M. Kobayashi introduced a notion of duality of weight systems. We tone this notion slightly down to a notion called coupling. We show that coupling induces a relation between the reduced zeta functions of the monodromy operators of the corresponding singularities generalizing an observation of K. Saito concerning Arnold's strange duality. We show that the weight systems of the mirror symmetric pairs of M. Reid's list of 95 families of Gorenstein K3 surfaces in weighted projective 3-spaces are strongly coupled. This includes Arnold's strange duality where the corresponding weight systems are strongly dual in Kobayashi's original sense. We show that the same is true for the extension of Arnold's strange duality found by the author and C. T. C. Wall.

This is a publication of an old preprint which extended Futaki character to an obstruction for Chern class forms to be harmonic. Although there were considerable developments on the subject, this paper is presented "as was".

In this paper we will perturb the scalar curvature of compact Kähler manifolds by incorporating it with higher Chern forms, and then show that the perturbed scalar curvature has many common properties with the unperturbed scalar curvature. In particular the perturbed scalar curvature becomes a moment map, with respect to a perturbed symplectic structure, on the space of all complex structures on a fixed symplectic manifold, which extends the results of Donaldson and Fujiki on the unperturbed case.

We give an explicit computation of the pointed harmonic volumes of hyperelliptic curves with Weierstrass base points, which are paraphrased into a combinatorial formula.

Let M be a semi-discrete linearly compact module over a commutative noetherian ring R and i a non-negative integer. We show that the set of co-associated primes of the local homology R-module $H^I_i$(M) is finite in either of the following cases: (i) The R-modules $H^I_j$(M) are finite for all j < i; (ii) I ⊆ Rad (Ann_R($H^I_j$(M))) for all j < i. By Matlis duality we extend some results for the finiteness of associated primes of local cohomology modules $H^I_i$(M).

An analytic function f(z)=z+a₂z² + … in the unit disk Δ = {z: |z| < 1} is said to be in $\mathscr{U}(\lambda, \mu)$ if $$\left|f'(z) \big( \frac{z}{f(z)} \big) ^{\mu +1}-1 \right|\le \lambda \quad (|z|<1)$$ for some λ ≥ 0 and μ > –1. For –1 ≤ α ≤ 1, we introduce a geometrically motivated $\mathscr{S}_p(\alpha)$-class defined by $${\mathscr S}_p(\alpha) = \big\{ f\in {\mathscr S}:\, \left |\frac{zf'(z)}{f(z)} -1\right |\leq {\rm Re}\, \frac{zf'(z)}{f(z)}-\alpha, z\in \Delta \big\} ,$$ where ${\mathscr S}$ represents the class of all normalized univalent functions in Δ. In this paper, the authors determine necessary and sufficient coefficient conditions for certain class of functions to be in $\mathscr{S}_p(\alpha)$. Also, radius properties are considered for $\mathscr{S}_p(\alpha)$-class in the class ${\mathscr S}$. In addition, we also find disks |z| < r:= r(λ,μ) for which $\frac{1}{r}f(rz)\in \mathscr{U(\lambda,\mu)}$ whenever $f\in \mathscr{S}$. In addition to a number of new results, we also present several new sufficient conditions for f to be in the class $\mathscr{U}(\lambda, \mu)$.

We build a variational theory of geodesics of the Tanaka-Webster connection ∇ on a strictly pseudoconvex CR manifold M. Given a contact form θ on M such that (M, θ) has nonpositive pseudohermitian sectional curvature (k_θ(σ) ≤ 0) we show that (M, θ) has no horizontally conjugate points. Moreover, if (M, θ) is a Sasakian manifold such that k_θ(σ) ≥ k₀ > 0 then we show that the distance between any two consecutive conjugate points on a lengthy geodesic of ∇ is at most $\pi/(2 \sqrt{k_0})$. We obtain the first and second variation formulae for the Riemannian length of a curve in M and show that in general geodesics of ∇ admitting horizontally conjugate points do not realize the Riemannian distance.

We provide generalizations of theorems of Myers and others to Riemannian manifolds with density and provide a minor correction to Morgan [8].

In [12] Ishida introduces a complex, denoted by $\tilde {\Omega}^{^.}_Y$, associated to a filtered semi-toroidal variety Y over Spec C and proves that it is quasi-isomorphic to the Du Bois complex $\underline{\underline{ \Omega}}^{^.}_Y$ ([5]). In this article we regard a filtered semi-toroidal variety Y as an ideally log smooth log scheme over Spec C, and we give an interpretation of the Ishida complex $\tilde {\Omega}^{^.}_Y$ in terms of logarithmic geometry. Therefore, given a log smooth log scheme X over Spec C, we use this logarithmic interpretation of the Ishida complex to construct the following distinguished triangle in the Du Bois derived category D_diff(X): $I_M \omega^{^.}_X \longrightarrow \underline{\underline{\Omega}}^{^.}_X \longrightarrow \underline{\underline{\Omega}}^{^.}_D \longrightarrow .$, where D = X − X_triv (X_triv being the trivial locus for the log structure M on X). Since the complex $I_M \omega^{^.}_X$ calculates the De Rham cohomology with compact supports of the smooth analytic space $X_{triv}^{an}$ ([20, Corollary 1.6]), this triangle is useful to give an interpretation of $H^{^.}_{DR,c}$(X_triv/C) as the hyper-cohomology of the simple complex $\underline{\underline{s}}[\underline{\underline{\Omega}}^{^.}_X \longrightarrow \underline{\underline{\Omega}}^{^.}_D]$.

We give relations of two meromorphic functions sharing 0, 1, ∞ and a set CM.