Volume 12, Issue 1

We revisit Koiso’s original examples of rigid infinitesimally deformable Einstein metrics. We show how to compute Koiso’s obstruction to the integrability of the infinitesimal deformations on CP 2 n × CP 1 {{\mathbb{CP}}}^{2n}\times {{\mathbb{CP}}}^{1} using completely elementary complex differential geometry.

We prove a formula for the first few terms of the asymptotic expansion of the holomorphic analytic torsion of the Dirac operator modified by the Clifford action of a real and closed three-form.

We survey some results and open questions related to various algebraic invariants of compact complex manifolds that can be obtained from differential forms.

We study infinite superelliptic curves as translation surfaces and explore their Veech groups. These objects are branched covering of the complex plane, branching over infinitely many points. We provide a criterion for isomorphism between a special family of infinite superelliptic curves. We describe the geometry of saddle connections and holonomy vectors on these infinite superelliptic curves. In addition, we prove that the Veech group of an infinite superelliptic curve consists of matrices arising from the differentials of the affine mappings from C {\mathbb{C}} to itself, which permutes the branched points. We obtain necessary and sufficient conditions to guarantee that the Veech group of an infinite superelliptic curve is uncountable. Furthermore, we establish a trichotomy on the holonomy vector set and precisely characterize certain countable groups that can appear as Veech groups of an infinite superelliptic curve. Finally, we also construct and study several examples of interesting infinite superelliptic curves illustrating our results.

We study the stability at blow-ups and deformations of a class of Hermitian metrics whose fundamental two-form ω \omega satisfies the condition ∂ ∂ ¯ ω k = 0 \partial \bar{\partial }{\omega }^{k}=0 , for any k k between 1 and n − 1 n-1 (where n n is the complex dimension of the manifold). We are motivated by the existence of compact complex manifolds supporting such metrics.

Let M i {M}_{i} , for i = 1 i=1 , 2, be a Kähler manifold, and let G G be a compact Lie group acting on M i {M}_{i} by Kähler isometries. Suppose that the action admits a momentum map μ i {\mu }_{i} , and let N i ≔ μ i − 1 ( 0 ) {N}_{i}:= {\mu }_{i}^{-1}\left(0) be a regular-level set. When the action of G G on N i {N}_{i} is proper and free, the Meyer-Marsden-Weinstein quotient P i ≔ N i ∕ G {P}_{i}:= {N}_{i}/G is a Kähler manifold and π i : N i → P i {\pi }_{i}:{N}_{i}\to {P}_{i} is a principal fiber bundle with base P i {P}_{i} and characteristic fiber G G . In this article, we define an almost-complex structure on the manifold N 1 × N 2 {N}_{1}\times {N}_{2} and give necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for N 1 × N 2 {N}_{1}\times {N}_{2} . As applications, we consider a nonintegrable almost-complex structure on the product of two complex Stiefel manifolds and the infinite Calabi-Eckmann manifolds S 2 n + 1 × S ( ℋ ) {{\mathbb{S}}}^{2n+1}\times S\left({\mathcal{ {\mathcal H} }}) , for n ≥ 1 n\ge 1 , where S ( ℋ ) S\left({\mathcal{ {\mathcal H} }}) denotes the unit sphere of an infinite-dimensional complex Hilbert space ℋ {\mathcal{ {\mathcal H} }} .

We explore submersions introduced by reducible holonomy representations of connections with parallel skew torsion. A submersion theorem extending previous, less general, results is given. As our main application, we show that parallel 3- ( α , δ ) \left(\alpha ,\delta ) -Sasaki manifolds admit one-dimensional submersions onto nearly Kähler orbifolds. As a secondary application, we reprove that a given class of nearly Kähler manifolds submerges onto quaternionic Kähler manifolds. This new proof gives a direct expression for the quaternionic structure on the base.

In this study, we extend the uniform L 2 {L}^{2} -estimate of ∂ ¯ \bar{\partial } -equations for flat nontrivial line bundles, proved for compact Kähler manifolds by Hashimoto and Koike, to compact complex manifolds. In the proof, by tracing the Dolbeault isomorphism in detail, we derive the desired L 2 {L}^{2} -estimate directly from Ueda’s lemma.

In this article, we introduce certain arity-3 operations on complex manifolds arising from homotopy transfer theory. Such operations are related to the triple A B C ABC -Massey products from Bott-Chern to Aeppli cohomology. We present a package for computing such products as well as other cohomological and homotopical invariants for complex nilmanifolds and give some examples.

In this article, we briefly survey the classical problem of understanding which Lie algebras admit a complex structure, put in the broader perspective of almost complex structures with special properties. We focus on the different behavior of invariant and non-invariant structures, with special attention to their canonical bundle and Kodaira dimension. We provide new examples of computations of the Kodaira dimension of invariant and non-invariant structures.