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In this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities in is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs , , , and , where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs , , and remain open now.
A domain admits the circular slit mapping for such that is regular at and . We call the -principal function and the -constant, and similarly, the radial slit mapping implies the -principal function and the -constant . We call the harmonic span for . We show the geometric meaning of . Hamano showed the variation formula for the -constant for the moving domain in with . We show the corresponding formula for the -constant for and combine these to prove that, if the total space is pseudoconvex in , then is subharmonic on . As a direct application, we have the subharmonicity of on , where is the Poincaré distance between and on .
In this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum link invariant, where is the set of fundamental representations of . In the case of an oriented link diagram composed of -crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general -crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.
We consider the following conjecture: if is a smooth and irreducible -dimensional projective variety over a field of characteristic zero, then there is a dense set of reductions to positive characteristic such that the action of the Frobenius morphism on is bijective. There is another conjecture relating certain invariants of singularities in characteristic zero (the multiplier ideals) with invariants in positive characteristic (the test ideals). We prove that the former conjecture implies the latter one in the case of ambient nonsingular varieties.