International Journal of Mathematics

If a finite group action α on a unital C*-algebra M is saturated, the canonical conditional expectation E : M → M^α onto the fixed point algebra is known to be of index finite type with Index(E) = |G| in the sense of Watatani. More generally, if a finite-dimensional Hopf *-algebra A acts on M and the action is saturated, the same is true with Index(E) = dim(A). In this paper, we prove that the converse is true. Especially in case M is a commutative C*-algebra C(X) and α is a finite group action, we give an equivalent condition in order that the expectation E : C(X) → C(X)^α is of index finite type, from which we obtain that α is saturated if and only if G acts freely on X. Actions by compact groups are also considered to show that the gauge action γ on a graph C*-algebra C*(E) associated with a locally finite directed graph E is saturated.

The space of Abelian functions of a principally polarized abelian variety (J,Θ) is studied as a module over the ring of global holomorphic differential operators on J. We construct a free resolution in case Θ is non-singular. As an application, in the case of dimensions 2 and 3, we construct a new linear basis of the space of abelian functions which are singular only on Θ in terms of logarithmic derivatives of the higher-dimensional σ-function.

We study Gorenstein almost Fano threefolds X with canonical singularities and pseudoindex > 1. We show that the maximal Picard number of X is 10 in general, 3 if X is Fano, and 8 if X is toric. Moreover, we characterize the boundary cases. In the Fano case, we prove that the generalized Mukai conjecture holds.

We introduce the pole diagram, which helps to retrieve information from a knot diagram when we smooth crossings. By using the notion, we define a bracket polynomial for the Miyazawa polynomial. The bracket polynomial gives a simple definition and evaluation for the Miyazawa polynomial. Then we show that the virtual crossing number of a virtualized alternating link is determined by its diagram. Furthermore, we construct infinitely many virtual link diagrams which attain the minimal real and virtual crossing numbers together.

Reflection walls of certain primitive vectors in the anti-invariant sublattice of the K3 lattice define Heegner divisors in the period space of Enriques surfaces. We show that depending on the norm of these primitive vectors, these Heegner divisors are either irreducible or have two irreducible components. The two components are obtained as the walls orthogonal to primitive vectors of the same norm but of different type as ordinary or characteristic.

We classify endomorphisms of the plane that preserve a pencil of curves.

Let G_ℝ be a real form of a complex, semisimple Lie group G. Assume is an even nilpotent coadjoint G_ℝ-orbit. We prove a limit formula, expressing the canonical measure on as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the negative chamber defined by the parabolic subalgebra associated with .

We study two natural questions about subvarieties of moduli spaces. In the first section, we study the locus of curves equipped with F-nilpotent bundles and its relationship to the p-rank zero locus of the moduli space of curves of genus g. In the second section, we study subvarieties of moduli spaces of vector bundles on curves. We prove an analogue of a result of F. Oort about proper subvarieties of moduli of abelian varieties.