In this paper we discuss three distance functions on the set of convex bodies. In particular we study the convergence of Delzant polytopes, which are fundamental objects in symplectic toric geometry. By using these observations, we derive some convergence theorems for symplectic toric manifolds with respect to the Gromov–Hausdorff distance.

We will give a precise and explicit asymptotic estimate for the characteristic of the Riemann zeta function $\zeta $ with an error term of order $O(\frac {\log r}{r})$ and a corresponding asymptotic estimate for the number of fixed points of $\zeta $.

We prove the coherence of multiplier submodule sheaves associated with Griffiths semi-positive singular hermitian metrics over holomorphic vector bundles on complex manifolds which have no nontrivial subvarieties, such as generic complex tori.

In this paper, we first prove that the totally real discs lying in certain Levi-flat hypersurfaces are polynomially convex. We also studied the polynomial convexity of totally real discs lying in the regular part of certain singular Levi-flat hypersurfaces. In particular, a necessary and sufficient condition for polynomial convexity of totally real discs lying in the non-singular part of the boundary of the Hartogs triangle is achieved. Sufficient conditions on general compact subsets lying on those hypersurfaces for polynomial convexity are also reported here.

We determine explicit generators for the ring of modular forms associated with the moduli spaces of K3 surfaces with automorphism group $(\mathbb {Z}/2\mathbb {Z})^2$ and of Picard rank 13 and higher. The K3 surfaces in question carry a canonical Jacobian elliptic fibration and the modular form generators appear as coefficients in the Weierstrass-type equations describing these fibrations.

We study the moduli stacks of slope-semistable torsion-free coherent sheaves that admit reflexive, respectively, locally free, Seshadri graduations on a smooth projective variety. We show that they are open in the stack of coherent sheaves and that they admit good moduli spaces when the field characteristic is zero. In addition, in the locally free case we prove that the resulting moduli space is a quasi-projective scheme.

This article focuses on the representation theory of algebras associated with $\mathfrak {sl}_2$, including the affine Lie algebra $\widehat {\mathfrak {sl}_2}$, the affine Kac–Moody algebra $\widetilde {\mathfrak {sl}_2}$, and the affine-Virasoro algebra $\mathfrak {Vir}\ltimes \widehat {\mathfrak {sl}_2}$. First, we classify certain modules over these algebras, which are free of rank one when restricted to some specific subalgebras. We demonstrate a connection between these modules and modules over the Weyl algebras, which allows us to construct large families of modules that are free of arbitrary finite rank when restricted to the Cartan subalgebra. We then investigate the simplicity of these modules. For reducible modules, we fully characterize their composition factors. Through a comparison with existing simple modules in the literature, we have identified a novel family of simple modules over the affine Kac–Moody algebra $\widetilde {\mathfrak {sl}_2}$. Finally, we turn our attention to a class of tensor product modules over the affine-Virasoro algebra $\mathfrak {Vir}\ltimes \widehat {\mathfrak {sl}_2}$. We derive a necessary and sufficient condition for the simplicity of these modules and determine their isomorphism classes.

It was conjectured by McKernan and Shokurov that for any Fano contraction $f:X \to Z$ of relative dimension r with X being $\epsilon $-lc, there is a positive $\delta $ depending only on $r,\epsilon $ such that Z is $\delta $-lc and the multiplicity of the fiber of f over a codimension one point of Z is bounded from above by $1/\delta $. Recently, this conjecture was confirmed by Birkar [9]. In this article, we give an explicit value for $\delta $ in terms of $\epsilon ,r$ in the toric case, which belongs to $O(\epsilon ^{2^r})$ as $\epsilon \rightarrow 0$. The order $O(\epsilon ^{2^r})$ is optimal in some sense.

In this article, we apply the Bestvina–Mess type formula for relatively hyperbolic groups, which is established by Tomohiro Fukaya, to automorphism groups of K3 surfaces, and we show that the virtual cohomological dimension of automorphism groups of K3 surfaces is determined by the covering dimension of the blown-up boundaries associated with their ample cones.

In this article, we show that any $\mathbb {A}^2$-fibration over a discrete valuation ring which is also an $\mathbb {A}^2$-form is necessarily a polynomial ring. Further, we show that separable $\mathbb {A}^2$-forms over principal ideal domains are trivial.