International Journal of Mathematics

We prove that the numerical invariant $3\mu -4\tau$ of a reduced irreducible plane curve singularity germ is non-negative, non-decreasing under blowups and strictly increasing unless the curve is non-singular. This provides a new perspective to understand the question posed by Dimca and Greuel. Moreover, our work can be put in the general framework of discovering monotonic invariants under blowups.

In this paper, we present explicit toric construction of moduli space of quasi maps from ${\mathbb{P}}^1$ with two marked points to ${\mathbb{P}}^1\times {\mathbb{P}}^1$, which was first proposed by Jinzenji and prove that it is a compact orbifold. We also determine its Chow ring and compute its Poincaré polynomial for some lower degree cases.

In this paper, we give an equivalent characterization of conformal vector fields on a Finsler manifold $M^n(n\ge 3)$, whose metric $F$ is defined by a Riemannian metric $\alpha$ and a 1-form $\beta$. This characterization contains all related results in [Z. Shen and Q. Xia, On conformal vector fields on Randers manifolds, Sci. China Math. 55(9) (2012) 1869–1882; Z. Shen and M. Yuan, Conformal vector fields on some Finsler manifolds, Sci. China Math. 59(1) (2016) 107–114; X. Cheng, Y. Li and T. Li, The conformal vector fields on Kropina manifolds, Diff. Geom. Appl. 56 (2018) 344–354] as special cases. Further, we determine conformal fields on some Finsler manifolds $(M,F)$ when $\alpha$ is of constant sectional curvature and $\beta$ is a conformal 1-form with respect to $\alpha$.

We prove that for any type $F_4$ unitary affine VOA $V_{{𝔣}_4}^l$, sufficiently many intertwining operators satisfy polynomial energy bounds. This finishes the Wassermann type analysis of intertwining operators for all WZW-models.

We study the following question: Given a vector bundle on a projective variety $X$ such that the restriction of $E$ to every closed curve $C\subset X$ is ample, under what conditions $E$ is ample? We first consider the case of an abelian variety $X$. If $E$ is a line bundle on $X$, then we answer the question in the affirmative. When $E$ is of higher rank, we show that the answer is affirmative under some conditions on $E$. We then study the case of $X=G/P$, where $G$ is a reductive complex affine algebraic group, and $P$ is a parabolic subgroup of $G$. In this case, we show that the answer to our question is affirmative if $E$ is $T$-equivariant, where $T\subset P$ is a fixed maximal torus. Finally, we compute the Seshadri constant for such vector bundles defined on $G/P$.

In this paper, we introduce the notion of generalized Kodaira dimension with multiplier ideal sheaves, and prove the subadditivity of these generalized Kodaira dimensions for certain Kähler fibrations, which was originally obtained for Kodaira dimensions of algebraic fiber spaces by Kawamata and Viehweg. Our method is analytic and based on some new results in recent years. The crucial step in our proof is to prove an $L^{\frac{2}{k}}$ extension theorem for twisted pluricanonical sections on compact Kähler manifolds. Moreover, we also discuss the relation between two previous optimal $L^2$ extension theorems with singular weights on weakly pseudoconvex Kähler manifolds.

In this paper, we consider the matrix of the Green function of the Stokes equations for a half space. To the best of author’s knowledge, the matrix of the Green function has been obtained for a half space in ${ℝ}^n(n=2,3)$. But the matrix of the Green function for a half space in ${ℝ}^n(n\ge 4)$ has not been obtained. Moreover, the author considers that the form of the matrix of the Green function in ${ℝ}^n(n=2,3)$ is irregular.

In this paper, we obtain the regular form of the matrix of the Green function of the Stokes equations for a half space in ${ℝ}^n(n\ge 2)$. Moreover we obtain the Green representation formula of the Stokes equations for a half space in ${ℝ}^n(n\ge 2)$.

Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin${}^c$ manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin${}^c$ case the result of Kowalski stating that, every totally umbilical hypersurface of an Einstein manifold of dimension greater or equal to $3$ is of constant mean curvature. As an application, we prove that there are no extrinsic hypersheres in complete Riemannian $\mathrm{\text{Spin}}$ manifolds of non-constant sectional curvature carrying a parallel, Killing or imaginary Killing spinor.

This paper concerns the continuity of the solution to the even $L_p$ Minkowski problem in the plane. When $0<p<1$, it is proved that the weak convergence of the even $L_p$ surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric. Moreover, the continuity of the solution to the even $L_p$ Minkowski problem with respect to $0<p<1$ is also obtained.

In this paper, we study a class of Borel measures on ${ℝ}^n$ that arises as the class of representing measures of Herglotz-Nevanlinna functions. In particular, we study product measures within this class where products with the Lebesgue measures play a special role. Hence, we give several characterizations of the $n$-dimensional Lebesgue measure among all such measures and characterize all product measures that appear in this class of measures. Furthermore, analogous results for the class of positive Borel measures on the unit poly-torus with vanishing mixed Fourier coefficients are also presented, and the relation between the two classes of measures with regard to the obtained results is discussed.

In this paper, we make a correction to Theorem 3.3 of the aforementioned paper (Internat. J. Math. 28(13) (2017) 1750099). We also provide a counterexample to the theorem and point out an error in the proof.