arXiv subjects
Publications and source records attributed to Shuho Kanda.
We construct a simply connected solvable Lie group $G$ admitting lattices and a left-invariant complex structure $J$ such that $(G,J)$ is not Stein. This provides a counterexample to Hasegawa's conjecture on the Stein property of simply connected unimodular solvable Lie groups with left-invariant complex structures.
We prove that every simply connected nilpotent Lie group endowed with a left-invariant nilpotent complex structure is biholomorphic to $\mathbb{C}^n$. Moreover, we construct such a biholomorphism explicitly by polynomial maps in exponential coordinates. As a consequence, every lattice in such a Lie group admits a free, properly discontinuous and cocompact action on $\mathbb{C}^n$ by holomorphic polynomial automorphisms. We interpret this consequence as a holomorphic analogue of polynomial crystallographic actions introduced by Dekimpe, Igodt, and Lee.
We investigate complex structures on the Oeljeklaus-Toma manifolds. The Oeljeklaus-Toma manifolds are defined using complex embeddings of number fields. By replacing these embeddings with their conjugates, one obtains other manifolds that share the same underlying differential structure. In this paper, we give an algebraic description of when such manifolds are biholomorphic. As a simple application, we obtain compact non-K\"{a}hler manifolds of dimension $2t+1$ that admit $2^t$ different rigid complex structures.
We show that for a certain class of solvable Lie groups, if they admit a left-invariant non-Vaisman locally conformally K\"{a}hler metric and a lattice, they must arise from the construction of Oeljeklaus-Toma manifolds. This result provides a natural explanation for why number-theoretic considerations play a role in the construction of Oeljeklaus-Toma manifolds.
An iterated circumcenter sequence (ICS) in dimension $d$ is a sequence of points in $\mathbb{R}^d$ where each point is the circumcenter of the preceding $d+1$ points. The purpose of this paper is to completely determine the parameter space of ICSs and its subspace consisting of periodic ICSs. In particular, we prove Goddyn's conjecture on periodic ICSs, which was independently proven recently by Ardanuy. We also prove the existence of a periodic ICS in any dimension.
We prove the hard Lefschetz duality for locally conformally almost K\"{a}hler manifolds. This is a generalization of that for almost K\"{a}hler manifolds studied by Cirici and Wilson. We generalize the K\"{a}hler identities to prove the duality. Based on the result, we introduce the hard Lefschetz condition for locally conformally symplectic manifolds. As examples, we give solvmanifolds which do not satisfy the hard Lefschetz condition.