Let $X$ denote a smooth projective variety of dimension $n$ defined over the field of complex numbers such that the anti-canonical line bundle $-K_X$ of $X$ is nef and big with $h^{0}(-K_{X})>0$, and let $L$ be a nef and big line bundle on $X$. In this paper, we consider the dimension of the global sections of $K_{X}+mL$ with $m\geq n-1$ for this case. In particular, under the assumption that $K_{X}+(n-1)L$ is nef, we prove that $h^{0}(K_{X}+(n-1)L)>0$ if $6\leq n\leq 9$ and $L$ is ample.

In this paper, we consider the non-linear general $p$-Laplacian equation $\Delta_{p,f}u+F(u)=0$ for a smooth function $F$ on smooth metric measure spaces. Assume that a Sobolev inequality holds true on $M$ and an integral Ricci curvature is small, we first prove a local gradient estimate for the equation. Then, as its applications, we prove several Liouville type results on manifolds with lower bounds of Ricci curvature. We also derive new local gradient estimates provided that the integral Ricci curvature is small enough.

In an author's joint work with Hoshi and Mochizuki, we introduced the notion of TKND-AVKF-field [concerning the divisible subgroups of the groups of rational points of semi-abelian varieties] and obtained an anabelian Grothendieck Conjecture-type result for higher dimensional configuration spaces associated to hyperbolic curves over TKND-AVKF-fields. On the other hand, every concrete example of TKND-AVKF-field that appears in this joint work is a $\times\mu$-indivisible field [i.e., a field such that any divisible element of the multiplicative group of the field is a root of unity]. In the present paper, we construct new examples of TKND-AVKF-fields that are not $\times\mu$-indivisible.

We introduce a notion of highly Kummer-faithful fields and study its relationship with the notion of Kummer-faithful fields. We also give some examples of highly Kummer-faithful fields. For example, if $k$ is a number field of finite degree over \mathbb{Q}, $g$ is an integer $> 0$ and $\mathbf{m}=(m_p)_p$ is a family of non-negative integers, where $p$ ranges over all prime numbers, then the extension field $k_{g,\mathbf{m}}$ obtained by adjoining to $k$ all coordinates of the elements of the $p^{m_p}$-torsion subgroup $A[p^{m_p}]$ of $A$ for all semi-abelian varieties $A$ over $k$ of dimension at most $g$ and all prime numbers $p$, is highly Kummer-faithful.

Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s \zeta_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta functions over $\mathbf Q(x)$, where $\zeta_N (s,x \ |\ w_1,\ldots, w_N)$ is the Barnes zeta function and $w$ is a positive rational number explicitly determined by $w_1,\ldots, w_N$. Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-positive integers for higher order derivatives of the Barnes zeta function in the case that $x$ is a positive rational number, involving the generalized Stieltjes constants and the values at positive integers of the Riemann zeta function. Our formulae also makes it possible to calculate an approximation in the case that $w_1, \ldots, w_N$ and $x$ are positive real numbers.

In this paper, we prove Souplet-Zhang type gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with the compact boundary under the Dirichlet boundary condition when the Bakry-Emery Ricci tensor and the weighted mean curvature are both bounded below. As an application, we obtain a new Liouville type result for some space-time functions on such smooth metric measure spaces. These results generalize previous linear equations to a nonlinear case.

Dunfield, Friedl and Jackson make a conjecture that the hyperbolic torsion polynomial determines the genus and fibering of hyperbolic knots. In this paper, we study a similar problem for the adjoint torsion polynomial, and show that it determines the genus and fibering of a large family of hyperbolic genus one two-bridge knots.

We investigate minimal surfaces in products of two-spheres $\mathbf{S}^2_p\times\mathbf{S}^2_p$, with the neutral metric given by $(g,-g)$. Here $\mathbf{S}^2_p\subset \mathbf{R}^{p,3-p}$, and $g$ is the induced metric on the sphere. We compute all totally geodesic surfaces and we give a relation between minimal surfaces and the solutions of the Gordon equations. Finally, in some cases we give a topological classification of compact minimal surfaces.

In this paper, we investigate the eigenvalues of the drifting Laplacian on the bounded domain in Hardamard manifolds. By using upper half-plane model, we establish a universal inequality for the drifting Laplacian with a specific class of potential functions on the hyperbolic space, which can be viewed as a rigidity result associated with such a class of functions. Furthermore, we consider general Hardamard manifolds with pinching condition of sectional curvature, and obtain an eigenvalue inequality without the condition of Bakry-Émery curvature. As a by-product, we obtain a universal bound for the case of radial drifting Laplacian satisfying certain growth condition along with the radial direction.

We give a construction of singular K3 surfaces with discriminant 3 and 4 as double coverings over the projective plane. Focusing on the similarities in their branching loci, we can generalize this construction, and obtain a three dimensional moduli space of certain K3 surfaces which admit infinite automorphism groups. Moreover, we show that these K3 surfaces are characterized in terms of the configuration of the singular fibres of a jacobian elliptic fibration and also in terms of periods.