In this paper we introduce the weighted complex Randers metric $F=h+\sum_{i=1}^{m}\lvert B_{i}\rvert^{1/i}$ on a complex manifold $M$, here $h$ is a Hermitian metric on $M$ and $B_{i}$, $i=1,\ldots, m$ are holomorphic symmetric forms of weights $i$ on $M$, respectively. These metrics are special case of jet metric studied in Chandler--Wong [6]. Our main theorem is that the holomorphic sectional curvature $\mathrm{hbsc}_{F}$ of $F$ is always less or equal to $\mathrm{hbsc}_{h}$. Using this result we obtain a rigidity result, that is, a compact complex manifold $M$ of complex dimension $n$ with a weighted complex Randers metric $F$ of positive constant holomorphic sectional curvature is isomorphic to $\mathbb{P}^{n}$.

There are natural actions of the braid group $B_{n}$ on $B_{m}^{n}$, the $n$-fold product of the braid group $B_{m}$, called the Hurwitz action. We first study the roots of centralizers in the braid groups. By using the structure of the roots, we provide a criterion for the Hurwitz orbit to be finite and give an upper bound of the size for a finite orbit in $n=2$ or $m=3$ case.

We give a geometric proof of the fact that any affine surface with trivial Makar-Limanov invariant has finitely many singular points. We deduce that a complete intersection surface with trivial Makar-Limanov invariant is normal.

In this paper we discuss the $C^{\infty}$ well-posedness for second order hyperbolic equations $Pu=\partial_{t}^{2}u-a(t,x) \partial_{x}^{2}u=f$ with two independent variables $(t,x)$. Assuming that the $C^{\infty}$ function $a(t,x) \geq 0$ verifies $\partial_{t}^{p}a(0,0)\neq 0$ with some $p$ and that the discriminant $\Delta(x)$ of $a(t,x)$ vanishes of finite order at $x=0$, we prove that the Cauchy problem for $P$ is $C^{\infty}$ well-posed in a neighbourhood of the origin.

We study rational surfaces having an even set of disjoint ($-4$)-curves. The properties of the surface $S$ obtained by considering the double cover branched on the even set are studied. It is shown, that contrarily to what happens for even sets of ($-2$)-curves, the number of curves in an even set of ($-4$)-curves is bounded (less or equal to 12). The surface $S$ has always Kodaira dimension bigger or equal to zero and the cases of Kodaira dimension zero and one are completely characterized. Several examples of this situation are given.

Let $E$ be an elliptic curve over $\mathbb{Q}_{p}$ which has potentially supersingular good reduction. Let $L/\mathbb{Q}_{p}$ be a totally ramified extension such that $E$ has good reduction over $L$ and $\tilde{E}$ be the reduction of $E$ mod $\pi$, where $\pi$ is a prime element of the ring of integers $\mathcal{O}_{L}$ of $L$. Let $\hat{E}$ be the formal group over $\mathcal{O}_{L}$ associated to $E/\mathcal{O}_{L}$. The multiplication by $p$ map $[p]\colon \hat{E} \to \hat{E}$ is written by power series $[p](x) = px +c_{2}x^{2} + \cdots + c_{p}x^{p}+ \cdots + c_{p^{2}} x^{p^{2}} + \cdots{} \in \mathcal{O}_{L}[[x]]$. By using the liftings over $\mathcal{O}_{L}$ of the Dieudonné module of $p$-divisible group $\tilde{E}(p)$ over $\mathbb{F}_{p}$, we determine the values of $v_{L}(c_{p})$.

Using Seifert fibered three-manifold examples of Boileau and Zieschang, we demonstrate that the Reshetikhin--Turaev quantum invariants may be used to provide a sharp lower bound on the Heegaard genus which is strictly larger than the rank of the fundamental group.

In this paper we are concerned with a variational problem for a functional related to the conformality of maps between Riemannian manifolds. We give the first variation formula, the second variation formula, a kind of the monotonicity formula and a Bochner type formula. We also consider a variational problem of minimizing the functional in each 3-homotopy class of the Sobolev space.

We give bounds for the number of morphisms $f\colon X\to Y$ where $X$ and $Y$ are compact Riemann surfaces. The target surface $Y$ is not necessarily fixed.

We develop a modification of the Zariski--van Kampen approach for the computation of the fundamental group of a trigonal curve with improper fibers. As an application, we list the deformation families and compute the fundamental groups of all irreducible maximizing simple sextics with a type $\mathbf{D}$ singular point.

The Fourier--Jacobi coefficients of vector valued Siegel modular forms of degree $n$ are more general functions than vector valued Jacobi forms defined by Ziegler [9] even when $n=2$. We define generalized vector valued Jacobi forms corresponding to the above coefficients when $n=2$ and prove that such a space is isomorphic to a certain product of spaces of usual scalar valued Jacobi forms of various weights. This isomorphism is realized by certain linear holomorphic differential operators. The half-integral weight case is also treated.

We study the ramifications in the extensions of number fields arising from an isogeny of elliptic curves. In particular, we start with an elliptic curve with a rational torsion point, and show that the extension is unramified if and ``only if'' the point which generates the extension is reduced into a nonsingular point (we need to assume certain conditions in order to prove the ``only if'' part). We also study a characterization of quadratic number fields with class numbers divisible by 5.

We determine certain exceptional surgeries on a 3-parametric family of hyperbolic 1-bridge genus one knots ($(1,1)$-knots, in short). In particular, we show that such knots admit two infinite series of lens space surgeries. Our work is related to a nice paper of Teragaito [16], since we represent his toroidal manifolds as 2-fold coverings of the 3-sphere branched over well-specified links.

Under the assumption of asymptotic relative Chow-stability for polarized algebraic manifolds $(M, L)$, a series of weighted balanced metrics $\omega_{m}$, $m \gg 1$, called polybalanced metrics, are obtained from complete linear systems $\lvert L^{m} \rvert$ on $M$. Then the asymptotic behavior of the weights as $m \to \infty$ will be studied.