International Journal of Mathematics

We completely classify K-stability of log del Pezzo hypersurfaces with index 2.

We give a generalized Thurston–Bennequin-type inequality for links in $S^3$ using a Bauer–Furuta-type invariant for four-manifolds with contact boundary introduced by the first aut hor. As a special case, we also give an adjunction inequality for smoothly embedded orientable surfaces with negative intersection in a closed oriented smooth four-manifold whose non-equivariant Bauer–Furuta invariant is nonzero.

We study when the twisted groupoid Banach *-algebra $L^1({𝒢},\sigma )$ is Hermitian. In particular, we prove that Hermitian groupoids satisfy the weak containment property. Furthermore, we find that for $L^1({𝒢},\sigma )$ to be Hermitian it is sufficient that $L^1({{𝒢}}_{\sigma })$ is Hermitian. Moreover, if ${𝒢}$ is ample, we find necessary conditions for $L^1({𝒢},\sigma )$ to be Hermitian in terms of the fibers ${{𝒢}}_x^x$.

Let $(M,{ℱ},g_0)$ be a Riemannian minimal foliation. The transverse Yamabe problem is to find a metric $g$ in the basic conformal class of $g_0$ such that the transverse scalar curvature of $g$ is constant. We first study the uniqueness of the solutions of the transverse Yamabe problem. As a generalization of the transverse Yamabe problem, we study the problem of prescribing transverse scalar curvature by using geometric flow. We then prove a version of conformal Schwarz lemma on $(M,{ℱ},g_0)$. Finally, we consider the transverse Yamabe soliton, which is the self-similar solution of the transverse Yamabe flow.

A Hermitian metric on a complex manifold is called SKT (strong Kähler with torsion) if the Bismut torsion 3-form $H$ is closed. As the conformal generalization of the SKT condition, we introduce a new type of Hermitian structure, called locally conformal SKT (or shortly LCSKT). More precisely, a Hermitian structure $(J,g)$ is said to be LCSKT if there exists a closed nonzero $1$-form $\alpha$ such that $dH=\alpha \wedge H$. In this paper, we consider nontrivial LCSKT structures, i.e. we assume that $dH\ne 0$ and we study their existence on Lie groups and their compact quotients by lattices.

In particular, we classify six-dimensional nilpotent Lie algebras admitting a LCSKT structure and we show that, in contrast to the SKT case, there exists a six-dimensional 3-step nilpotent Lie algebra admitting a nontrivial LCSKT structure. Moreover, we show a characterization of even dimensional almost abelian Lie algebras admitting a nontrivial LCSKT structure, which allows us to construct explicit examples of six-dimensional unimodular almost abelian Lie algebras admitting a nontrivial LCSKT structure. The compatibility between the LCSKT and the balanced condition is also discussed, showing that a Hermitian structure on a six-dimensional nilpotent or a $2n$-dimensional almost abelian Lie algebra cannot be simultaneously LCSKT and balanced, unless it is Kähler.

In this paper, we make a correction to Theorem 1.2 of the aforementioned paper [J. Liu, Fano manifolds containing a negative divisor isomorphic to a rational homogeneous space of Picard number one, Int. J. Math. 31(9) (2020) 2050066].

This is a corrigendum to the authors paper [A reconstruction theorem for complex polynomials, Int. J. Math. 26(9) (2015) 1550073]. Here we present a completely rewritten and self-contained Sec. 4 of the aforementioned paper and provide the reader with the corrected proof of the main result that briefly states: “Given a generic polynomial $P$ of degree at least $2,$ there exist positive integers $N,M$ such that $\text{Re}(P^k(z))=\text{Re}(P^k(w))$ for all $0\le k\le N$ implies $P^M(z)=P^M(w)$ ”.