In this paper we will introduce a new notion of geometric structures defined by systems of closed differential forms in term of the Clifford algebra of the direct sum of the tangent bundle and the cotangent bundle on a manifold. We develop a unified approach of a deformation problem and establish a criterion of unobstructed deformations of the structures from a cohomological point of view. We construct the moduli spaces of the structures by using the action of b-fields and show that the period map of the moduli space is locally injective under a cohomological condition (the local Torelli type theorem). We apply our approach to generalized Calabi-Yau (metrical) structures and obtain an analog of the theorem by Bogomolov-Tian-Todorov. Further we prove that deformations of generalized Spin(7) structures are unobstructed.
For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension $4n$ non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk{ä}hler moment map. We use the complex part of this moment map to construct a holomorphic Lagrangian fibration with generic fiber diffeomorphic to $(\mathbb{C}^*)^n$, and study the singular fibers.
Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{ä}hler manifold are sufficient to glue the desired toric hyperk{ä}hler manifold.
We characterize those complete K{ä}hler manifolds supporting a nonconstant real-valued function with critical points whose Hessian is complex linear, has pointwise two eigenvalues and whose gradient is a Hessian-eigenvector.
We define the quantum correction of the Teichmüller space $\mathcal{T}$ of Calabi-Yau manifolds. Under the assumption of no weak quantum correction, we prove that the Teichmüller space $\mathcal{T}$ is a locally symmetric space with the Weil-Petersson metric. For Calabi-Yau threefolds, we show that no strong quantum correction is equivalent to that, with the Hodge metric, the image $ Φ(\mathcal{T})$ of the Teichmüller space $\mathcal{T}$ under the period map $Φ$ is an open submanifold of a globally Hermitian symmetric space $W$ of the same dimension as $\mathcal{T}$. Finally, for Hyperkähler m…
Recently, a metric construction for the Calabi-Yau 3-folds from a four-dimensional hyperkahler space by adding a complex line bundle was proposed. We extend the construction by adding a U(1) factor to the holomorphic (3,0)-form, and obtain the explicit formalism for a generic hyperkahler base. We find that a discrete choice arises: the U(1) factor can either depend solely on the fibre coordinates or vanish. In each case, the metric is determined by one differential equation for the modified Kahler potential. As explicit examples, we obtain the generalized resolutions (up to orbifold singularit…
In this paper we describe QCH Kähler surfaces $(M,g,J)$ of generalized orthotoric type. We introduce a distinguished orthonormal frame on $(M,g)$ and give the structure equations for $(M,g,J)$. In the case when $I$ is conformally Kähler and $(M,g,J)$ is not hyperkähler we integrate these structure equations and rediscover the orthotoric Kähler surfaces. We also investigate the hyperkäler case. We prove in a simple way that if $(M,g,J)$ is a hyperkähler surface with a degenerate Weyl tensor $W^-$ (i.e. QCH hyperkähler surface) then among all hyperkäler structures on $(M,g)$ there exists a Kähle…
A method for constructing explicit Calabi-Yau metrics in six dimensions in terms of an initial hyperkahler structure is presented. The equations to solve are non linear in general, but become linear when the objects describing the metric depend on only one complex coordinate of the hyperkahler 4-dimensional space and its complex conjugated. This situation in particular gives a dual description of D6-branes wrapping a complex 1-cycle inside the hyperkahler space, which was studied by Fayyazuddin. The present work generalize the construction given by him. But the explicit solutions we present co…
We consider the natural generalization of the parabolic Monge-Ampère equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex manifold is locally flat and admits a hyperkähler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge-Ampère equation introduced by Alesker and Verbitsky. The result gives an alternative proof of a theorem of Alesker.
In the present work the local form of certain Calabi-Yau metrics possessing a local Hamiltonian Killing vector is described in terms of a single non linear equation. The main assumptions are that the complex $(3,0)$-form is of the form $e^{ik}\widetildeΨ$, where $\widetildeΨ$ is preserved by the Killing vector, and that the space of the orbits of the Killing vector is, for fixed value of the momentum map coordinate, a complex 4-manifold, in such a way that the complex structure of the 4-manifold is part of the complex structure of the complex 3-fold. The link with the solution generating techn…
We prove the $C^0$ estimate for the quaternionic Monge-Ampère equation on compact hyperKähler with torsion manifolds. Our goal is to provide a simpler proof than the one presented by Alesker and Shelukhin.
The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture for any hyperkähler manifold that admits a deformation with two Lagrangian fibrations and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperkähler manifolds are semi-ample. We prove that the Kobayashi pseudom…
In this paper, we provide new examples of Levi-Civita Ricci-flat Hermitian metrics on certain compact non-Kähler Calabi-Yau manifolds, including every compact Hermitian Weyl-Einstein manifold, every compact locally conformal hyperKähler manifold, certain suspensions of Brieskorn manifolds, and every generalized Hopf manifold provided by suspensions of exotic spheres. These examples generalize previous constructions on Hopf manifolds. Additionally, we also construct new examples of compact Hermitian manifolds with nonnegative first Chern class that admit constant strictly negative Riemannian sc…
We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines on the Calabi family of a hyperkähler manifold. We also prove convexity properties of the 1-cycle space of the twistor space of a 4-dimensional anti-self-dual Einstein manifold of non-positive scalar curvature and of the 1-cycle space of the twistor space of a quaternionic Kähle…
We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of pola…
Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they deviate from being deformable to a family of special Lagrangian deformations by examining in detail Joyce's criteria on this family. The result suggests a potential image-support rigidity of A-branes that wrap around a special Lagrangian 3-sphere in a Calabi-Yau 3-fold, which res…
We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M over B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics on M that collapses to the original Calabi-Yau metric on the base B as the parameter converges to 0. The G2-metrics we construct have controlled asymptotic ge…
In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both Kähler and non-Kähler complex structures. Such examples were constructed independently by M. Atiyah, A. Blanchard and E. Calabi in the $1950$'s. In the $1980$'s V. Tsanov gave an example of a simply connected manifold that admits both Kähler and non-Kähler complex structures - the twistor space of a $K3$ surface. Here we show that the quaternion twistor space of a hyperkähler manifold has the same property.
Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperkähler; thus, when four-dimensional, they are self-dual gravitational instantons. We find all monowalls with lowest number of moduli. Their moduli spaces can be identified, on the one hand, with Coulomb branches of five-dimensional supersymmetric quantum field theories on $\mathbb{R}^3\times T^2$ and, on the other hand, with moduli spaces of local Calabi-Yau metrics on the canonical bundle of a del Pezzo surface. We explore the asymptotic metric of these moduli spaces and compare our results with Seib…
We construct the moduli space of contact instantons, an analogue of Yang-Mills instantons defined for contact metric $5$-manifolds and initiate the study of their structure. In the $K$-contact case we give sufficient conditions for smoothness of the moduli space away from reducible connections and show the dimension is given by the index of an operator elliptic transverse to the Reeb foliation. The moduli spaces are shown to be Kähler when the $5$-manifold $M$ is Sasakian and hyperKähler when $M$ is transverse Calabi-Yau. We show how the transverse index can be computed in various cases, in pa…
We study the transverse Kähler holonomy groups on Sasaki manifolds $(M,{\oldmathcal S})$ and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti number $b_1(M)$ and the basic Hodge number $h^{0,2}_B({\oldmathcal S})$ vanish, then ${\oldmathcal S}$ is stable under deformations of the transverse Kähler flow. In addition we show that an irreducible transverse hyperkähler Sasakian structure is ${\oldmathcal S}$-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian…
We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special coordinates which can be constructed as Fock-Goncharov coordinates associated with foliations of quadratic differentials on a Riemann surface. A wall crossing formula of Kontsevich and Soibelman arises both as a crucial consistency condition and an effective computational tool. …
