We are concerned with the question how the capacity of the ideal boundary of a subsurface of a covering Riemann surface over a Riemann surface varies according to the variation of its branch points. In the present paper we treat the most primitive but fundamental situation that the covering surface is a two sheeted sphere with two branch points one of which is fixed and the other is moving and the subsurface is given as the complement of two disjoint continua each in different sheets of the covering surface whose projections are two disjoint continua in the base plane given in advance not touching the projections of branch points. We will derive a variational formula for the capacity and as one of its many useful consequences expected we will show that the capacity changes smoothly as one branch point moves in the subsurface.

We obtain characterizations of positive Borel measures $\mu$ on $\mathbf{B}^{n}$ so that some weighted Hardy-Sobolev are imbedded in $L^{p}(d\mu)$, where $w$ is an $A_{p}$ weight in the unit sphere of $\mathbf{C}^{n}$.

In this paper we study the averaging formula for Nielsen coincidence numbers of pairs of maps $(f, g) : M \to N$ between closed smooth manifolds of the same dimension. Suppose that $G$ is a normal subgroup of $\Pi = \pi_{1}(M)$ with finite index and $H$ is a normal subgroup of $\Delta = \pi_{1}(N)$ with finite index such that $f_{*}(G) \subset H$ and $g_{*}(G) \subset H$. Then we investigate the conditions for which the following averaging formula holds

N(f, g) = \frac{1}{[\Pi:G]} \sum_{\bar{\alpha} \in \Delta/H} N(\bar{\alpha} \bar{f}, \bar{g}),

where $(\bar{f}, \bar{g}) : M_{G} \to N_{H}$ is any pair of fixed liftings of $(f, g)$. We prove that the averaging formula holds when $M$ and $N$ are orientable infra-nilmanifolds of the same dimension, and when $M = N$ is a non-orientable infra-nilmanifold with holonomy group $\mathbb{Z}_{2}$ and $(f, g)$ admits a pair of liftings $(\bar{f}, \bar{g}) : \bar{M} \to \bar{M}$ on the nil-covering $\bar{M}$ of $M$.

We study the Saito-Ikeda infinitesimal invariant of the cycle defined by curves in their Jacobians using rank $k+1$ vector bundles. We give a criterion for which the higher cycle class map is not trivial. When $k = 2$, this turns out to be strictly linked to the Petri map for vector bundles. In this case we can improve a result of Ikeda: an explicit construction on a curve of genus $g \geq 10$ shows the existence of a non trivial element in the higher Griffiths group.

The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on $n$-dimensional smooth projective varieties $X$ with an $n$-block collection $\mathcal{B}$ which generates the bounded derived category $\mathcal{D}^{b}({\mathcal{O}}_{X}{\textit -mod})$. To this end, we use the theory of $n$-blocks and Beilinson type spectral sequence to define the notion of regularity of a coherent sheaf $F$ on $X$ with respect to the $n$-block collection $\mathcal{B}$. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we compare our definition of regularity with previous ones. In particular, we show that in case of coherent sheaves on $\mathbb{P}^{n}$ and for the $n$-block collection $\mathcal{B} = (\mathcal{O}_{\mathbb{P}^{n}}, \mathcal{O}_{\mathbb{P}^{n}}(1), \dots, \mathcal{O}_{\mathbb{P}^{n}}(n))$ on $\mathbb{P}^{n}$ Castelnuovo-Mumford regularity and our new definition of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a multiprojective space $\mathbb{P}^{n_{1}} \times \dots \times \mathbb{P}^{n_{r}}$ with respect to a suitable $n_{1}+\dots+n_{r}$-block collection and we compare it with the multigraded variant of the Castelnuovo-Mumford regularity given by Hoffman and Wang in [14].

We show that the Bergman metric of a bounded homogeneous domain has a potential function whose gradient has a constant norm with respect to the Bergman metric, and further that this constant is independent of the choice of such a potential function.

Let $R$ be a Dedekind domain, $G$ an affine flat $R$-group scheme, and $B$ a flat $R$-algebra on which $G$ acts. Let $A \rightarrow B^{G}$ be an $R$-algebra map. Assume that $A$ is Noetherian. We show that if the induced map $K \otimes A \rightarrow (K \otimes B)^{K \otimes G}$ is an isomorphism for any algebraically closed field $K$ which is an $R$-algebra, then $S \otimes A \rightarrow (S \otimes B)^{S \otimes G}$ is an isomorphism for any $R$-algebra $S$.

We study the asymptotic behavior in Sobolev norm of the local time of the $d$-dimensional fractional Brownian motion with $N$-parameters when the space variable tends to zero, both for the fixed time case and when simultaneously time tends to infinity and space variable to zero.