Complex Codimension Research Articles

On transversely holomorphic foliations with homogeneous transverse structure

In this paper we study transversely holomorphic foliations of complex codimension one with a transversely homogeneous complex transverse structure. We prove that the only cases are the transversely additive, affine and projective cases. We shall focus on the transversely affine case and describe the holonomy of a leaf which is "at the infinity" with respect to this structure and prove this is a solvable group. Using this we are able to prove linearization results for the foliation under the assumption of existence of some hyperbolic map in the holonomy group. Such foliations will then be given by simple-poles closed transversely meromorphic one-forms.

A foliated Hitchin-Kobayashi correspondence

We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with transverse holomorphic structure and prove that such bundles admit a basic Hermitian-Einstein connection if and only if they are polystable. Our proof is obtained by adapting the proof by Uhlenbeck and Yau to the foliated setting. We relate the transverse Hermitian-Einstein equations to higher dimensional instanton equations and in particular we look at the relation to higher contact instantons on Sasakian manifolds. For foliations of complex codimension 1, we obtain a transverse Narasimhan-Seshadri theorem. We also demonstrate that the weak Uhlenbeck compactness theorem fails in general for basic connections on a foliated bundle. This shows that not every result in gauge theory carries over to the foliated setting.

Positive Hermitian curvature flow on nilpotent and almost-abelian complex Lie groups

We study the positive Hermitian curvature flow on the space of left-invariant metrics on complex Lie groups. We show that in the nilpotent case, the flow exists for all positive times and subconverges in the Cheeger-Gromov sense to a soliton. We also show convergence to a soliton when the complex Lie group is almost abelian. That is, when its Lie algebra admits a (complex) co-dimension one abelian ideal. Finally, we study solitons in the almost-abelian setting. We prove uniqueness and completely classify all left-invariant, almost-abelian solitons, giving a method to construct examples in arbitrary dimensions, many of which admit co-compact lattices.

Dessin d’enfant and a description of mutually nonlocal 7-branes without using branch cuts

We consider the special roles of the zero loci of the Weierstrass invariants $g_2(\tau(z))$, $g_3(\tau(z))$ in F-theory on an elliptic fibration over $P^1$ or a further fibration thereof. They are defined as the zero loci of the coefficient functions $f(z)$ and $g(z)$ of a Weierstrass equation. They are thought of as complex co-dimension one objects and correspond to the two kinds of critical points of a dessin d'enfant of Grothendieck. The $P^1$ base is divided into several cell regions bounded by some domain walls extending from these planes and D-branes, on which the imaginary part of the $J$-function vanishes. This amounts to drawing a dessin with a canonical triangulation. We show that the dessin provides a new way of keeping track of mutual non-localness among 7-branes without employing unphysical branch cuts or their base point. With the dessin we can see that weak- and strong-coupling regions coexist and are located across an $S$-wall from each other. We also present a simple method for computing a monodromy matrix for an arbitrary path by tracing the walls it goes through.

Transverse fully nonlinear equations on Sasakian manifolds and applications

We prove a priori estimates for a class of transverse fully nonlinear equations on Sasakian manifolds and give some geometric applications such as the transversion Calabi-Yau theorem for transverse balanced and (strongly) Gauduchon metrics. We also explain that similar results hold on compact oriented, taut, transverse Hermitian foliated manifold of complex co-dimension n, and give some geometric applications such as the transverse Calabi-Yau theorems for transverse Hermitian and (strongly) Gauduchon metrics.

Blow-ups in generalized complex geometry

Generalized complex geometry is a theory that unifies complex geometry and symplectic geometry into one single framework. It was introduced by Hitchin and Gualtieri around 2002. In this thesis we address the following question: given a generalized complex manifold together with a submanifold, does the blow-up of that submanifold admit again a generalized complex structure? We give the following answers to this question: If the directions that are normal to the submanifold are purely complex, then the blow-up admits a generalized complex structure for which the blow-down map is generalized holomorphic if and only if the induced Lie algebra on the conormal bundle of the submanifold is degenerate. This is a rather restrictive condition, but it is always satisfied if the submanifold is of complex codimension two. On the other extreme, if all the normal directions are symplectic, then we show that the blow-up always admits a generalized complex structure, provided the submanifold is compact. We then also give an example of a submanifold that is not of the above two types, and show that it can not be blown up. Finally, we also address the analogous question in generalized Kahler geometry. Given a generalized Kahler manifold together with a submanifold, we show that under certain geometric constraints on the submanifold the blow-up is again generalized Kahler.

Transversely holomorphic branched Cartan geometry

In Biswas and Dumitrescu (2018), we introduced and studied the concept of holomorphic branched Cartan geometry. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic dimension d admits, away from a closed analytic subset of positive codimension, a nonsingular holomorphic foliation of complex codimension d endowed with a transversely flat branched complex projective geometry (equivalently, a ℂPd-geometry). We also prove that transversely branched holomorphic Cartan geometries on compact complex projective rationally connected varieties and on compact simply connected Calabi–Yau manifolds are always flat (consequently, they are defined by holomorphic maps into homogeneous spaces).

Local Demailly-Bouche's holomorphic morse inequalities

Let $(X,\omega)$ be a Hermitian manifold and let $(E,h^E)$, $(F,h^F)$ be two Hermitian holomorphic line bundle over $X$. Suppose that the maximal rank of the Chern curvature $c(E)$ of $E$ is $r$, and the kernel of $c(E)$ is foliated, i.e. there is a foliation $Y$ of $X$, of complex codimension $r$, such that the tangent space of the leaf at each point $x\in X$ is contained in the kernel of $c(E)$. In this paper, local versions of Demailly-Bouche's holomorphic Morse inequalities (which give asymptotic bounds for cohomology groups $H^{q}(X,E^k\otimes F^l)$ as $k,l,k/l\rightarrow \infty$) are presented. The local version holds on any Hermitian manifold regardless of compactness and completeness. The proof is a variation of Berman's method to derive holomorphic Morse inequalities on compact complex manifolds with boundary.

Global Existence of Bi-Hamiltonian Structures on Orientable Three-Dimensional Manifolds

In this work, we show that an autonomous dynamical system defined by a nonvanishing vector field on an orientable three-dimensional manifold is globally bi-Hamiltonian if and only if the first Chern class of the normal bundle of the given vector field vanishes. Furthermore, the bi-Hamiltonian structure is globally compatible if and only if the Bott class of the complex codimension one foliation defined by the given vector field vanishes.

Gromov hyperbolicity, the Kobayashi metric, and $\mathbb {C}$-convex sets

In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $\mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.

Compressible Taylor–Couette flow – instability mechanism and codimension 3 points

AbstractTaylor–Couette flow in a compressible perfect gas is studied. The onset of instability is examined as a function of the Reynolds numbers of the inner and outer cylinder, the Mach number of the flow and the Prandtl number of the gas. We focus on the case of a wide gap, with radius ratio 0.5. We find new modes of instability at high Prandtl number, which can allow oscillatory axisymmetric modes to onset first. We also find that onset can occur even when the angular momentum increases outwards, so that the classical Rayleigh criterion can be violated in the compressible case. We have also considered the case of counter-rotating cylinders, where the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}m=0$ and $m=1$ modes can onset simultaneously to give a codimension 2 bifurcation, leading to the formation of complex flow patterns. In compressible flow we also find codimension 3 points. The Mach number and the critical inner and outer Reynolds numbers can be adjusted so that the two neutral curves for the $m=0$ and $m=1$ modes touch rather than cross. Complex codimension 3 points occur more readily in the compressible case than in the Boussinesq case, and they are expected to lead to a rich nonlinear behaviour.

Recent advances in F‐theory compactifications

AbstractWe review recent progress in the construction of F‐theory compactifications on elliptically fibered Calabi‐Yau 4‐folds. Of special importance is the resolution of the singularities in complex codimension one, two and three. These encode 7‐branes along holomorphic divisors carrying non‐abelian gauge groups, localised matter at the intersection of pairs of 7‐branes and Yukawa interactions, respectively. Toric methods admit the explicit construction of a large class of resolved 4‐folds. In particular the matter surfaces and interaction points can be analysed in great detail via resolution by blow‐up. Engineering chiral matter spectra requires non‐trivial gauge bundles described as certain types of G4‐fluxes. The recent construction of such fluxes paves the way for globally defined 3‐generation models e.g. of SU(5) GUT type, possibly including extra U(1) factors.

T-branes and monodromy

We introduce T-branes, or "triangular branes," which are novel non-abelian bound states of branes characterized by the condition that on some loci, their matrix of normal deformations, or Higgs field, is upper triangular. These configurations refine the notion of monodromic branes which have recently played a key role in F-theory phenomenology. We show how localized matter living on complex codimension one subspaces emerge, and explain how to compute their Yukawa couplings, which are localized in complex codimension two. Not only do T-branes clarify what is meant by brane monodromy, they also open up a vast array of new possibilities both for phenomenological constructions and for purely theoretical applications. We show that for a general T-brane, the eigenvalues of the Higgs field can fail to capture the spectrum of localized modes. In particular, this provides a method for evading some constraints on F-theory GUTs which have assumed that the spectral equation for the Higgs field completely determines a local model.

A criterion for holomorphic extension of principal bundles

Let G be a complex affine algebraic group, and let EG be a holomorphic principal G-bundle on the complement M$\backslash$S, where S is a closed complex analytic subset, of complex codimension at least two, of a connected complex manifold M. We give a criterion for EG to extend to M as a holomorphic principal G-bundle. Two applications of this criterion are given.

A Fatou-Julia decomposition of transversally holomorphic foliations

A Fatou-Julia decomposition of transversally holomorphic foliations of complex codimension one was given by Ghys, Gomez-Mont and Saludes. In this paper, we propose another decomposition in terms of normal families. Two decompositions have common properties as well as certain differences. It will be shown that the Fatou sets in our sense always contain the Fatou sets in the sense of Ghys, Gomez-Mont and Saludes and the inclusion is strict in some examples. This property is important when discussing a version of Duminy’s theorem in relation to secondary characteristic classes. The structure of Fatou sets is studied in detail, and some properties of Julia sets are discussed. Some similarities and differences between the Julia sets of foliations and those of mapping iterations will be shown. An application to the study of the transversal Kobayashi metrics is also given.

Foliations of Codimension Greater than One with a Kupka Component

We consider holomorphic foliations of complex codimension ≥ 2 with a compact connected component of the Kupka set.

Extension Phenomena for Holomorphic Geometric Structures

The most commonly encountered types of complex analytic G-structures and Cartan geometries cannot have singularities of complex codimension 2 or more.

A Takayama-type extension theorem

AbstractWe prove a theorem on the extension of holomorphic sections of powers of adjoint bundles from submanifolds of complex codimension 1 having non-trivial normal bundle. The first such result, due to Takayama, considers the case where the canonical bundle is twisted by a line bundle that is a sum of a big and nef line bundle and a $\mathbb {Q}$-divisor that has Kawamata log terminal singularities on the submanifold from which extension occurs. In this paper we weaken the positivity assumptions on the twisting line bundle to what we believe to be the minimal positivity hypotheses. The main new idea is an L2 extension theorem of Ohsawa–Takegoshi type, in which twisted canonical sections are extended from submanifolds with non-trivial normal bundle.

Toric resolutions of heterotic orbifolds

We investigate resolutions of heterotic orbifolds using toric geometry. Our starting point is provided by the recently constructed heterotic models on explicit blowup of ${\mathbb{C}}^{n}/{\mathbb{Z}}_{n}$ singularities. We show that the values of the relevant integrals, computed there, can be obtained as integrals of divisors (complex codimension one hypersurfaces) interpreted as (1, 1)-forms in toric geometry. Motivated by this we give a self-contained introduction to toric geometry for nonexperts, focusing on those issues relevant for the construction of heterotic models on toric orbifold resolutions. We illustrate the methods by building heterotic models on the resolutions of ${\mathbb{C}}^{2}/{\mathbb{Z}}_{3}$, ${\mathbb{C}}^{3}/{\mathbb{Z}}_{4}$, and ${\mathbb{C}}^{3}/{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{\ensuremath{'}}$. We are able to obtain a direct identification between them and the known orbifold models. In the ${\mathbb{C}}^{3}/{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{\ensuremath{'}}$ case we observe that, in spite of the existence of two inequivalent resolutions, fully consistent blowup models of heterotic orbifolds can only be constructed on one of them.

Norm expansion along a zero variety

The reproducing kernel function of a weighted Bergman space over domains in C d is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of (complex) codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kernel is thus reduced to the same kind of problem when one of the two entries is on the subvariety. A complete expansion of the reproducing kernel may be achieved in this manner. We carry this out in dimension d = 2 for certain classes of weighted Bergman spaces over the bidisk (with the diagonal z 1 = z 2 as subvariety) and the ball (with z 2 = 0 as subvariety), as well as for a weighted Bargmann–Fock space over C 2 (with the diagonal z 1 = z 2 as subvariety).