehler曲面的提升Futaki不变量为零当且仅当该曲面具有Einstein-Kaehler度规。(2)完全交的某个一般自同构的提升Futaki不变量为零。定义了椭圆算子的行列式，得到了行列式的显式计算公式，证明了Witten完整公式的特例，并利用椭圆算子的行列式定义了李群同态，它是Futaki-Morita不变多项式(Futaki不变量的推广)的推广，得到了李群同态的显式计算公式。

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.