arXiv research

Study continuity equation on Hopf and Inoue surfaces, proving estimates and convergence.

We show that any hyperbolic Inoue surface (or Inoue-Hirzebruch surface of even type) admits anti-self-dual bihermitian structures. The same result also holds for any of its small deformations as far as its anti-canonical system is non-empty. Similar results are obtained for parabolic Inoue surfaces. Our method also yie…

Study automorphism groups of Inoue surfaces using quadratic number fields.

The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.

We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics, on Inoue surfaces. These are non-Kahler compact complex surfaces of type Class VII. We show that, after an initial conformal change, the flow always collapses the Inoue surface to a circle at infinite time, in the sense of Gromov-Hausdorff…

New complex manifolds found with flat structure.

We determine explicitly the structure of the automorphism group of a parabolic Inoue surface. We also describe the quotients of the surface by typical cyclic subgroups of the automorphism group.

Study on spectral points of Inoue surfaces with Tricerri metric.

Oeljeklaus-Toma manifolds are complex non-Kähler manifolds constructed by Oeljeklaus and Toma from certain number fields. These manifolds generalize Inoue surfaces of type $S_m$. In this work it is shown that Oeljeklaus-Toma manifolds could not contain any compact complex submanifolds of dimension 2 (surfaces) except I…

Yamabe invariants of certain non-Kähler surfaces are zero.

The abstract discusses compact quotients of Riemannian products by discrete subgroups, generalizing Inoue-Bombieri surfaces.

We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kähler metrics. We present explicit computations for the Inoue surfaces $\mathcal{S}^0$, $\mathcal{S}^+$, $\mathcal{S}^-$ and classify the locally conformally Kähler (and the tamed loc…

We prove that the natural $(\operatorname{Aff} _2(\mathbf{C}),\mathbf{C}^2)$-structure on an Inoue surface is the unique $(\operatorname{Bir}(\mathbb{P}^2),\mathbb{P}^2(\mathbf{C}))$-structure, generalizing a result of Bruno Klingler which asserts that the natural $(\operatorname{Aff}_2(\mathbf{C}),\mathbf{C}^2)$-struc…

This study examines torsion homology in Oeljeklaus-Toma manifolds, extending knot theory concepts.

In this note we discuss the problem of existence of para-hyperhermitian structures on compact complex surfaces. We construct examples of para-hypercomplex structures on Inoue surfaces of type $S^{-}$ which do not admit compatible metrics.

Study on properties of special Kähler metrics and their interplay.

In this paper, we study how the notions of geometric formality according to Kotschick and other geometric formalities adapted to the Hermitian setting evolve under the action of the Chern-Ricci flow on class VII surfaces, including Hopf and Inoue surfaces, and on Kodaira surfaces.

Inspired by a construction due to Hitchin, we produce strongly bihermitian metrics on certain Hopf complex surfaces, which integrate the locally conformally Kaehler metrics found by Gauduchon and Ornea. We also show that the Inoue complex surfaces with zero second Betti number do not admit bihermitian metrics. This com…

Oeljeklaus-Toma manifolds are complex non-Kähler manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces $S_m$. We prove that Oeljeklaus-Toma manifolds contain no compact complex curves.

Streets and Tian introduced a parabolic flow of pluriclosed metrics. We classify the long time behavior of homogeneous solutions of this flow on closed complex surfaces including minimal Hopf, Inoue, Kodaira, and non-Kahler, properly elliptic surfaces. We also construct expanding soliton solutions to the flow on the un…

This paper is about a generalization of famous Inoue's surfaces. Let $M$ be a matrix in $SL(2n+1,\mathbb{Z})$ having only one real eigenvalue which is simple. We associate to $M$ a complex manifold $T_M$ of complex dimension $n+1$. This manifold fibers over $S^1$ with the fiber $\mathbb{T}^{2n+1}$ and monodromy $M^\top…

Classifies meromorphic affine connections on complex surfaces.

Researchers compute Dolbeault cohomology of Endo-Pajitnov manifolds.

The last years have seen striking improvements on Vaisman's question about existence of locally conformally Kähler (lcK) metrics on compact complex surfaces. The aim of this paper is two-fold. We review results of different authors which, for all known examples of compact complex surfaces, give a complete answer to Vai…

Study bi-Hermitian metrics on complex surfaces and solve geometric PDEs.

We discuss the Morse-Novikov cohomology of a compact manifold, associated to a closed one--form whose free abelian group generated by its periods $\langle \int_γη\mid [γ] \in π_1(M)\rangle$ is of rank 1, the focus being on locally conformally symplectic manifolds. In particular, we provide an explicit computation for t…

In this paper we construct a family of complex analytic manifolds that generalize Inoue surfaces and Oeljeklaus-Toma manifolds. To a matrix $M$ in $SL(N,\mathbb{Z})$ satisfying some mild conditions on its characteristic polynomial we associate a manifold $T(M,\mathbf{D})$ (depending on an auxiliary parameter $\mathbf{D…

Study on complex submanifolds in Endo-Pajitnov manifolds.

We study the Morse-Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz Condition. We consider solvmanifolds and Oeljeklaus-Toma manifolds. In…

New metrics found on complex solvmanifolds.

The Chern-Ricci flow is an evolution equation of Hermitian metrics by their Chern-Ricci form, first introduced by Gill. Building on our previous work, we investigate this flow on complex surfaces. We establish new estimates in the case of finite time non-collapsing, anologous to some known results for the Kahler-Ricci …

Given a braid presentation $D$ of a hyperbolic knot, Hikami and Inoue consider a system of polynomial equations arising from a sequence of cluster mutations determined by $D$. They show that any solution gives rise to shape parameters and thus determines a boundary-parabolic $\mathrm{PSL}(2,\mathbb{C})$-representation …

The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces $S_m$. On each OT-manifold we construct a holomorphic line bundle with semipositive curvature form and trivial Chern class. Using this form, we prove that the OT-m…

We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkähler are analogs of the hypercomplex, hyperhermitian and hyperkähler structures in the definite case. We show that …

We propose a simple method to produce quandle cocycles from group cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between their third homologies. For example, all Mochizuki's quandle 3-cocycles are sho…

We prove that the deRham cohomology classes of Lee forms of locally conformally symplectic structures taming the complex structure of a compact complex surface $S$ with first Betti number equal to $1$ is either a non-empty open subset of $H^1_{dR}(S, \mathbb R)$, or a single point. In the latter case, we show that $S$ …

We revisit Brunella's proof of the fact that Kato surfaces admit locally conformally K\" ahler metrics, and we show that it holds for a large class of higher dimensional complex manifolds containing a global spherical shell. On the other hand, we construct manifolds containing a global spherical shell which admit no lo…

In this article we develop a new approach to the problem of the stability of locally conformally Kähler structures (l.c.k structures) under small deformations of complex structures and deformations of flat line bundles. We show that under the certain cohomological condition the stability of l.c.k structures does hold. …

The Oeljeklaus-Toma (OT-) manifolds are compact, complex, non-Kahler manifolds constructed by Oeljeklaus and Toma, and generalizing the Inoue surfaces. Their construction uses the number-theoretic data: a number field $K$ and a torsion-free subgroup $U$ in the group of units of the ring of integers of $K$, with rank of…

This study examines arithmetic properties of GIB manifolds and their monodromy representations.

We describe explicitly the moduli spaces $M^{pst}_g(S,E)$ of polystable holomorphic structures $E$ with $\det E\cong K$ on a rank 2 vector bundle $E$ with $c_1(E)=c_1(K)$ and $c_2(E)=0$ for all minimal class VII surfaces $S$ with $b_2(S)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are …

Paper constructs representations for virtual braids and flat braids.

We discuss the fundamental (relative) 3-classes of knots (or hyperbolic links), and provide diagrammatic descriptions of the push-forwards with respect to every link-group representation. The point is an observation of a bridge between the relative group homology and quandle homology from the viewpoints of Inoue--Kabay…

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module $M$ over the Laurent polynomial ring $Λ_μ=\mathbb{Z}[t_1^{\pm1},\dots,t_μ^{\pm1}]$. If $D$ is a diagram of a link $L$ with $μ$ components, then the colorings of $D$ with values in $M$ form a $Λ_μ$-module…

Study of special Kato manifolds derived from toric geometry.

Characterizes pluriclosed metrics on Oeljeklaus-Toma manifolds.

Stability of weighted extremal manifolds proven through blowups.

One of the main themes of this long article is the study of projective varieties which are K(H,1)'s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera-de Franchis varieties, the qu…