Hatice Coban, Cagri Haciyusufoglu, Mainak Poddar
We produce infinite families of SKT manifolds by using methods of toric geometry like the $J$-construction. These SKT manifolds are total spaces of certain principal $G$-bundles over smooth projective toric varieties, where $G$ is an even dimensional compact connected Lie group.
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SKT Hyperbolic and Gauduchon Hyperbolic Compact Complex Manifolds
本文为紧复流形引入SKT双曲性与Gauduchon双曲性，证明前者蕴含Kobayashi/Brody双曲性，后者蕴含除子双曲性，并给出SKT双曲流形万有覆盖上L^2调和空间的消失定理。
We give a construction of integrable complex structures on the total space of a smooth principal bundle over a complex manifold, with an even dimensional compact Lie group as structure group, under certain conditions. This generalizes the constructions of complex structure on compact Lie groups by Samelson and Wang, and on principal torus bundles by Calabi-Eckmann and others. It also yields large classes of new examples of non-Kähler compact complex manifolds. Moreover, under suitable restrictions on the base manifold, the structure group, and characteristic classes, the total space of the pri…
The equality between the balanced and the Gauduchon cones is discussed in several situations. In particular, it is shown that equality does not hold on many twistor spaces, and it holds on Moishezon manifolds. Moreover, it is proved that a SKT manifold of dimension three on which the balanced cone equals the Gauduchon cone is in fact Kahler.
Given a $\partial\bar\partial$-manifold $X$ with trivial canonical bundle and carrying a metric $ω$ such that $\partial\bar\partialω=0$, we introduce the concept of small deformations of $X$ polarised by the Aeppli cohomology class $[ω]_A$ of an SKT metric $ω$. There is a correspondence between the manifolds polarised by $[ω]_A$ in the Kuranishi family of $X$ and the Bott-Chern classes that are primitive in a sense that we define. We also investigate the existence of a primitive element in an arbitrary Bott-Chern primitive class and compare the metrics on the base space of the subfamily of man…
The main result of this paper is to study the local deformations of Calabi-Yau $\partial\bar{\partial}$-manifold that are co-polarised by the Gauduchon metric by considering the subfamily of co-polarised fibres by the class of Aeppli/De Rham-Gauduchon cohomology of Gauduchon metric given at the beginning on the central fibre. In the latter part, we prove that the $p$-SKT $h$-$\partial\bar{\partial}$-property is deformation open by constructing and studying a new notion called $hp$-Hermitian symplectic ($hp$-HS) form.
We prove that any invariant strong Kahler structure with torsion (SKT structure) on a flag manifold M=G/K of a semisimple compact Lie group G is Kahler. As an application we describe invariant generalized Kahler structures on M.
A Variational Approach to SKT and Balanced Metrics
本文针对SKT与平衡度量定义新泛函，证明其临界点恰为Kähler度量，为Kähler性障碍研究提供新途径。
In this paper, we introduce the notions of $p$-Hermitian-symplectic and $p$-pluriclosed compact complex manifolds as generalisations for an arbitrary positive integer $p$ not exceeding the complex dimension of the manifold of the standard notions of Hermitian-symplectic and SKT manifolds that correspond to the case $p=1$. We then notice that these two properties are equivalent on $\partial\bar{\partial}$-manifolds and go on to prove that in (smooth) complex analytic families of $\partial\bar{\partial}$-manifolds, they are deformation open. Concerning closedness results, we prove that the cones…
On a complex manifold an Hermitian metric which is simultaneously SKT and balanced has to be necessarily Kähler. It has been conjectured that if a compact complex manifold (M,J) has an SKT metric and a balanced metric both compatible with J, then (M, J) is necessarily Kähler. We show that the conjecture is true for nilmanifolds.
We use tools from generalized complex geometry to develop the theory of SKT (a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with special holonomy with respect to a metric connection with closed skew-symmetric torsion. We develop Hodge theory on such manifolds showing how the reduction of the holonomy group causes a decomposition of the twisted cohomology. For SKT manifolds this decomposition is accompanied by an identity between different Laplacian operators and forces the collapse of a spectral sequence at the first page. Further we study the deformation theory of SKT st…
CYT and SKT manifolds with parallel Bismut torsion
本文研究具有平行Bismut挠率的SKT和CYT紧复流形，将其万有覆盖刻画为Kähler Ricci平坦流形与Bismut平坦流形的乘积，并通过映射环面构造给出非Bismut平坦的例子，同时探讨广义Kähler结构的存在性。
For an almost contact metric manifold $N$, we find conditions for which either the total space of an $S^1$-bundle over $N$ or the Riemannian cone over $N$ admits a strong Kähler with torsion (SKT) structure. In this way we construct new 6-dimensional SKT manifolds. Moreover, we study the geometric structure induced on a hypersurface of an SKT manifold, and use such structures to construct new SKT manifolds via appropriate evolution equations. Hyper-Kähler with torsion (HKT) structures on the total space of an $S^1$-bundle over manifolds with three almost contact structures are also studied.
Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics $\{ω_t\}_t$ which equals a fixed SKT metric $ω$ for $t=0$, along a differentiable family of complex manifolds $\{M_t\}_t$.
Non-Kähler metrics on complex manifolds of LVMB type
本文研究LVMB型复流形上的非Kähler度量，通过给出全纯丛特征类的组合公式，探讨平衡度量与SKT度量的存在性，并构造了新的平衡度量例子。
A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not admit compact four-dimensional quotients. It also shows that there are solvable groups in dimension four that admit invariant complex structures but have no invariant SKT structure.
Almost Hermitian geometry and generalized Ricci flow
本文研究全斜对称Nijenhuis张量且Bismut挠率闭合的殆埃尔米特流形（almost-SKT），引入超定椭圆性条件保证受约束Bismut–Ricci流短时存在，并在六维紧流形上构造非可积例子，揭示其与广义Ricci流的联系。
We show a bijective correspondence between compact toric locally conformally symplectic manifolds which admit a compatible complex structure and pairs $(C,a)$, where $C$ is a good cone in the dual Lie algebra of the torus and $a$ is a positive real number. Moreover, we prove that any toric locally conformally Kähler metric on a compact manifold admits a positive potential.
We study Hermitian metrics whose Bismut connection $\nabla^B$ satisfies the first Bianchi identity in relation to the SKT condition and the parallelism of the torsion of the Bimut connection. We obtain a characterization of complex surfaces admitting Hermitian metrics whose Bismut connection satisfy the first Bianchi identity and the condition $R^B(x,y,z,w)=R^B(Jx,Jy,z,w)$, for every tangent vectors $x,y,z,w$, in terms of Vaisman metrics. These conditions, also called Bismut Kähler-like, have been recently studied in [D. Angella, A. Otal, L. Ugarte, R. Villacampa, On Gauduchon connections with…
We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled Kähler-Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau-Tian-Donaldson conjecture. We also obtain a necessary and sufficient condition for existence of torus-invariant solutions to a system of soliton type equations on toric Fano manifolds. Some of these solutions provide natural candidates for the large time limits of a certain geometric flow generalizing the Kähler-Ricci flow.
We introduce and study a special class of Kato manifolds, which we call toric Kato manifolds. Their construction stems from toric geometry, as their universal covers are open subsets of toric algebraic varieties of non-finite type. This generalizes previous constructions of Tsuchihashi and Oda, and in complex dimension 2, retrieves the properly blown-up Inoue surfaces. We study the topological and analytical properties of toric Kato manifolds and link certain invariants to natural combinatorial data coming from the toric construction. Moreover, we produce families of flat degenerations of any …
Vaisman manifolds are strongly related to Kähler and Sasaki geometry. In this paper we introduce toric Vaisman structures and show that this relationship still holds in the toric context. It is known that the so-called minimal covering of a Vaisman manifold is the Riemannian cone over a Sasaki manifold. We show that if a complete Vaisman manifold is toric, then the associated Sasaki manifold is also toric. Conversely, a toric complete Sasaki manifold, whose Kähler cone is equipped with an appropriate compatible action, gives rise to a toric Vaisman manifold. In the special case of a strongly r…
In this paper, we prove the Yau-Tian-Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles.
In \cite{btoric}, Guillemin et al. proved a Delzant-type theorem which classifies $b$-symplectic toric manifolds. More generally, in \cite{torus} they proved a similar convexity result for general Hamiltonian torus action on $b$-symplectic manifolds. In this paper, we provide a new way to construct $b$-symplectic toric manifolds from usual toric manifolds. Conversely, through this way, we can also decompose a $b$-symplectic toric manifolds to usual toric manifolds. Finally, we will try to prove that this kind of decomposition is useful, although the symplectic structure for our decomposition o…
We give criterions for the existence of toric conical Kahler-Einstein and Kahler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry-Emery-Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical Kahler-Einstein metrics in the Gromov-Hausdorff topology.
In the present paper we introduce and study a new notion of toric manifold in the quaternionic setting. We develop a construction with which, starting from appropriate $m$-dimensional Delzant polytopes, we obtain manifolds of real dimension $4m$, acted on by $m$ copies of the group ${\rm Sp}(1)$ of unit quaternions. These manifolds are quaternionic regular and can be endowed with a $4$-plectic structure and a generalized moment map. Convexity properties of the image of the moment map are studied. Quaternionic toric manifolds appear to be a large enough class of examples where one can test and …
We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $-\infty$ and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.
We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to a weakly convex good cone by the cutting construction. (Note that these toric contact manifolds can not be constructed by Delzant construction.) We further prove there are no toric Sasakian structures on these contact manifolds. From this, contact toric manifolds of toric K-cont…
In this paper, we discuss the relative $K$-stability and the modified $K$-energy associated to the Calabi's extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds for both the relative $K$-stability and the properness of modified $K$-energy. In particular, our results hold for toric Fano manifolds with vanishing Futaki-invariant. We also verify our results on the toric Fano surfaces.
A series of examples of toric Sasaki-Einstein 5-manifolds is constructed. These are submanifolds of toric 3-Sasaki 7-manifolds and such a Sasaki-Einstein 5-manifold corresponds uniquely to a toric 3-Sasaki 7-manifold. This produces examples of quasi-regular Sasaki-Einstein structures on every #k(S^2 xS^3), for k odd. Toric geometry is used to construct examples of positive Ricci curvature toric Sasaki structures on non-spin 5-manifolds. Then the join construction is used to construct infinitely many quasi-regular toric Sasaki-Einstein manifolds with arbitrarily high second Betti number in ever…
We study toric nearly Kähler manifolds, extending the work of Moroianu and Nagy. We give a description of the global geometry using multi-moment maps. We then investigate polynomial and radial solutions to the toric nearly Kähler equation.
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.
This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-fold
We prove that a compact toric locally conformally Kähler manifold which is not Kähler admits a toric Vaisman structure, a fact which was conjectured in \cite{mmp}. This is the final step leading to the classification of compact toric locally conformally Kähler manifolds started in \cite{p} and \cite{mmp}. We also show, by constructing an example, that unlike in the symplectic case, toric locally conformally symplectic manifolds are not necessarily toric locally conformally Kähler.
In this short note, based on the work of Wang-Zhu, we determine the greatest lower bounds on Ricci curvature for all toric Fano manifolds.
Strong formality of toric and homogeneous compact Kähler manifolds
本文研究紧 Kähler 流形的强形式性，证明完备光滑复环面簇与齐性紧 Kähler 流形能以相容方式同时满足理性形式性和强形式性。
We show that any $(\C ^*)^n$-invariant stably complex structure on a topological toric manifold of dimension $2n$ is integrable. We also show that such a manifold is weakly $(\C ^*)^n$-equivariantly isomorphic to a toric manifold.
Weyl group of the group of holomorphic isometries of a Kähler toric manifold
本文计算了具有实解析凯勒度量的凯勒环流形全纯等距群的Weyl群（Segal意义下）。
We give a correspondence between toric 3-Sasaki 7-manifolds S and certain toric Sasaki-Einstein 5-manifolds M. These 5-manifolds are all diffeomorphic to k#(S^2\times S^3), where k=2b_2(S)+1, and are given by a pencil of Sasaki embeddings of M in S and are given concretely by the zero set of a component of the 3-Sasaki moment map. It follows that there are infinitely many examples of these toric Sasaki-Einstein manifolds M for each odd b_2(M)>1. As an application of the proof of the above, we prove that the local deformation space of ASD structures on a compact toric ASD Einstein orbifold is g…
We give a characterization of relative Ding stable toric Fano manifolds in terms of the behavior of the modified Ding functional. We call the corresponding behavior of the modified Ding functional the pseudo-boundedness from below. We also discuss the pseudo-boundedness of the Ding / Mabuchi functional of general Fano manifolds.
We compute the first and second homotopy groups of a class of contact toric manifolds in terms of the images of the associated moment map.
In this paper we generalize examples of Hamiltonian stationary Lagrangian submanifolds constructed by Lee and Wang in $\mathbb{C}^m$ to toric almost Calabi-Yau manifolds. We construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized Lagrangian mean curvature flows starting from these examples. We allow these flows to have some singularities and topological changes.
We complete the classification of compact connected contact toric manifolds initiated by Banyaga and Molino and by Galicki and Boyer. As an application we prove the conjectures of Toth and Zelditch on toric integrable systems on the n-torus and the 2-sphere.
We give a complete criterion for the existence of generalized Kähler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of Kähler metrics.
We show that on a Kahler manifold whether the J-flow converges or not is independent of the chosen background metric in its Kahler class. On toric manifolds we give a numerical characterization of when the J-flow converges, verifying a conjecture of Lejmi and the second author in this case. We also strengthen existing results on more general inverse $σ_{k}$ equations on Kahler manifolds.
The logarithmic Chow semistability is a notion of Geometric Invariant Theory for the pair consists of varieties and its divisors. In this paper we introduce a obstruction of semistability for polarized toric manifolds and its toric divisors. As its application, we show the implication from the asymptotic log Chow semistability to the log K-semistability by combinatorial arguments. Furthermore we give a non-semistable example which has a conical Kahler Einstein metric.
Conformally Kähler, Einstein-Maxwell metrics and $f$-extremal metrics are generalization of canonical metrics in Kähler geometry. We introduce uniform K-stability for toric Kähler manifolds, and show that uniform K-stability is necessary condition for the existence of $f$-extremal metrics on toric manifolds. Furthermore, we show that uniform K-stability is equivalent to properness of relative K-energy.
We prove a conjecture saying that complex projective space has maximal volume (degree) among all toric Kaehler-Einstein manifolds of dimension n. The proof is inspired by our recent work on sharp Moser-Trudinger and Brezis-Merle type inequalities for the complex Monge-Ampere operator, but is essentially self-contained.
Toric Einstein 4-manifolds with nonnegative sectional curvature
本文证明了4-流形上具有非负截面曲率的$T^2$-不变爱因斯坦度量是局部对称的。
In this short note, we will prove a volume stability theorem which says that if an n-dimensional toric manifold $M$ admits a $\mathbb{T}^n$ invariant Kähler metric $ω$ with Ricci curvature no less than 1 and its volume is close to the volume of $\mathbb{CP}^n$, $M$ is bi-holomorphic to $\mathbb{CP}^n$.
In this note, we prove that on polarized toric manifolds the relative $K$-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and also we show that the modified $K$-energy is proper in the space of $G_0$-invariant Kähler metrics for the case of toric surfaces which admit the extremal metrics.
3-Manifolds admitting toric integrable geodesic flows
本文利用余球面丛在环面可积时具有的接触环面流形结构，通过比较Betti数，得到紧连通3维流形上测地流环面可积的必要条件。
The aim of this paper is to solve a uniform version of the Yau-Tian-Donaldson conjecture for polarized toric manifolds. Also, we show a combinatorial sufficient condition for uniform relative K-polystability.
We study the J-flow on the toric manifolds, through study the transition map between the moment maps induced by two Kähler metrics, which is a diffeomorphism between polytopes. This is similar to the work of Fang-Lai, under the assumption of Calabi symmetry, they study the monotone map between two intervals. We get a partial bound of the derivatives of transition map.
We determine the greatest lower bounds on the transverse Ricci curvature of compact toric Sasaki manifolds with positive basic first Chern class and with the first Chern class of the contact bundle being trivial. This is based on Wang-Zhu's and Futaki-Ono-Wang's works, and is an analogue of C. Li's work on toric Fano manifolds.
In this note, we prove that on an $n$-dimensional compact toric manifold with positive first Chern class, the Kähler-Ricci flow with any initial $(S^1)^n$-invariant Kähler metric converges to a Kähler-Ricci soliton. In particular, we give another proof for the existence of Kähler-Ricci solitons on a compact toric manifold with positive first Chern class by using the Kähler-Ricci flow.
We show that every toric Sasaki-Einstein manifold $S$ admits a special Legendrian submanifold $L$ which arises as the link ${\rm fix}(τ)\cap S$ of the fixed point set ${\rm fix}(τ)$ of an anti-holomorphic involution $τ$ on the cone $C(S)$. In particular, an irregular toric Sasaki-Einstein manifold $S^{2}\times S^{3}$ has a special Legendrian torus $S^{1}\times S^{1}$. Moreover, we also obtain a special Legendrian submanifold in $\sharp m(S^{2}\times S^{3})$ for each $m\ge 1$.
Using mirror symmetry as described by Hori and Vafa, we compute the quantum equivariant cohomology ring of toric manifolds. This ring arises naturally in topological gauged sigma-models and is related to the Hamiltonian Gromov-Witten invariants of the target manifold.
The global holomorphic α-invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metric. We apply the result of Tian, Lu and Zelditch on polarized Kahler metrics to approximate plurisubharmonic functions and compute the α-invariant of toric Fano manifolds.
We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We introduce the tropical momentum map, which takes values in a generalization of affine space called a log affine manifold. Using this momentum map, we obtain a complete classification of such manifolds in terms of decorated log affine polytopes, hence extending the classification of symplectic toric manifolds achieved by Atiyah, Guillemin-Sternberg, Kostant, and Delzant.
Abreu-Sena-Dias have constructed two distinct families of scalar-flat Kähler non-compact toric metrics using Donaldson's rephrasing of Joyce's construction in action-angle coordinates. In this paper and using the same set-up, we show that these are the only J-complete scalar-flat Kähler metrics on any given strictly unbounded toric surface. We also show that the asymptotic behaviour of such a metric determines it uniquely.
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