Compact Complex Manifold Research Articles

Functionals for the study of lcK metrics on compact complex manifolds

We propose an approach to the existence problem for locally conformally Kähler metrics on compact complex manifolds by introducing and studying a functional that is different according to whether the complex dimension of the manifold is 2 or higher.

Poincaré metric of holomorphic foliations with non-degenerate singularities

Consider a Brody hyperbolic foliation [Formula: see text] with non-degenerate singularities on a compact complex manifold. We show that its leafwise Poincaré metric is transversally Hölder continuous with a logarithmic slope towards the singular set of [Formula: see text].

Relative connections on principal bundles and relative equivariant structures

We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over a complex analytic family. We also introduce the notion of relative equivariant bundles and establish its relation with relative holomorphic connections on principal bundles.

A variational approach to SKT and balanced metrics

We investigate compact complex manifolds endowed with SKT or balanced metrics. In each case, we define a new functional whose critical points are proved to be precisely the Kähler metrics, if any, on the manifold. As general manifolds of either type need not admit Kähler metrics, this provides an approach to new obstructions to Kählerianity within these two families of metrics.

Scalar curvature and deformations of complex structures

Abstract We study a system of equations on a compact complex manifold, that couples the scalar curvature of a Kähler metric with a spectral function of a first-order deformation of the complex structure. The system comes from an infinite-dimensional Kähler reduction, which is a hyperkähler reduction for a particular choice of the spectral function. The main tool for studying the system is a flat connection on the space of first-order deformations of the complex structure, that allows to obtain a formal complexification of the moment map equations. Using this connection, we describe a variational characterization of the equations, a Futaki invariant for the system, and a generalization of K-stability that is conjectured to characterize the existence of solutions.

Irrational pencils and Betti numbers

We study irrational pencils with isolated critical points on compact aspherical complex manifolds. We prove that if the set of critical points is nonempty, the homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated. This generalizes a result by Dimca, Papadima and Suciu. By considering self-products of the Cartwright-Steger surface, this allows us to build new examples of smooth projective varieties whose fundamental group has a non-finitely generated homology.

Compatibility of Balanced and SKT Metrics on Two-Step Solvable Lie Groups

AbstractIt has been conjectured by Fino and Vezzoni that a compact complex manifold admitting both a compatible SKT and a compatible balanced metric also admits a compatible Kähler metric. Using the shear construction and classification results for two-step solvable SKT Lie algebras from our previous work, we prove this conjecture for compact two-step solvmanifolds endowed with an invariant complex structure which is either (a) of pure type or (b) of dimension six. In contrast, we provide two counterexamples for a natural generalisation of this conjecture in the homogeneous invariant setting. As part of the work, we obtain further classification results for invariant SKT, balanced and Kähler structures on two-step solvable Lie groups. In particular, we give the full classification of left-invariant SKT structures on two-step solvable Lie groups in dimension six.

On the Curvature of the Bismut Connection: Bismut–Yamabe Problem and Calabi–Yau with Torsion Metrics

We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a conformal Hermitian structure on a compact complex manifold, there exists a metric with constant Bismut scalar curvature in that class when the expected constant scalar curvature is non-negative. A similar result is given in the general case of Gauduchon connections. We then study an Einstein-type condition for the Bismut Ricci curvature tensor on principal bundles over Hermitian manifolds with complex tori as fibers. Thanks to this analysis, we construct explicit examples of Calabi–Yau with torsion Hermitian structures and prove a uniqueness result for them.

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [1] that a compact complex manifold admitting both a balanced metric and an SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [2] and of the anomaly flow [3] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.

Relative internality and definable fibrations

We first elaborate on the theory of relative internality in stable theories from [13], focusing on the notion of uniform relative internality (called collapse of the groupoid in [13]), and relating it to orthogonality, triviality of fibrations, the strong canonical base property, differential Galois theory, and GAGA.We prove that DCF0 does not have the strong canonical base property, correcting a proof in [20]. We also prove that the theory CCM of compact complex manifolds does not have the strong CBP, and initiate a study of the definable Galois theory of projective bundles. In the rest of the paper we study definable fibrations in DCF0, where the general fibre is internal to the constants, including differential tangent bundles, and geometric linearizations. We obtain new examples of higher rank types orthogonal to the constants.

Balanced hyperbolic and divisorially hyperbolic compact complex manifolds

Balanced hyperbolic and divisorially hyperbolic compact complex manifolds

Deformations of Compact Complex Manifolds with Ample Canonical Bundles

Deformations of Compact Complex Manifolds with Ample Canonical Bundles

Intersection theory and Chern classes in Bott–Chern cohomology

In this article, we investigate an axiomatic approach introduced by Grivaux for the study of rational Bott-Chern cohomology, and use it in that context to define Chern classes of coherent sheaves. This method also allows us to derive a Riemann-Roch-Grothendieck formula for a projective morphism between smooth complex compact manifolds. In the general case of complex spaces, the Poincar\'e and Dolbeault-Grothendieck lemmas are not always valid. For this reason, and to simplify the exposition, we only consider non singular complex spaces. The appendix presents a calculation of integral Bott-Chern cohomology in top degree for a connected compact manifold.

The Dirichlet problem for the k-Hessian equation on a complex manifold

We solve the Dirichlet problem for $k$-Hessian equations on compact complex manifolds with boundary, given the existence of a subsolution. Our method is based on a second order a priori estimate of the solution on the boundary with a particular gradient scale. The scale allows us to apply a blow-up argument to obtain control on all necessary norms of the solution.

P-Kähler and balanced structures on nilmanifolds with nilpotent complex structures

Let (X, J) be a nilmanifold with an invariant nilpotent complex structure. We study the existence of p-Kähler structures (which include Kähler and balanced metrics) on X. More precisely, we determine an optimal p such that there are no p-Kähler structures on X. Finally, we show that, contrarily to the Kähler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Frölicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.

Floer cohomology and flips

We show that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. These results are part of a conjectural decomposition of the Fukaya category of a compact symplectic manifold with a singularity-free running of the minimal model program, analogous to the description of Bondal-Orlov (Derived categories of coherent sheaves, 2002) and Kawamata (Derived categories of toric varieties, 2006) of the bounded derived category of coherent sheaves on a compact complex manifold.

Group actions on local moduli space of holomorphic vector bundles

AbstractWe prove that actions of complex reductive Lie groups on a holomorphic vector bundle over a complex compact manifold are locally extendable to its local moduli space.

Canonical metrics on holomorphic Courant algebroids

The solution of the Calabi Conjecture by Yau implies that every Kähler Calabi–Yau manifold X $X$ admits a metric with holonomy contained in SU ( n ) $\mathrm{SU}(n)$ , and that these metrics are parameterized by the positive cone in H 1 , 1 ( X , R ) $H^{1,1}(X,\mathbb {R})$ . In this work, we give evidence of an extension of Yau's theorem to non-Kähler manifolds, where X $X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid Q $Q$ of Bott–Chern type. The equations that define our notion of best metric correspond to a mild generalization of the Hull–Strominger system, whereas the role of H 1 , 1 ( X , R ) $H^{1,1}(X,\mathbb {R})$ is played by an affine space of ‘Aeppli classes’ naturally associated to Q $Q$ via Bott–Chern secondary characteristic classes.

On linear combinations of cohomological invariants of compact complex manifolds

We prove that there are no unexpected universal integral linear relations and congruences between Hodge, Betti and Chern numbers of compact complex manifolds and determine the linear combinations of such numbers which are bimeromorphic or topological invariants. This extends results in the Kähler case by Kotschick and Schreieder. We then develop a framework to tackle the more general questions taking into account ‘all’ cohomological invariants (e.g. the dimensions of the higher pages of the Frölicher spectral sequence, Bott-Chern and Aeppli cohomology). This allows us to reduce the general questions to specific construction problems. We solve these problems in many cases. In particular, we obtain full answers to the general questions concerning universal relations and bimeromorphic invariants in low dimensions.

COMPATIBILITY BETWEEN NON-KÄHLER STRUCTURES ON COMPLEX (NIL)MANIFOLDS

We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds (locally conformally Kähler, k-Gauduchon, balanced, and locally conformally balanced) and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a left-invariant k-Gauduchon metric is necessarily a torus. Combined with the main result in [FV16], this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics—balanced, pluriclosed and locally conformally Kähler—is a torus. Moreover, we construct a family of compact nilmanifolds in any dimension carrying both balanced and locally conformally balanced metrics and finally we show a compact complex nilmanifold does not support a left-invariant locally conformally hyper-Kähler structure.