Complex Submanifolds Research Articles

The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds.

The asymptotics of the optimal holomorphic extensions of holomorphic jets along submanifolds

For a complex submanifold in a complex manifold, we consider the operator which for a given holomorphic jet of a vector bundle along the submanifold associates the L2-optimal holomorphic extension of it to the ambient manifold. When the vector bundle is given by big tensor powers of a positive line bundle, we give an asymptotic formula for this extension operator.

Complex submanifolds of indefinite complex space forms

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al.

Algebraic cones of LCK manifolds with potential

A complex manifold X, dimX>2, is called “an LCK manifold with potential”, if it can be realized as a complex submanifold of a Hopf manifold. Let X˜ be its Z-covering, considered as a complex submanifold in Cn﹨0. We prove that X˜ is algebraic. We call the manifolds obtained this way the algebraic cones, and show that the affine algebraic structure on X˜ is independent from the choice of X. We give several intrinsic definitions of an algebraic cone, and prove that these definitions are equivalent.

Formality of differential graded algebras and complex Lagrangian submanifolds

Let be a compact Kähler Lagrangian in a holomorphic symplectic variety X/C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{X}/\ extbf{C}$$\\end{document}. We use deformation quantisation to show that the endomorphism differential graded algebra RHom(i∗KL1/2,i∗KL1/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{RHom}\\big (i_*\ extrm{K}_\ extrm{L}^{1/2},i_*\ extrm{K}_\ extrm{L}^{1/2}\\big )$$\\end{document} is formal. We prove a generalisation to pairs of Lagrangians, along with auxiliary results on the behaviour of formality in families of A∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {A}}_{\\infty }$$\\end{document}-modules.

Bimeromorphic geometry of LCK manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold M M which has a Kähler structure on its cover, such that the deck transform group acts on it by homotheties. Assume that the Kähler form is exact on the minimal Kähler cover of M M . We prove that any bimeromorphic map M ′ → M M’\rightarrow M is in fact holomorphic; in other words, M M has a unique minimal model. This can be applied to a wide class of LCK manifolds, such as the Hopf manifolds, their complex submanifolds and to OT manifolds.

A twistor construction of Hopf real hypersurfaces in complex hyperbolic space

It is very well known that Hopf real hypersurfaces in the complex projective space can be locally characterized as tubes over complex submanifolds. This also holds true for some, but not all, Hopf real hypersurfaces in the complex hyperbolic space. The main goal of this paper is to show, in a unified way, how to construct Hopf real hypersurfaces in the complex hyperbolic space from a horizontal submanifold in one of the three twistor spaces of the indefinite complex 2-plane Grassmannian with respect to the natural para-quaternionic Kähler structure. We also identify these twistor spaces with the sets of circles in totally geodesic complex hyperbolic lines in the complex hyperbolic space. As an application, we describe all classical Hopf examples. We also solve the remarkable and long-standing problem of the existence of Hopf real hypersurfaces in the complex hyperbolic space, different from the horosphere, such that the associated principal curvature is 2. We exhibit a method to obtain plenty of them.

Expected local topology of random complex submanifolds

Let n ≥ 2 n\geq 2 and r ∈ { 1 , ⋯ , n − 1 } r\in \{1, \cdots , n-1\} be integers, M M be a compact smooth Kähler manifold of complex dimension n n , E E be a holomorphic vector bundle with complex rank r r and equipped with a Hermitian metric h E h_E , and L L be an ample holomorphic line bundle over M M equipped with a metric h h with positive curvature form. For any d ∈ N d\in \mathbb N large enough, we equip the space of holomorphic sections H 0 ( M , E ⊗ L d ) H^0(M,E\otimes L^d) with the natural Gaussian measure associated to h E h_E , h h and its curvature form. Let U ⊂ M U\subset M be an open subset with smooth boundary. We prove that the average of the ( n − r ) (n-r) -th Betti number of the vanishing locus in U U of a random section s s of H 0 ( M , E ⊗ L d ) H^0(M,E\otimes L^d) is asymptotic to ( n − 1 r − 1 ) d n ∫ U c 1 ( L ) n {n-1 \choose r-1} d^n\int _U c_1(L)^n for large d d . On the other hand, the average of the other Betti numbers is o ( d n ) o(d^n) . The first asymptotic recovers the classical deterministic global algebraic computation. Moreover, such a discrepancy in the order of growth of these averages is new and contrasts with all known other smooth Gaussian models, in particular the real algebraic one. We prove a similar result for the affine complex Bargmann-Fock model.

Holomorphic Retractions of Bounded Symmetric Domains onto Totally Geodesic Complex Submanifolds

Holomorphic Retractions of Bounded Symmetric Domains onto Totally Geodesic Complex Submanifolds

Umehara algebra and complex submanifolds of indefinite complex space forms

The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some applications. In particular, if a complex manifolds admits a holomorphic polynomial isometric immersion to one indefinite complex space form, then it cannot admits a holomorphic isometric immersion to another indefinite complex space form of different type. Other consequences include the non-existence of the common complex submanifolds for indefinite complex projective space or hyperbolic space and a complex manifold with a distinguished metric, such as homogeneous domains, the Hartogs triangle, the minimal ball, the symmetrized polydisc, etc, equipped with their intrinsic Bergman metrics, which generalizes more or less all existing results.

Complete complex hypersurfaces in the ball come in foliations

In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n (n \geq 2)$ is a level set of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose level sets are complete. This shows that $\mathbb{B}_n$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in $\mathbb{B}_n$ of this type can be embedded into such a foliation. We establish a more general result in which neither completeness nor smoothness of the given hypersurface is required. Furthermore, we obtain a similar result for complex submanifolds of arbitrary positive codimension and prove the existence of a nonsingular holomorphic submersion foliation of $\mathbb{B}_n$ by smooth complete closed complex submanifolds of any pure codimension $q \in \lbrace 1, \dotsc , n-1 \rbrace$.

Wild holomorphic foliations of the ball

We prove that the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ admits a nonsingular holomorphic foliation $\mathcal F$ by closed complex hypersurfaces such that both the union of the complete leaves of $\mathcal F$ and the union of the incomplete leaves of $\mathcal F$ are dense subsets of $\mathbb{B}_n$. In particular, every leaf of $\mathcal F$ is both a limit of complete leaves of $\mathcal F$ and a limit of incomplete leaves of $\mathcal F$. This gives the first example of a holomorphic foliation of $\mathbb{B}_n$ by connected closed complex hypersurfaces having a complete leaf that is a limit of incomplete ones. We obtain an analogous result for foliations by complex submanifolds of arbitrary pure codimension $q$ with $1\le q<n$.

Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids

Abstract The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.

Extending Nirenberg–Spencer’s question on holomorphic embeddings to families of holomorphic embeddings

Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first-order neighborhoods to have an affirmative answer to the reformulated question. When the submanifolds do have nonzero vector fields, we obtain an affirmative answer to the question under the additional assumption that submanifolds have certain nice deformation properties, which is applicable to free rational curves. As an application, we obtain a stronger version of the Cartan–Fubini-type extension theorem for Fano manifolds of Picard number 1. We also propose a potential application on hyperplane sections of projective K3 surfaces.

Euclidean domains in complex manifolds

In this paper we find big Euclidean domains in complex manifolds. We consider open neighbourhoods of sets K∪M in a complex manifold X, where K is a compact holomorphically convex set in an open Stein neighbourhood, M is an embedded Stein submanifold, and K∩M is compact O(M)-convex. We prove a Docquier–Grauert type theorem concerning biholomorphic equivalence of neighbourhoods of such sets, and we give sufficient conditions for the existence of Stein neighbourhoods of K∪M, biholomorphic to domains in Cn with n=dim⁡X, such that M is mapped onto a closed complex submanifold of Cn.

The orbit type stratification of the moduli space of Higgs bundles

The moduli space of Higgs bundles can be constructed as a quotient of an infinite-dimensional space and hence admits an orbit type decomposition. In this paper, we show that the orbit type decomposition is a complex Whitney stratification such that each stratum is a complex symplectic submanifold and hence admits a complex Poisson bracket. Moreover, these Poisson brackets glue to a Poisson bracket on the structure sheaf of the moduli space so that the moduli space is a stratified complex symplectic space.

On the structure of double complexes

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of ‘universal’ quasi-isomorphism and the behaviour of the decomposition under tensor product and compute the Grothendieck ring of the category of bounded double complexes over a field with finite cohomologies up to such quasi-isomorphism (and some variants). Applying the theory to the double complexes of smooth complex valued forms on compact complex manifolds, we obtain a Poincaré duality for higher pages of the Frölicher spectral sequence, construct a functorial three-space decomposition of the middle cohomology, give an example of a map between compact complex manifolds which does not respect the Hodge filtration strictly, compute the Bott–Chern and Aeppli cohomology for Calabi–Eckmann manifolds, introduce new numerical bimeromorphic invariants, show that the non-Kählerness degrees are not bimeromorphic invariants in dimensions higher than three and that the ∂ ∂ ¯ -lemma and some related properties are bimeromorphic invariants if, and only if, they are stable under restriction to complex submanifolds.

Periodic cycles of attracting Fatou components of type $${\mathbb {C}}\times ({\mathbb {C}}^{*})^{d-1}$$ in automorphisms of $${\mathbb {C}}^{d}$$

We generalise a recent example by F. Bracci, J. Raissy and B. Stensønes to construct automorphisms of \({\mathbb {C}}^{d}\) admitting an arbitrary finite number of non-recurrent Fatou components, each biholomorphic to \({\mathbb {C}}\times ({\mathbb {C}}^{*})^{d-1}\) and all attracting to a common boundary fixed point. These automorphisms can be chosen such that each Fatou component is invariant or such that the components are grouped into periodic cycles of any common period. We further show that no orbit in these attracting Fatou components can converge tangent to a complex submanifold, and that every stable orbit near the fixed point is contained either in these attracting components or in one of d invariant hypersurfaces tangent to each coordinate hyperplane on which the automorphism acts as an irrational rotation.

The Gauss maps of transversally complex submanifolds of a quaternion projective space

We study a kind of complex submanifolds in a quaternion projective space, which we call transversally complex submanifolds, from the viewpoint of quaternionic differential geometry. We treat them applying the theory of the quaternionic vector bundles. For a transversally complex immersion, we define a Gauss map whose values are complex structures of a quaternionic vector space. It is a generalization of “the mean curvature sphere” in the theory by Burstall, Ferus, Leschke, Pedit, and Pinkall. The Gauss map is a key notion for our theory. As an application, we show a characterization of complex projective spaces which are transversally complex submanifolds of a quaternion projective space.

Strongly homotopy Lie algebras and deformations of calibrated submanifolds

For an element $\Psi$ in the graded vector space $\Omega^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $\Psi$-submanifold is defined as a submanifold $N$ of $M$ such that $\Psi_{|N} \in \Omega^*(N, TN)$. The class of $\Psi$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space $\Omega^*(M, TM)$ carries a natural graded Lie algebra structure, given by the Frolicher-Nijenhuis bracket $[-,- ]^{FN}$. When $\Psi$ is an odd degree element with $[ \Psi, \Psi]^{FN} =0$, we associate to a $\Psi$-submanifold $N$ a strongly homotopy Lie algebra, which governs the formal and (under certain assumptions) smooth deformations of $N$ as a $\Psi$-submanifold, and we show that under certain assumptions these deformations form an analytic variety. As an application we revisit formal and smooth deformation theory of complex closed submanifolds and of $\varphi$-calibrated closed submanifolds, where $\varphi$ is a parallel form in a real analytic Riemannian manifold.