Study continuity equation on Hopf and Inoue surfaces, proving estimates and convergence.
problem Analyzing the continuity equation on specific complex surfaces.
method Extended La Nave-Tian's continuity equation to Hermitian setting, proving estimates and Gromov-Hausdorff convergence.
result Proved a priori estimates for solutions on Hopf and Inoue surfaces, and convergence of Inoue surfaces to a circle.
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.
problem Understanding automorphism groups of Inoue surfaces.
method Construction and description of automorphism groups using quadratic number fields.
result Automorphism groups of Inoue surfaces S(+)/S(−) described in terms of quadratic number fields. The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
problem Proving the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
method Using the ∂∂-class, the study proves the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces. result Uniform convergence of the normalized Chern-Ricci flow starting at any Gauduchon metric on all Inoue-Bombieri surfaces, with smooth convergence and bounded curvature for initial metrics in the ∂∂-class of the Tricerri/Vaisman metric. 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…
Unique birational structure proven on Inoue surfaces.
problem Proving uniqueness of birational structures on Inoue surfaces.
method Generalizing a result by Bruno Klingler, proving uniqueness of structures.
result The natural (Aff2(C),C2)-structure on an Inoue surface is the unique (Bir(P2),P2(C))-structure. New complex manifolds found with flat structure.
problem Finding compact complex manifolds with flat affine structure.
method Using Lie groups with left-invariant complex structure.
result Retrieved Inoue surfaces S+ in 2D. Authors compute Morse-Novikov cohomology for Inoue surfaces and prove nonexistence of certain metrics.
problem Computing and classifying locally conformally Kähler metrics on complex surfaces.
method Review and computation of Morse-Novikov cohomology for known surfaces.
result Nonexistence of LCK metrics with potential on Inoue surfaces and Oeljeklaus-Toma manifolds.
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.
problem Identifying spectral points on Inoue surfaces.
method Analyzing chiral Dirac operators twisted by flat C∗-connections. result No spectral points inside the annulus α−1/4<∣z∣<α1/4, with spectral points on boundary. Oeljeklaus-Toma manifolds are complex non-Kähler manifolds constructed by Oeljeklaus and Toma from certain number fields. These manifolds generalize Inoue surfaces of type Sm. 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.
problem Determining the sign of Yamabe invariants for non-Kähler surfaces.
method Analyzing Inoue surfaces and Kodaira surfaces, their blowups, and applying Seiberg-Witten theory.
result Yamabe invariants of Inoue surfaces and their blowups are all zero.
The abstract discusses compact quotients of Riemannian products by discrete subgroups, generalizing Inoue-Bombieri surfaces.
problem Compact quotients of Riemannian products by discrete subgroups.
method Study of compact quotients of a Riemannian product Rqimes(N,gN) by discrete subgroups Γ of Sim(Rq)imesIsom(N). result The construction is equivalent to LCP manifolds and provides a Bieberbach-type rigidity result.
The paper generalizes Inoue surfaces using matrices and complex manifolds.
problem Characterizing and understanding new complex manifolds.
method Using matrices in SL(2n+1, Z) to construct complex manifolds.
result Some Oeljeklaus-Toma manifolds are biholomorphic to constructed manifolds.
This study examines torsion homology in Oeljeklaus-Toma manifolds, extending knot theory concepts.
problem Investigating torsion homology in Oeljeklaus-Toma manifolds.
method Adapting knot theory concepts to Oeljeklaus-Toma manifolds and extending to higher homology groups.
result Torsion grows exponentially in H1 and H2 for Inoue surfaces of type S0. The paper constructs new complex manifolds generalizing Inoue and OT manifolds.
problem Creating new complex manifolds that generalize existing ones.
method Associate manifolds T(M,D) to matrices M in SL(N,Z). result For certain non-diagonalizable matrices, these manifolds do not admit Kähler structures or are not homeomorphic to OT manifolds.
Study Morse-Novikov cohomology for 1-forms on rank 1 manifolds.
problem Analyzing cohomology of closed one-forms on manifolds.
method Explicit computation and discussion of locally conformally symplectic manifolds.
result Explicit computation for Inoue surface S^0.
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.
problem Properties and interplay of Strong Kähler with torsion and astheno-Kähler metrics.
method Analyzes families of astheno-Kähler nilmanifolds, studies complex blowups, and investigates interplay between metrics.
result Existence of astheno-Kähler metrics is not preserved by blowup, and some metrics are geometrically Bott-Chern formal.
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 Sm. 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…
The paper examines geometric formality on specific surfaces under the Chern-Ricci flow.
problem Understanding geometric formality on class VII surfaces.
method Analysis of Chern-Ricci flow on specific surfaces.
result Evolution of geometric formality under the Chern-Ricci flow.
Classifies meromorphic affine connections on complex surfaces.
problem Investigating uniformization in higher dimensions with singularities.
method Extending work on holomorphic connections, classifying meromorphic connections on compact surfaces.
result Classification of meromorphic affine connections on compact complex surfaces.
Researchers compute Dolbeault cohomology of Endo-Pajitnov manifolds.
problem Computing Dolbeault cohomology for a new class of non-Kähler manifolds.
method Computed Dolbeault cohomology using the Hodge decomposition.
result Endo-Pajitnov manifolds satisfy the Hodge decomposition at the level of dimensions.
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.
problem Construct canonical metrics on complex surfaces with split tangent bundle.
method Introduced new fully non-linear geometric PDEs and established smooth solutions.
result Solved the prescribed Bismut Ricci problem on complex surfaces.
Study on complex submanifolds in Endo-Pajitnov manifolds.
problem Existence and characterization of complex submanifolds in Endo-Pajitnov manifolds.
method Identification of a class of Endo-Pajitnov manifolds containing compact complex submanifolds and establishment of an algebraic condition for the absence of compact complex curves.
result Established an algebraic condition for the absence of compact complex curves in Endo-Pajitnov manifolds.
New metrics found on complex solvmanifolds.
problem Characterizing new types of metrics on complex solvmanifolds.
method Investigated higher-dimensional analogues of Inoue surfaces, provided solvmanifold structure, and characterized metrics.
result Found new examples of special metrics in all complex dimensions.
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 …
The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces Sm. 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…
Study on fundamental 3-classes of knot representations using diagrams and algorithms.
problem Understanding fundamental 3-classes of knot group representations.
method Diagrammatic descriptions and algebraic algorithms.
result Algebraic description of fundamental 3-class for hyperbolic knots.
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…
Study cohomologies on manifolds with locally conformally symplectic structures.
problem Understanding cohomologies on manifolds with locally conformally symplectic structures.
method Introduced lcs cohomologies, studied elliptic Hodge theory, dualities, and Hard Lefschetz Condition.
result Oeljeklaus-Toma manifolds with precisely one complex place and under an arithmetic condition satisfy the Mostow property.
Paper proves conjecture linking knot braid length to representation existence.
problem Existence of geometric knot representations from braid presentations.
method Analyzes cluster mutations and polynomial equations from braid presentations.
result Hikami-Inoue conjecture holds if and only if braid length is odd.
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. …
Study on Lee classes for complex surfaces, proving cohomology properties.
problem Characterizing Lee classes for complex surfaces.
method Analyzing deRham cohomology of Lee forms for locally conformally symplectic and Kähler structures.
result Characterizes Lee classes and cohomology groups for complex surfaces.
We describe explicitly the moduli spaces Mgpst(S,E) of polystable holomorphic structures E with detE≅K on a rank 2 vector bundle E with c1(E)=c1(K) and c2(E)=0 for all minimal class VII surfaces S with b2(S)=1 and with respect to all possible Gauduchon metrics g. These surfaces S are …
This study examines arithmetic properties of GIB manifolds and their monodromy representations.
problem Arithmetic structure of generalized Inoue--Bombieri manifolds.
method Study of monodromy representations and their arithmetic properties.
result The image of the monodromy representation is a subgroup of a cocompact arithmetic lattice.
Paper constructs representations for virtual braids and flat braids.
problem Calculating hyperbolic volumes of knot complements.
method Cluster algebra approach for virtual braid group.
result Forbidden relations do not hold in virtual braid group representation.
Study of Kato manifolds and their locally conformally Kähler properties.
problem Characterize Kato manifolds and their locally conformally Kähler metrics.
method Revisit Brunella's proof and construct new examples of Kato manifolds.
result Found a class of Kato manifolds that admit locally conformally Kähler metrics and another class that do not.
Flat affine subvarieties found in OT-manifolds.
problem Characterizing subvarieties in Oeljeklaus-Toma manifolds.
method Analyzing the structure of Oeljeklaus-Toma manifolds using number-theoretic data.
result Any complex subvariety of smallest possible positive dimension in an OT-manifold is flat affine.
Study of special Kato manifolds derived from toric geometry.
problem Characterize and study properties of Kato manifolds.
method Construction from toric geometry, topological and analytical properties, combinatorial data, flat degenerations, Hermitian geometry.
result No Kato manifold supports balanced or pluriclosed metrics.
Characterizes pluriclosed metrics on Oeljeklaus-Toma manifolds.
problem Characterizing existence of pluriclosed metrics on OT manifolds.
method Purely number-theoretical conditions.
result Explicit examples of pluriclosed OT manifolds in arbitrary complex dimension.
Extends link colorings to modules over Laurent polynomial rings, showing isomorphisms and dimensions.
problem Extending link colorings to modules over Laurent polynomial rings.
method Using Alexander quandles and modules over Laurent polynomial rings, showing isomorphisms and dimensions.
result Dimension of colorings as vector spaces over fields is determined by ring homomorphisms.
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…
Stability of weighted extremal manifolds proven through blowups.
problem Stability of weighted extremal manifolds.
method Blowup technique to analyze weighted extremal Kähler manifolds.
result Proves weighted extremal manifolds are relatively weighted K-polystable.