Non-compact Complex Manifolds Research Articles

The L2 Aeppli-Bott-Chern Hilbert complex

We analyse the L2 Hilbert complexes naturally associated to a non-compact complex manifold, namely the ones which originate from the Dolbeault and the Aeppli-Bott-Chern complexes. In particular we define the L2 Aeppli-Bott-Chern Hilbert complex and examine its main properties on general Hermitian manifolds, on complete Kähler manifolds and on Galois coverings of compact complex manifolds. The main results are achieved through the study of self-adjoint extensions of various differential operators whose kernels, on compact Hermitian manifolds, are isomorphic to either Aeppli or Bott-Chern cohomology.

$L^2$-estimates for the Dirac-Dolbeault operator and Bergman kernel asymptotics on some classes of non-compact complex manifolds

$L^2$-estimates for the Dirac-Dolbeault operator and Bergman kernel asymptotics on some classes of non-compact complex manifolds

Asymptotic normality of divisors of random holomorphic sections on non-compact complex manifolds

We prove a central limit theorem for divisors of Gaussian i.i.d. centered holomorphic sections of tensor powers of a Hermitian holomorphic line bundle over a non-compact Hermitian manifold.

Estimates of Picard modular cusp forms

Abstract In this article, for n ≥ 2 {n\geq 2} , we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU ⁢ ( ( n , 1 ) , ℂ ) {\mathrm{SU}((n,1),\mathbb{C})} . The main result of the article is the following result. Let Γ ⊂ SU ⁢ ( ( 2 , 1 ) , 𝒪 K ) {\Gamma\subset\mathrm{SU}((2,1),\mathcal{O}_{K})} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬ Γ k {{{\mathcal{B}_{\Gamma}^{k}}}} denote the Bergman kernel associated to the 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} , complex vector space of weight-k cusp forms with respect to Γ. Let 𝔹 2 {\mathbb{B}^{2}} denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X Γ := Γ \ 𝔹 2 {X_{\Gamma}:=\Gamma\backslash\mathbb{B}^{2}} denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | ⋅ | pet {|\cdot|_{\mathrm{pet}}} denote the point-wise Petersson norm on 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} . Then, for k ≫ 1 {k\gg 1} , we have the following estimate: sup z ∈ X Γ ⁡ | ℬ Γ k ⁢ ( z ) | pet = O Γ ⁢ ( k 5 2 ) , \sup_{z\in X_{\Gamma}}|{{\mathcal{B}_{\Gamma}^{k}}}(z)|_{\mathrm{pet}}=O_{% \Gamma}(k^{\frac{5}{2}}), where the implied constant depends only on Γ.

SINGULAR HOLOMORPHIC MORSE INEQUALITIES ON NON-COMPACT MANIFOLDS

We obtain asymptotic estimates of the dimension of cohomology on possibly non-compact complex manifolds for line bundles endowed with Hermitian metrics with algebraic singularities. We give a unified approach to establishing singular holomorphic Morse inequalities for hyperconcave manifolds, pseudoconvex domains, q-convex manifolds and q-concave manifolds, and we generalize related estimates of Berndtsson. We also consider the case of metrics with more general than algebraic singularities.

Non-degeneracy of Cohomological Traces for General Landau–Ginzburg Models

We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair $(X,W)$, where $X$ is a non-compact complex manifold with trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$, assuming only that the critical locus of $W$ is compact (but may not consist of isolated points). These results can be viewed as certain "deformed" versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of $W$, while the second is equivalent with the statement that a certain power of the shift functor is a Serre functor on the even subcategory of the $\mathbb{Z}_2$-graded category of topological D-branes of such models.

The Geometry of the Space of BPS Vortex\u2013Antivortex Pairs

The gauged sigma model with target {mathbb {P}}^1, defined on a Riemann surface Sigma , supports static solutions in which k_{+} vortices coexist in stable equilibrium with k_{-} antivortices. Their moduli space is a noncompact complex manifold {textsf {M}}_{(k_{+},k_{-})}(Sigma ) of dimension k_{+}+k_{-} which inherits a natural Kähler metric g_{L^2} governing the model’s low energy dynamics. This paper presents the first detailed study of g_{L^2}, focussing on the geometry close to the boundary divisor D=partial , {textsf {M}}_{(k_{+},k_{-})}(Sigma ). On Sigma =S^2, rigorous estimates of g_{L^2} close to D are obtained which imply that {textsf {M}}_{(1,1)}(S^2) has finite volume and is geodesically incomplete. On Sigma ={mathbb {R}}^2, careful numerical analysis and a point-vortex formalism are used to conjecture asymptotic formulae for g_{L^2} in the limits of small and large separation. All these results make use of a localization formula, expressing g_{L^2} in terms of data at the (anti)vortex positions, which is established for general {textsf {M}}_{(k_{+},k_{-})}(Sigma ). For arbitrary compact Sigma , a natural compactification of the space {{textsf {M}}}_{(k_{+},k_{-})}(Sigma ) is proposed in terms of a certain limit of gauged linear sigma models, leading to formulae for its volume and total scalar curvature. The volume formula agrees with the result established for mathrm{Vol}(mathsf{M}_{(1,1)}(S^2)), and allows for a detailed study of the thermodynamics of vortex-antivortex gas mixtures. It is found that the equation of state is independent of the genus of Sigma , and that the entropy of mixing is always positive.

Cohomology Dimension Growth for Nakano q-Semipositive Line Bundles

We study the cohomology with high tensor powers of Nakano $q$-semipositive line bundles on complex manifolds. We obtain the asymptotic estimates for the dimension of cohomology with high tensor powers of semipositive line bundles over q-convex manifolds and various possibly non-compact complex manifolds, in which the order of estimates are optimal. Besides, estimates for the modified Dirac operator on Nakano $q$-positive line bundle on almost complex manifolds are given.

Chern–Ricci flows on noncompact complex manifolds

In this work, we obtain existence criteria for Chern-Ricci flows on noncompact manifolds. We generalize a result by Tossati-Wienkove on Chern-Ricci flows to noncompact manifolds and at the same time generalize a result for Kahler-Ricci flows by Lott-Zhang to Chern-Ricci flows. Using the existence results, we prove that any complete noncollapsed Kahler metric with nonnegative bisectional curvature on a noncompact complex manifold can be deformed to a complete Kahler metric with nonnegative and bounded bisectional curvature which will have maximal volume growth if the initial metric has maximal volume. Combining this result with the result of Chau-Tam, we give another proof that a complete noncompact Kahler manifold with nonnegative bisectional curvature (not necessarily bounded) and maximal volume growth is biholomorphic to the complex Euclidean space. This last result has already been proved by Gang Liu recently using other methods. This last result is partial confirmation of a uniformization conjecture of Yau.

Residue formulas for logarithmic foliations and applications

In this work we prove a Baum–Bott type formula for noncompact complex manifold of the form X ~ = X − D \tilde {X}=X-{\mathcal D} , where X X is a complex compact manifold and D {\mathcal D} is a normal crossing divisor on X X . As applications, we provide a Poincaré–Hopf type theorem and an optimal description for a smooth hypersurface D {\mathcal D} invariant by an one-dimensional foliation F {\mathscr F} on P n \mathbb {P}^n satisfying Sing ( F ) ⊊ D \textrm {Sing}({\mathscr F}) \subsetneq {\mathcal D} .

An Equivalence of Scalar Curvatures on Hermitian Manifolds

For a Kahler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kahler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu–Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kahler. However, we prove that a certain class of non-compact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.

Equidistribution of Zeros of Holomorphic Sections in the Non-compact Setting

We consider N-tensor powers of a positive Hermitian line bundle L over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed with respect to the natural measure coming from the curvature of L, as N tends to infinity. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of SL2(Z) and to the hyperbolic measure, the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity. We also give estimates for the speed of convergence of the currents of integration on the zero-divisors.

Balanced metrics on Cartan and Cartan–Hartogs domains

This paper consists of two results dealing with balanced metrics (in Donaldson terminology) on noncompact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan–Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in Loi and Zedda (Mathematische Annalen, 2011, to appear) we also provide the first example of complete, Kähler-Einstein and projectively induced metric g such that α g is not balanced for all α > 0.

On the geometry of C3/∆27 and del Pezzo surfaces

We clarify some aspects of the geometry of a resolution of the orbifold X = C3/D_27, the noncompact complex manifold underlying the brane quiver standard model recently proposed by Verlinde and Wijnholt. We explicitly realize a map between X and the total space of the canonical bundle over a degree 1 quasi del Pezzo surface, thus defining a desingularization of X. Our analysis relys essentially on the relationship existing between the normalizer group of D_27 and the Hessian group and on the study of the behaviour of the Hesse pencil of plane cubic curves under the quotient.

Indefinite locally conformal Kähler manifolds

We study the basic properties of an indefinite locally conformal Kähler (l.c.K.) manifold. Any indefinite l.c.K. manifold M with a parallel Lee form ω is shown to possess two canonical foliations F and F c , the first of which is given by the Pfaff equation ω = 0 and the second is spanned by the Lee and the anti-Lee vectors of M. We build an indefinite l.c.K. metric on the noncompact complex manifold Ω + = ( Λ + ∖ Λ 0 ) / G λ (similar to the Boothby metric on a complex Hopf manifold) and prove a CR extension result for CR functions on the leafs of F when M = Ω + (where Λ + ∖ Λ 0 ⊂ C s n is − | z 1 | 2 − ⋯ − | z s | 2 + | z s + 1 | 2 + ⋯ + | z n | 2 > 0 ). We study the geometry of the second fundamental form of the leaves of F and F c . In the degenerate cases (corresponding to a lightlike Lee vector) we use the technique of screen distributions and (lightlike) transversal bundles developed by A. Bejancu et al. [K.L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol. 364, Kluwer Academic, Dordrecht, 1996].

Classes de variétés localement conformément Kählériennes, non Kählériennes

In this Note, we construct two classes of non-compact complex manifolds, locally conformally Kähler but non-Kähler. The construction is inspired by the results of Loeb. We give two examples in dimensions 2 and 3. To cite this article: J. Renaud, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Generically strongly $q$-convex complex manifolds

Suppose ϕ is a real analytic plurisubharmonic exhaustion function on a connected noncompact complex manifold X. The main result is that if the real analytic set of points at which ϕ is not strongly q-convex is of dimension at most 2q+1, then almost every sufficiently large sublevel of ϕ is strongly q-convex as a complex manifold. For X of dimension 2, this is a special case of a theorem of Diederich and Ohsawa. A version for ϕ real analytic with corners is also obtained.

INDEX THEORY OF PERTURBED DOLBEAULT OPERATORS: SMOOTH POLAR DIVISORS

This paper generalizes the index-theoretic content of the physical models studied in [3, 7]. The paper calculates the homology Chern character, and thus the index formula, for a broad class of perturbed Dolbeault operators on complete noncompact complex manifolds. The manifolds studied are complements of smooth polar divisors of meromorphic sections of vector bundles over closed complex manifolds. These sections define complexes of Koszul type that are used to construct the perturbations of the Dolbeault operators.

A correction to “The Dirichlet problem for complex Monge–Ampère equations and regularity of the pluri-complex Green function”

(1.1) det(uzj zk) = ψ(z, u,∇u) in Ω, u = φ on ∂Ω and related questions. When Ω is a strongly pseudoconvex domain, this problem has received extensive study. In [4]-[6], E. Bedford and B. A. Taylor established the existence, uniqueness and global Lipschitz regularity of generalized pluri-subharmonic solutions. S.-Y. Cheng and S.-T. Yau [8], in their work on complete Kahler-Einstein metrics on non-compact complex manifolds, solved (1.1) for ψ = e and φ = +∞, obtaining a solution in C∞(Ω). In 1985, L. Caffarelli, J. J. Kohn, L. Nirenberg and J. Spruck [7] proved the existence of classical pluri-subharmonic solutions of (1.1) for the non-degenerate case ψ > 0, under suitable conditions on ψ. The degenerate case ψ ≥ 0 has also attracted a lot of attention, and counterexamples have been found showing that there need not be a C solution (see [3], [11]). It is of interest in complex analysis to ask whether C regularity holds for the degenerate case; see [1] for related results and further references. In [20], S.-Y. Li studied the Neumann problem for complex Monge-Ampere equations. In this paper we treat the Dirichlet problem (1.1) for general domains which are not necessarily pseudoconvex. We shall prove

Berezin Quantization and Reproducing Kernels on Complex Domains

LetΩ\Omegabe a non-compact complex manifold of dimensionnn,ω=∂∂¯Ψ\omega =\partial \overline \partial \Psia Kähler form onΩ\Omega, andKα(x,y¯)K_\alpha ( x,\overline y)the reproducing kernel for the Bergman spaceAα2A^2_\alphaof all analytic functions onΩ\Omegasquare-integrable against the measuree−αΨ|ωn|e^{-\alpha \Psi } |\omega ^n|. Under the condition\[Kα(x,x¯)=λαeαΨ(x)K_\alpha ( x,\overline x)= \lambda _\alpha e^{\alpha \Psi (x)}\]F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109–1163] was able to establish a quantization procedure on(Ω,ω)(\Omega ,\omega )which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are justΩ=Cn\Omega = \mathbf {C} ^nandΩ\Omegaa bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, asα→+∞\alpha \to +\infty. This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains inCn\mathbf {C}^n. Along the way, we also fix two gaps in Berezin’s original paper, and discuss, forΩ\Omegaa domain inCn\mathbf {C}^n, a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure|ωn||\omega ^n|.