In connection with Serre's problem, we consider a locally trivial analytic fibration $\pi: E \longrightarrow B $ of complex spaces with typical fiber $X$. We show that if $X$ is a Stein curve and $B$ is $q$-complete, then $E$ is $q$-complete.

We show that for the second Painlevé equation $y'' = 2y^{3}+ty+\alpha$, the Bäcklund transformation group $G$, which is isomorphic to the extended affine Weyl group of type $\hat{A}_{1}$, operates regularly on the natural projectification ${\mathcal X}(c)/\mathbb{C}(c, t)$ of the space of initial conditions, where $c = \alpha-1/2$. ${\cal X}(c)/\mathbb{C}(c, t)$ has a natural model ${\mathcal X}[c]/\mathbb{C}(t)[c]$. The group $G$ does not operate, however, regularly on $\mathcal {X}[c]/\mathbb{C}(t)[c]$. To have a family of projective surfaces over $\mathbb{C}(t)[c]$ on which $G$ operates regularly, we have to blow up the model ${\mathcal X}[c]$ along the projective lines corresponding to the Riccati type solutions.

First we show that any complex Lie group is complete Kähler. Moreover we obtain a plurisubharmonic exhaustion function on a complex Lie group as follows. Let ${\frak k}$ the real Lie algebra of a maximal compact real Lie subgroup $K$ of a complex Lie group $G$. Put $q:=\dim_ {\Bbb C} {\frak k} \cap \sqrt{-1} {\frak k}$. Then we obtain that there exists a plurisubharmonic, strongly $(q + 1)$-pseudoconvex in the sense of Andreotti-Grauert and $K$-invariant exhaustion function on $G$.

We prove a vanishing theorem for the $\overline{\partial_b}$-cohomology in top degree on 1-concave $CR$ generic manifolds.

We denote by $FPMC$ the class of all non-singular projective algebraic surfaces $X$ over $\mathbb{c}$ with finite polyhedral Mori cone $NE(X)$ $\subset NS(X)\otimes \mathbb{R}$. If $\rho(X)=rk \,\, NS(X)\ge 3$, then the set Exc$(X)$ of all exceptional curves on $X\in FPMC$ is finite and generates NE$(X)$. Let $\delta_E(X)$ be the maximum of $(-C^2)$ and $p_E(X)$ the maximum of $p_a(C)$ respectively for all $C\in \,$Exc$(X)$. For fixed $\rho\ge 3$, $\delta_E$ and $p_E$ we denote by $FPMC_{\rho,\delta_E,p_E}$ the class of all algebraic surfaces $X\in FPMC$ such that $\rho(X)=\rho$, $\delta_E(X)=\delta_E$ and $p_E(X)=p_E$. We prove that the class $FPMC_{\rho,\delta_E,p_E}$ is bounded in the following sense: for any $X\in FPMC_{\rho,\delta_E,p_E}$ there exist an ample effective divisor $h$ and a very ample divisor $h'$ such that $h^2\le N(\rho,\,\delta_E)$ and ${h'}^2\le N'(\rho,\,\delta_E,\,p_E)$ where the constants $N(\rho,\,\delta_E)$ and $N'(\rho,\,\delta_E,\,p_E)$ depend only on $\rho,\,\delta_E$ and $\rho,\,\delta_E,\,p_E$ respectively.

One can consider Theory of surfaces $X\in FPMC$ as Algebraic Geometry analog of the Theory of arithmetic reflection groups in hyperbolic spaces.

Kaneko [2] proved a new recurrence formula for the Bernoulli numbers and gave two proofs. One of them was due to Don Zagier. We shall apply Zagier's idea to the $q$-Bernoulli numbers attached to formal group.

For real $x \ge y \ge 2$ and positive integers $a$, $q$, let $\Phi(x, y; a, q)$ denote the number of positive integers${} \le x$, free of prime factors${} \le y$ and satisfying $n \equiv a$ $\pmod q$. By the fundamental lemma of sieve, it follows that for $(a, q) = 1$, $\Phi(x, y; a, q) = \varphi(q)^{-1}$, $\Phi(x, y) \{1 + O(\exp(-u (\log u - \log_{2} 3 u - 2))) + O(\exp(-\sqrt{\log x} / 2))\}$ ($u = \log x / \log y$) holds uniformly in a wider ranges of $x$, $y$ and $q$.

Let $\chi$ be any character to the modulus $q$, and $L(s, \chi)$ be the corresponding $L$-function. Let $\tilde{\chi}$ be a (`exceptional') real character to the modulus $q$ for which $L(s,\tilde\chi)$ have a (`exceptional') real zero $\tilde\beta$ satisfying $\tilde\beta>1-c_0/\log q$. In the paper, we prove that in a slightly short range of $q$ the above first error term can be replaced by $\tilde\chi(a)\varphi(q)^{-1} \cdot x^{\tilde\beta}\rho'(u)(\tilde\beta\log y)^{-1} (1 + O((\log y)^{-1/2}))$, where $\rho(u)$ is Dickman function, and $\rho'(u)=d\rho(u)/du$.

The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that if suppose that $1<q<(\log q)^{A}$.

In the present work we describe 3-dimensional complex $SL_2$-varieties where the generic $SL_2$-orbit is a surface. We apply this result to classify the minimal 3-dimensional projective varieties with Picard-number 1 where a semisimple group acts such that the generic orbits are 2-dimensional.

This is an ingredient of the classification [Keb99] of the 3-dimensional relatively minimal quasihomogeneous varieties where the automorphism group is not solvable.

In the present work we classify the relatively minimal 3-dimensional quasihomogeneous complex projective varieties under the assumption that the automorphism group is not solvable. By relatively minimal we understand varieties $X$ having at most $\mathbb{Q}$-factorial terminal singularities and allowing an extremal contraction $X \to Y$ where $\dim \, Y < 3$.

The Dirichlet problem of prescribed mean curvature equations is well posed, if the boundery is H-convex. In this article we eliminate the H-convexity condition from a portion $\Gamma$ of the boundary and prove the existence theorem, where the boundary condition is satisfied on $\Gamma$ in the weak sense.

The joint universality theorem for Lerch zeta-functions $L(\lambda_l,\alpha_l,s)$ ($1\leq l \leq n$) is proved, in the case when $\lambda_l$s are rational numbers and $\alpha_l$s are transcendental numbers. The case $n=1$ was known before ([12]); the rationality of $\lambda_l$s is used to establish the theorem for the "joint" case $n\geq 2$. As a corollary, the joint functional independence for those functions is shown.