International Journal of Mathematics

In this paper, we study the structure theory of two classes of Lie superalgebras ${𝒮}(q)$ of Block type, where $q$ is a positive integer. In particular, the automorphism groups, derivation superalgebras and central extensions of ${𝒮}(q)$ are completely determined.

We give a new proof of the fact that the Maaß space is invariant under all Hecke operators. It is based on the characterization of the Maaß space by a symmetry relation and certain commutation relations of the Hecke algebra for the Jacobi group.

We give a concrete formula for the unique faithful trace on the finite simple non-AF labeled graph $C^{\ast }$-algebra $C^{\ast }(E_{\mathbb{Z}},{ℒ},{\overline{{ℰ}}}_{\mathbb{Z}})$ associated to the Thue–Morse sequence $(E_{\mathbb{Z}},{ℒ})$. Our result provides an alternative proof of the existence of a labeled graph $C^{\ast }$-algebra that is not Morita equivalent to any graph $C^{\ast }$-algebras. Furthermore, we compute the $K$-groups of $C^{\ast }(E_{\mathbb{Z}},{ℒ},{\overline{{ℰ}}}_{\mathbb{Z}})$ using the path structure of the Thue–Morse sequence.

In this paper, we study biconservative hypersurfaces in the four-dimensional Minkowski space ${𝔼}_1^4$. We give the complete explicit classification of biconservative hypersurfaces with diagonalizable shape operator in ${𝔼}_1^4$.

We first present a Skoda-type division theorem for holomorphic sections of line bundles on a projective variety which is essentially the most general, compared to previous ones. Then we revisit Geometric Effective Nullstellensatz and observe that even this general Skoda division is far from sufficient to yield stronger GEN such as ‘vanishing order $1$ division’, which could be used for finite generation of section rings by the basic finite generation lemma. To resolve this problem, we develop a notion of pseudo-division and show that it can replace the usual division in the finite generation lemma. We also give a vanishing order 1 pseudo-division result when the line bundle is ample.

The unimodality conjecture posed by Tolman in [L. Jeffrey, T. Holm, Y. Karshon, E. Lerman and E. Meinrenken, Moment maps in various geometries, http://www.birs.ca/workshops/2005/05w5072/report05w5072.pdf] states that if $(M,\omega )$ is a $2n$-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, then the sequence of Betti numbers $\{b_0(M),b_2(M),\dots ,b_{2n}(M)\}$ is unimodal, i.e. $b_i(M)\le b_{i+2}(M)$ for every $i<n$. Recently, the author and Kim [Y. Cho and M. Kim, Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points, Math. Res. Lett. 21(4) (2014) 691–696] proved that the unimodality holds in eight-dimensional case by using equivariant cohomology theory. In this paper, we generalize the idea in [Y. Cho and M. Kim, Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points, Math. Res. Lett. 21(4) (2014) 691–696] to an arbitrary dimensional case. We prove the conjecture in arbitrary dimension under the assumption that the moment map $H:M\to ℝ$ is index-increasing, which means that $\mathrm{\text{ind}}(p)<\mathrm{\text{ind}}(q)$ implies $H(p)<H(q)$ for every pair of critical points $p$ and $q$ of $H$, where $\mathrm{\text{ind}}(p)$ is the Morse index of $p$ with respect to $H$.

We prove that for any closed orientable connected $3$-manifold $L$ and any Lagrangian immersion of the connected sum $L\#(S^1\times S^2)$ either into the complex projective $3$-space $ℂP^3$ or into the product $ℂP^1\times ℂP^2$ of the complex projective line and the complex projective plane, there exists a Lagrangian embedding which is homotopic to the initial Lagrangian immersion. To prove this, we show that Eliashberg–Murphy’s $h$-principle for Lagrangian embeddings with a concave Legendrian boundary and Ekholm–Eliashberg–Murphy–Smith’s $h$-principle for self-transverse Lagrangian immersions with the minimal or near-minimal number of double points hold for six-dimensional simply connected compact symplectic manifolds.

Let $C$ be an integral and projective curve whose canonical model $C^{\prime }$ lies on a rational normal scroll $S$ of dimension $n$. We mainly study some properties on $C$, such as gonality and the kind of singularities, in the case where $n=2$ and $C$ is non-Gorenstein, and in the case where $n=3$, the scroll $S$ is smooth, and $C^{\prime }$ is a local complete intersection inside $S$. We also prove that the canonical model of a rational monomial curve with just one singular point lies on a surface scroll if and only if the gonality of the curve is at most $3$, and that it lies on a threefold scroll if and only if the gonality is at most $4$.

We construct a family of vertex algebras associated to the affine Lie algebra of polynomial current algebras of finite-dimensional abelian Lie algebras, along with their modules and logarithmic modules. These vertex algebras and their (logarithmic) modules are strongly $\mathbb{N}$-graded and quasi-conformal. We then show that matrix elements of products and iterates of logarithmic intertwining operators among these logarithmic modules satisfy certain systems of differential equations. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory developed by Huang, Lepowsky and Zhang.

Let $\pi :Z\to X$ be a holomorphic submersion from a complex manifold $Z$ onto a 1-convex manifold $X$ with exceptional set $S$ and $a:X\to Z$ a holomorphic section. Let $\phi :X\to [0,\infty )$ be a plurisubharmonic exhaustion function which is strictly plurisubharmonic on $X\backslash S$ with ${\phi }^{-1}(0)=S.$ For every holomorphic vector bundle $E\to Z$ there exists a neighborhood $V$ of $a(U\backslash S)$ for $U={\phi }^{-1}([0,c)),$ conic along $a(S),$ such that $E{|}_V$ can be endowed with Nakano strictly positive Hermitian metric. Let $g:X\to ℂ,$ $g^{-1}(0)\supset S$ be a given holomorphic function. There exist finitely many bounded holomorphic vector fields defined on a Stein neighborhood $V$ of $a(\overline{U}\backslash g^{-1}(0)),$ conic along $a(g^{-1}(0))$ with zeroes of arbitrary high order on $a(g^{-1}(0))$ and such that they generate $\mathrm{\text{ker}}D\pi {|}_{a(\overline{U}\backslash g^{-1}(0))}.$ Moreover, there exists a smaller neighborhood $V^{\prime }\subset V$ such that their flows remain in $V^{\prime }$ for sufficiently small times thus generating a local dominating spray.

In this paper, we give the boundedness of solutions to Ginzburg–Landau fractional Laplacian equation, which extends the Herve–Herve theorem into the nonlinear fractional Laplacian equation. We follow Brezis’ idea to use the Kato inequality. A related linear fractional Schrödinger equation is also studied.