Kahler Orbifold Research Articles

Deformations of complex orbifolds and the period maps

We consider the deformations of complex orbifolds with the underlying smooth structures being fixed. As a corollary, we can prove that the deformations of a Calabi-Yau orbifold is unobstructed by using standard arguments. Then we consider the period map for a family of complex Kahler orbifolds. We prove that the period map is holomorphic, horizontal and consistent with our Kodaira-Spencer map.

On the Kummer construction for Kcsc metrics

Given a compact constant scalar curvature Kahler orbifold, with nontrivial holomorphic vector fields, whose singularities admit a local ALE Kahler Ricci-flat resolution, we find sufficient conditions on the position of the singular points to ensure the existence of a global constant scalar curvature Kahler desingularization. We also give complete proofs of a number of analytic results which have been used in this context by various authors. A series of explicit examples is discussed.

Almost-Kähler smoothings of compact complex surfaces with $A_1$ singularities

This paper is concerned with the existence of metrics of constant Hermitian scalar curvature on almost-Kahler manifolds obtained as smoothings of a constant scalar curvature Kahler orbifold, with A1 singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphic vector fields, we show that an almost-Kahler smoothing (Me, ωe) admits an almost-Kahler structure (Je, ge) of constant Hermitian curvature. Moreover, we show that for e > 0 small enough, the (Me, ωe) are all symplectically equivalent to a fixed symplectic manifold (M , ω) in which there is a surface S homologous to a 2-sphere, such that [S] is a vanishing cycle that admits a representant that is Hamiltonian stationary for ge.

Unobstructed symplectic packing by ellipsoids for tori and hyperkähler manifolds

Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids are unobstructed for all even-dimensional tori equipped with Kahler symplectic forms and all closed hyperkahler manifolds of maximal holonomy, or, more generally, for closed Campana simple manifolds (that is, Kahler manifolds that are not unions of their complex subvarieties), as well as for any closed Kahler manifold which is a limit of Campana simple manifolds in a smooth deformation. The proof involves the construction of a Kahler resolution of a Kahler orbifold with isolated singularities and relies on the results of Demailly–Paun and Miyaoka on Kahler cohomology classes.

Sasakian quiver gauge theories and instantons on Calabi–Yau cones

We consider SU(2)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form $M\times S^3/\Gamma$, where $M$ is a smooth manifold and $S^3/\Gamma$ is a three-dimensional Sasaki-Einstein orbifold. We obtain new quiver gauge theories on $M$ whose quiver bundles are based on the affine ADE Dynkin diagram associated to $\Gamma$. We relate them to those arising through translationally-invariant dimensional reduction over the associated Calabi-Yau cones $C(S^3/\Gamma)$ which are based on McKay quivers and ADHM matrix models, and to those arising through SU(2)-equivariant dimensional reduction over the leaf spaces of the characteristic foliations of $S^3/\Gamma$ which are Kahler orbifolds of $\mathbb{C} P^1$ whose quiver bundles are based on the unextended Dynkin diagram corresponding to $\Gamma$. We use Nahm equations to describe the vacua of SU(2)-equivariant quiver gauge theories on the cones as moduli spaces of spherically symmetric instantons. We relate them to the Nakajima quiver varieties which can be realized as Higgs branches of the worldvolume quiver gauge theories on D$p$-branes probing D$(p+4)$-branes which wrap an ALE space, and to the moduli spaces of spherically symmetric solutions in putative non-abelian generalizations of two-dimensional affine Toda field theories.

Asymptotically conical Calabi–Yau metrics on quasi-projective varieties

Let X be a compact Kahler orbifold without $${\mathbb{C}}$$ -codimension-1 singularities. Let D be a suborbifold divisor in X such that $${D \supset {\rm Sing}(X)}$$ and −pK X = q[D] for some $${p, q \in \mathbb{N}}$$ with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kahler–Einstein, then, applying results from our previous paper (Conlon and Hein, Duke Math J, 162:2855–2902, 2013), we show that each Kahler class on $${X \setminus D}$$ contains a unique asymptotically conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r −1-e ) if X is smooth. This provides a definitive version of a theorem of Tian and Yau (Invent Math, 106:27–60, 1991). (2) We introduce new methods to prove an analogous statement (with rate O(r −0.0128)) when $${X = {\rm Bl}_p \mathbb{P}^{3}}$$ and $${D = {\rm Bl}_{p_1,p_2} \mathbb{P}^{2}}$$ is the strict transform of a smooth quadric through p in $${\mathbb{P}^3}$$ . Here D is no longer Kahler–Einstein, but the normal $${\mathbb{S}^1}$$ -bundle to D in X admits an irregular Sasaki–Einstein structure which is compatible with its canonical CR structure. This provides the first example of an affine Calabi–Yau manifold of Euclidean volume growth with irregular tangent cone at infinity.

Toric generalized Kähler–Ricci solitons with Hamiltonian 2-form

We show that the generalized Kahler–Ricci soliton equation on \(4\)-dimensional toric Kahler orbifolds reduces to ODEs assuming there is a Hamiltonian \(2\)-form. This leads to an explicit resolution of this equation on labelled triangles and convex labelled quadrilaterals. In particular, we give the explicit expression of the Kahler–Ricci solitons of weighted projective planes as well as new examples.

A remark on Weighted Bergman kernels on orbifolds

In this note, we explain that Ross-Thomas' result (4, Theorem 1.7) on the weighted Bergman kernels on orbifolds can be directly deduced from our previous result (1). This result plays an important role in the companion paper (5) to prove an orbifold version of Donaldson theorem. In two very interesting papers (4, 5), Ross-Thomas describe a notion of ampleness for line bundles on Kahler orbifolds with cyclic quotient singularities which is related to embeddings in weighted projective spaces. They then apply (4, Theorem 1.7) to prove an orbifold version of Donaldson theorem (5). Namely, the existence of an orbifold Kahler metric with constant scalar curvature implies certain stability condition for the orbifold. In these papers, the result (4, Theorem 1.7) on the asymptotic expansion of Bergman kernels plays a crucial role. In this note, we explain how to directly derive Ross-Thomas' result (4, Theorem 1.7) from Dai-Liu-Ma (1, (5.25)), provided Ross-Thomas condition (4, (1.8)) on ci holds. Since in (1, Section 5), we state our results for general symplectic orbifolds, in what follows, we will just use the version from (2, Theorem 5.4.11), where Ma- Marinescu wrote them in detail for Kahler orbifolds. We will use freely the notation in (2, Section 5.4). We assume also the auxiliary vector bundle E therein is C.

Topology of Locally Conformally Kahler Manifolds with Potential

Locally conformally Kahler (LCK) manifolds with poten- tial are those which admit a Kahler covering with a proper, automorphic, global potential. Existence of a potential can be characterized cohomologically as vanishing of a certain cohomology class, called the Bott-Chern class. Compact LCK manifolds with potential are stable at small deforma- tions and admit holomorphic embeddings into Hopf man- ifolds. This class strictly includes the Vaisman manifolds. We show that every compact LCK manifold with poten- tial can be deformed into a Vaisman manifold. Therefore, every such manifold is diffeomorphic to a smooth ellip- tic fibration over a Kahler orbifold. We show that the pluricanonical condition on LCK manifolds introduced by G. Kokarev is equivalent to vanishing of the Bott-Chern class. This gives a simple proof of some of the results on topology of pluricanonical LCK-manifolds, discovered by Kokarev and Kotschick.

Hearing the weights of weighted projective planes

Which properties of an orbifold can we “hear,” i.e., which topological and geometric properties of an orbifold are determined by its Laplace spectrum? We consider this question for a class of four-dimensional Kahler orbifolds: weighted projective planes $$M := {\mathbb{C}}P^2(N_1, N_2, N_3)$$ with three isolated singularities. We show that the spectra of the Laplacian acting on 0- and 1-forms on M determine the weights N 1, N 2, and N 3. The proof involves analysis of the heat invariants using several techniques, including localization in equivariant cohomology. We show that we can replace knowledge of the spectrum on 1-forms by knowledge of the Euler characteristic and obtain the same result. Finally, after determining the values of N 1, N 2, and N 3, we can hear whether M is endowed with an extremal Kahler metric.