In this paper, using the group structures of the spheres $S^{1}$, $S^{3}$ and the results of Atiyah-Patodi-Singer, Donnelly and Morita, we introduce a fixed point formula for periodic automorphisms of compact almost complex manifolds. Our main result is Theorem 1.3. The theorem is refined for a certain case if the almost complex manifold admits an Einstein-Kähler metric.

Let $I$ be an ideal in a Gorenstein local ring A with the maximal ideal $\mathfrak{m}$. Then $I$ is said to be an equimultiple good ideal if I contains a reduction $Q = (a_{1}, a_{2}, \ldots ,a_{s})$ generated by s elements in $A$ and if the associated graded ring $\mathrm{G}(I) = \bigoplus _{n\geq 0} I^{n}/I^{n+1}$ of $I$ is a Gorenstein ring with $\mathrm{a}(\mathrm{G}(I))=1-s$, where $s = \mathrm{ht}_{A}I$ and $\mathrm{a}(\mathrm{G}(I))$ denotes the $a$-invariant of $\mathrm{G}(I)$. The purpose is to explore the structure of the sets $\chi _{A,s}$ $(s\geq 0)$ of equimultiple good ideals I with $\mathrm{ht}_{A}I=s$. Goto, Iai and Watanabe [GIW] already studied $\mathfrak{m}$-primary good ideals $I$, that is the case where $\mathrm{ht}_{A}I = \dim A$. Some of their results are successfully generalized to those of equimultiple case with improvements.

It is known that the ring of modular forms of rational number weights for $\Gamma (5)$ is isomorphic to a polynomial ring of two variables. In this paper, we describe differential relations between logarithmic derivatives of the generators of that ring.

We give a new proof of the formulas for the twining character of the Verma module $M(\lambda )$ of symmetric highest weight $\lambda$ and for the twining character of the irreducible highest weight module $L(\Lambda )$ of symmetric, dominant integral highest weight $\Lambda$ over a symmetrizable generalized Kac-Moody algebra g, by using the Bernstein-Gelfand-Gelfand resolution of $L(\Lambda )$.

Morrow [9] classified all weighted dual graphs of the boundary of the minimal normal compactifications of the affine plane $\mathbf{A}^{2}$ by using a result of Ramanujam [10] that any minimal normal compactification of $\mathbf{A}^{2}$ has a linear chain as the graph of the boundary divisor. In this article, we give a new proof of the above-mentioned results of Ramanujam-Morrow [9] from a different point of view and by the purely algebro-geometric arguments. Moreover, we show that the affine plane $\mathbf{A}^{2}$ is characterized by the weighted dual graph of the boundary divisor.

Let $G$ be a compact, 1-connected, simple Lie group of exceptional type, $g$ its Lie algebra, and $p$ an odd prime. In this paper, the mod $p$ homology of the Kac-Moody group $K(g^{(1)})$ and the mod $p$ cohomology of the 3-connective cover over $G$ are determined as Hopf algebras over the Steenrod algebra for every case that the integral homology of $G$ has $p$-torsion.