On pluricanonical maps for threefolds of general type, II

This note mainly studies the generic finiteness of \phi_m of a complex projective 3-fold of general type. A new result on the classification to bicanonical pencil for Gorenstein 3-folds is attached in the last section.Comment: 16 pages, Amstex, The final version, Accepted for publication in Osaka Journal of Mathematic

Theory of multi-fans

We introduce the notion of a multi-fan. It is a generalization of that of a fan in the theory of toric variety in algebraic geometry. Roughly speaking a toric variety is an algebraic variety with an action of algebraic torus of the same dimension as that of the variety, and a fan is a combinatorial object associated with the toric variety. Algebro-geometric properties of the toric variety can be described in terms of the associated fan. We develop a combinatorial theory of multi-fans and define ``topological invariants'' of a multi-fan. A smooth manifold with an action of a compact torus of half the dimension of the manifold and with some orientation data is called a torus manifold. We associate a multi-fan with a torus manifold, and apply the combinatorial theory to describe topological invariants of the torus manifold. A similar theory is also given for torus orbifolds. As a related subject a generalization of the Ehrhart polynomial concerning the number of lattice points in a convex polytope is discussed

Remark on the rank of elliptic curves

A covariant functor on the elliptic curves with complex multiplication is constructed. The functor takes values in the noncommutative tori with real multiplication. A conjecture on the rank of an elliptic curve is formulated.Comment: 13 pages, 2 figures; to appear Osaka J. Mathematics 46 (2009), No.2; http://projecteuclid.org/euclid.ojm/124541568

Abundance theorem for semi log canonical surfaces in positive characteristic

We prove the abundance theorem for semi log canonical surfaces in positive characteristic.Comment: 33 pages. v2: I added Section 3, changed the definition of slc surfaces, and adopted the one of Kollar. v3: minor change

On homology 3-spheres defined by two knots

We show that if each of $K_1$ and $K_2$ is a trefoil knot or figure eight knot, the homology 3-sphere defined by the Kirby diagram which is a simple link of $K_1$ and $K_2$ with framing (0, n) is represented by an n-twisted Whitehead double of $K_2$ .Comment: 21 pages, 102 figure

Logarithmic Jet Bundles and Applications

We generalize Demailly's construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. We establish this logarithmic generalization explicitly via coordinates, just as Noguchi's generalization of the jets used by Green-Griffiths. As a first application, we give a metric proof for the logarithmic version of Lang's conjecture concerning the hyperbolicity of complements of divisors in a semi-abelian variety as well as for the corresponding big Picard theorem.Comment: 49 pages, Late

Chow-stability and Hilbert-stability in Mumford's Geometric Invariant Theory

In this note, we shall show that the Chow-stability and the Hilbert-stability in GIT asymptotically coincide.Comment: I slightly changed the title of the paper. Moreover, a proof using an equivariant version of Serre's Conjecture is given. This new version appeared in Osaka J. Math. 45 (2008

The quandle of the trefoil knot as the Dehn quandle of the torus

We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the symplectic space $\Z \oplus \Z$. The last quandle can be identified with the Dehn quandle of the torus and the cord quandle on a 2-sphere with four punctures. We also show that the fundamental quandle of the long trefoil knot is isomorphic to the cord quandle on a 2-sphere with a hole and three punctures.Comment: 14 pages, 5 figure

Reconstructible graphs, simplicial flag complexes of homology manifolds and associated right-angled Coxeter groups

In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if $\Gamma$ is a finite graph which is the 1-skeleton of some simplicial flag complex $L$ which is a homology manifold of dimension $n \ge 1$, then the graph $\Gamma$ is reconstructible.Comment: 7 page

On symplectic quandles

We study the structure of symplectic quandles, quandles which are also R-modules equipped with an antisymmetric bilinear form. We show that every finite dimensional symplectic quandle over a finite field F or arbitrary field F of characteristic other than 2 is a disjoint union of a trivial quandle and a connected quandle. We use the module structure of a symplectic quandle over a finite ring to refine and strengthen the quandle counting invariant.Comment: 11 pages. v2: typo corrections suggested by referee. To appear in Osaka J. Mat