arXiv research

We define the Kodaira dimension for $3$-dimensional manifolds through Thurston's eight geometries, along with a classification in terms of this Kodaira dimension. We show this is compatible with other existing Kodaira dimensions and the partial order defined by non-zero degree maps. For higher dimensions, we explore th…

Defines and classifies Thurston geometries and connects simplicial volume to Kodaira dimension.

Study Kodaira dimensions on almost complex manifolds, proving integrability and structural descriptions.

Study on surface bundles, CAT(0) groups, and mapping class groups.

We construct classes of Kähler groups that do not have finite classifying spaces and are not commensurable to subdirect products of surface groups. Each of these groups is the fundamental group of the generic fibre of a holomorphic map from a product of Kodaira fibrations onto an elliptic curve.

Analyzes complex structure deformations using cohomology contraction methods.

Study curvature of direct image bundles in deformations of maps.

A question of Griffiths-Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for the class of algebraic surfaces known as Atiyah-Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in …

The fundamental group $π$ of a Kodaira fibration is, by definition, the extension of a surface group $Π_b$ by another surface group $Π_g$, i.e. \[ 1 \rightarrow Π_g \rightarrow π\rightarrow Π_b \rightarrow 1. \] Conversely, we can inquire about what conditions need to be satisfied by a group of that sort in order to be…

This is the first of a series of papers, in which we study the plurigenera, the Kodaira dimension and more generally the Iitaka dimension on compact almost complex manifolds. Based on the Hodge theory on almost complex manifolds, we introduce the plurigenera, Kodaira dimension and Iitaka dimension on compact almost com…

Study of Fubini-Study forms on surfaces with punctures.

We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of non-zero degree.

New classification for Vaisman manifolds with specific properties.

Extends Kodaira Spencer to higher-dimensional Calabi-Yau manifolds.

Study on constant mean curvature 1-immersions into hyperbolic 3-manifolds.

Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.

Study on Kodaira dimension of SU(m)-structures on almost complex manifolds.

Automorphisms of Kodaira surfaces are shown to be affine transformations.

Computational techniques calculate dimensions of complex structures.

In this note we show that the Lagrangian Luttinger surgery preserves the symplectic Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.

Study extra-special quotients of surface braid groups and construct double Kodaira fibrations.

Study on Vaisman metrics on Kodaira-Thurston surface using pluriclosed flow.

We bound the index of a subgroup in iterated Kodaira fibrations.

New findings on Frobenius structures on Kodaira manifolds.

Generalizes Kodaira vanishing theorem to Kahler Lie algebroids.

Let S be a closed surface of genus g >= 2 and z in S a marked point. We prove that the subgroup of the mapping class group Map(S,z) corresponding to the fundamental group pi_1(S,z) of the closed surface does not lift to the group of diffeomorphisms of S fixing z. As a corollary, we show that the Atiyah-Kodaira surface …

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective…

The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost Kähler manifolds, providing an explicit computation for a family of almost Kähler threefolds on the differentiable manifold underlying a Nakamura manifold. We concent…

We develop a non-relativistic twistor theory, in which Newton--Cartan structures of Newtonian gravity correspond to complex three-manifolds with a four-parameter family of rational curves with normal bundle ${\mathcal O}\oplus{\mathcal O}(2)$. We show that the Newton--Cartan space-times are unstable under the general K…

The paper explores Kodaira dimension on almost complex manifolds.

The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.

Method solves $\bar\partial$-harmonic forms on Kodaira-Thurston manifold.

We define analogue of theta-functions on the Kodaira--Thurston manifold which is a compact 4-dimensional symplectic manifold and use them to construct canonical symplectic embedding of the Kodaira--Thurston manifold into the complex projective space (analogue of the Lefshetz theorem).

We provide infinitely many examples of pairs of diffeomorphic, non simply connected K\" ahler manifolds of complex dimension three with different Kodaira dimensions. Also, in any possible Kodaira dimension we find infinitely many pairs of non deformation equivalent, diffeomorphic K\" ahler threefolds.

We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier …

The paper examines Kodaira dimensions of specific complex 4-manifolds with torsion first Chern class.

Historically tensor calculus emerged in an attempt to formalize Rie- mann's ideas. We show that tensor calculus can be based also on Lie's idea of a transformation group and this approach leads quite naturally to the concept of deformation of a transformation group and the Kodaira- Spencer map.

Study on Kodaira dimension of specific solvmanifolds without complex structures.

Abstract: Almost Kähler Hodge numbers vary with metric choices.

Analyzes Kodaira-Iitaka dimension and multiplicity using intersection theory.

The paper explores curvature constraints on Kodaira dimension for specific almost Hermitian manifolds.

A Kodaira fibration is a compact, complex surface admitting a holomorphic submersion onto a complex curve, such that the fibers have nonconstant moduli. We consider Kodaira fibrations X with nontrivial invariant rational cohomology in degree 1, proving that if the dimension of the holomorphic invariants is 1 or 2, then…

We will consider locally conformally balanced manifolds. We prove that a locally conformally balanced condition is not stable under a small deformation. We prove that locally conformally balanced condition is stable under any proper modification. We prove that symmetric products of the Kodaira surface can be resolve to…

Study of section conjecture analogues over complex numbers and Kodaira fibrations.

Paper extends Kodaira dimension's role in Yamabe invariant for most complex surfaces.

Study on the Kodaira dimension of real parallelizable manifolds with almost complex structures.

In this paper, by using analytical methods we obtain a generalization of the famous Kodaira embedding theorem.

This is a survey on the various notions of Kodaira dimension in low dimensional topology. The focus is on progress after the 2006 survey [78].