We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the $\partial\overline\partial$-Lemma, is characterized in terms of the strongly Gauduchon cone and of the first $\partial\overline\partial$-degree measuring the difference of Aeppli and Bott-Chern cohomologies with respect to the Betti number $b_1$.
We will consider locally conformally balanced manifolds. We prove that a locally conformally balanced condition is not stable under a small deformation. We prove that locally conformally balanced condition is stable under any proper modification. We prove that symmetric products of the Kodaira surface can be resolve to locally conformally balanced manifolds by Hilbert-Chow map.
In this note we prove that, under a weak condition, small deformations of a compact balanced manifold are also balanced. This condition is satisfied on the twistor space over a compact self-dual four manifold.
Small deformations of the complex structure do not always preserve special metric properties in the Hermitian non-Kähler setting. In this paper, we find necessary conditions for the existence of smooth curves of balanced metrics $\{ω_t\}_t$ which start with a fixed balanced metric $ω$ for $t=0$, along a differentiable family of complex manifolds $\{M_t\}_t$.
In this paper, we consider a natural map from the Kahler cone to the balanced cone of a Kahler manifold. We study its injectivity and surjecticity. We also give an analytic characterization theorem on a nef class being Kahler.
In this note we present, for every $n \geq 4$, a non-Kähler compact complex manifold $X$ of complex dimension $n$ admitting a balanced metric and an astheno-Kähler metric which is in addition $k$-th Gauduchon for any $1\leq k\leq n-1$.
We study the intrinsic geometrical structure of hypersurfaces in 6-manifolds carrying a balanced Hermitian SU(3)-structure, which we call {\em balanced} SU(2)-{\em structures}. We provide conditions which imply that such a 5-manifold can be isometrically embedded as a hypersurface in a manifold with a balanced SU(3)-structure. We show that any 5-dimensional compact nilmanifold has an invariant balanced SU(2)-structure as well as new examples of balanced Hermitian SU(3)-metrics constructed from balanced SU(2)-structures. Moreover, for $n=3,4$, we present examples of compact manifolds, endowed w…
A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, the manifold is said to be hyperkaehler. In the paper "Non-Hermitian Yang-Mills connections", Kaledin and Verbitsky proved that the twistor space of a hyperkaehler manifold admits a balanced metric; these were first studied in the article "On the existence of special metrics in com…
Balanced Bismut torsion-parallel manifold with constant holomorphic sectional curvature
Halıcıoğlu Data Science Institute, University of California San Diego（加州大学圣地亚哥分校 Halıcıoğlu 数据科学研究所）；School of Mathematical Sciences, Chongqing Normal University（重庆师范大学数学科学学院）
A manifold (M,I,J,K) is called hypercomplex if I,J,K are complex structures satisfying quaternionic relations. A quaternionic Hermitian metric is called HKT (hyperkaehler with torsion) if $Idω_I = Jd ω_J=Kdω_K$, where $ω_I,ω_J, ω_K$ are Hermitian forms associated with I, J, K. A Hermitian metric $ω$ on a complex manifold is called balanced if $d^*ω=0$. We show that balanced HKT metrics are precisely the quaternionic Calabi-Yau metrics defined in terms of the quaternionic Monge-Ampere equation. In particular, a balanced HKT-metric is unique in its cohomology class, and it always exists if the q…
In this article, we study Hermitian manifolds whose Bismut-Strominger connection has parallel torsion tensor, which will be called {\em Bismut torsion parallel manifolds,} or {\em BTP} manifolds for short. We obtain a necessary and sufficient condition characterizing this class in terms of the curvature tensor. In particular, Bismut flat or Bismut Kähler-like manifolds are {\em BTP} manifolds, known by our earlier results. We also obtain structural results for non-balanced {\em BTP} manifolds, and classification theorems for non-balanced {\em BTP} threefolds and balanced {\em BTP} threefolds.
The Bott-Chern cohomology of 6-dimensional nilmanifolds endowed with invariant complex structure is studied with special attention to the cases when balanced or strongly Gauduchon Hermitian metrics exist. We consider complex invariants introduced by Angella and Tomassini and by Schweitzer, which are related to the $\partial \bar\partial$-lemma condition and defined in terms of the Bott-Chern cohomology, and show that the vanishing of some of these invariants is not a closed property under holomorphic deformations. In the balanced case, we determine the spaces that parametrize deformations in t…
We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds: it locally conformally Kähler, $k$-Gauduchon, balanced and locally conformally balanced and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a $k$-Gauduchon metric is necessarily a torus. Combined with a result of Fino and Vezzoni from 2016, this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics: balanced, pluriclosed and locally conformally Kähler is a torus. Moreover, w…
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.
Given a compact complex $n$-fold $X$ satisfying the $\partial\bar\partial$-lemma and supposed to have a trivial canonical bundle $K_X$ and to admit a balanced (=semi-Kähler) Hermitian metric $ω$, we introduce the concept of deformations of $X$ that are {\bf co-polarised} by the balanced class $[ω^{n-1}]\in H^{n-1,\,n-1}(X,\,\C)\subset H^{2n-2}(X,\,\C)$ and show that the resulting theory of balanced co-polarised deformations is a natural extension of the classical theory of Kähler polarised deformations in the context of Calabi-Yau or even holomorphic symplectic compact complex manifolds. The c…
We study many properties concerning weak Kählerianity on compact complex manifolds which admits a holomorphic submersion onto a Kähler or a balanced manifold. We get generalizations of some results of Harvey and Lawson (the Kähler case), Michelson (the balanced case), Popovici (the sG case) and others.
We study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, SKT and astheno-Kähler metrics. We prove that the twistor spaces of compact hyperkähler and negative quaternionic-Kähler manifolds do not admit astheno-Kähler metrics. Then we provide examples of astheno-Kähler structures on toric bundles over Kähler manifolds. In particular, we find examples of compact complex non-Kähler manifolds which admit a balanced and an astheno-Kähler metrics, thus answering to a question in [52] (see also [24]). One of these…
Let $(X,J)$ be a nilmanifold with a left-invariant nilpotent complex structure. We study the existence of $p$-Kähler structures (which include Kähler and balanced metrics) on $X$. More precisely, we determine an optimal $p$ such that there are no $p$-Kähler structures on $X$. Finally, we show that, contrarily to the Kähler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Frölicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.
The goal of this article is to generalise the Witten deformation to even dimensional conic manifolds and a class of functions called admissible Morse functions.
We prove several vanishing theorems for the cohomology of balanced hyperbolic manifolds that we introduced in our previous work and for the $L^2$ harmonic spaces on the universal cover of these manifolds. Other results include a Hard Lefschetz-type theorem for certain compact complex balanced manifolds and the non-existence of certain $L^1$ currents on the universal covering space of a balanced hyperbolic manifold.
We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations preserving 2, 1, or none of the 3 independent Killing spinors. Toric 3-Sasakian manifolds provide non-trivial examples with integrable deformation preserving precisely 2 Killing spinors. Thus the dimension of Killing spinors is not preserved under Einstein deformations but is only …
Nearly Kähler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly Kähler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly Kähler manifolds, showing that it is obstructed in general. More precisely, we show that the infinitesimal deformations of the homogeneous nearly Kähler structure on the flag manifold are all obstructed to second order.
We construct a type of transverse deformations of a Vaisman manifold, which preserves the canonical foliation. For this construction we only need a basic 1-form with certain properties. We show that such basic 1-forms exist in abundance.
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.
Manifolds admitting Killing spinors are Einstein manifolds. Thus, a deformation of a Killing spinor entails a deformation of Einstein metrics. In this paper, we study infinitesimal deformations of Killing spinors on nearly parallel $\mathrm{G}_2$-manifolds. Since there is a one-to-one correspondence between nearly parallel $\mathrm{G}_2$-structures and Killing spinors on 7-dimensional spin manifolds, our results imply that infinitesimal deformations of nearly parallel $\mathrm{G}_2$-structures are examined in terms of Killing spinors. Applying the same technique, we identify that the space of …
Balanced Bismut torsion-parallel fourfolds with constant holomorphic sectional curvature
Halıcıoğlu Data Science Institute, University of California San Diego（加州大学圣地亚哥分校哈利奥格鲁数据科学研究所）；School of Mathematical Sciences, Chongqing Normal University（重庆师范大学数学科学学院）
In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki-Einstein manifolds. In general, special Legendrian deformations have the obstruction. However, we show that the deformation space of special Legendrian submanifolds is the intersection of two larger smooth deformation spaces of different types. We also prove that any special Legendrian submanifold admits smooth deformations, whi…
This paper extends the nonabelian Hodge correspondence for Kaehler manifolds to a larger class of hermitian metrics on complex manifolds called balanced of Hodge-Riemann type. Essentially, it grows out of a few key observations so that the known results, especially the Donaldson-Uhlenbeck-Yau theorem and Corlette's theorem, can be applied in our setting. Though not necessarily Kaehler, we show that the Sampson-Siu Theorem proving that harmonic maps are pluriharmonic remains valid for a slightly smaller class by using the known argument. Special important examples include those balanced metrics…
We show that an $n-$dimensional Moishezon manifold is uniruled if and only if it supports a balanced metric $ω^{n-1}$ of positive total scalar Chern curvature. A similar statement also holds true for class $\mathcal C$ manifolds of dimension three.
In this paper, we prove several formulas related to Hodge theory, and using them to prove the deformations of a compact $H$-twisted generalized Calabi-Yau manifold are unobstructed and $L^2$ convergence in a neighborhood in another power series . And if we assume that the deformation is smooth in a fixed neighborhood, and assume the existence of a global canonical family of deformation, we also construct the global canonical family of the deformations of generalized Kähler manifolds.
We study the asymptotic behavior of quantized Ding functionals along Bergman geodesic rays and prove that the slope at infinity can be expressed in terms of Donaldson-Futaki invariants and Chow weights. Based on the slope formula, we introduce a new algebro-geometric stability on Fano manifolds and show that the existence of anti-canonically balanced metrics implies our stability. The relation between our stability and others is also discussed. As another application of the slope formula, we get the lower bound estimate on the Calabi like functionals on Fano manifolds.
We show that if a Fano manifold has discrete automorphism group and admits a polarized Kähler-Einstein metric, then there exists a sequence of anticanonically balanced metrics converging smoothly to the Kähler-Einstein metric. Our proof is based on a simplification of Donaldson's proof of the analogous result for balanced metrics, replacing a delicate geometric argument by the use of Berezin-Toeplitz quantization. We then apply this result to compute the asymptotics of the optimal rate of convergence to the fixed point of Donaldson's iterations in the anticanonical setting.
We show that there is no radial balanced metric on the unit ball of the Kepler manifold with not too wild boundary behavior. Additionally, we identify explicitly the weights corresponding to radial metrics with such boundary behavior which satisfy the balanced condition as far as germs at the boundary are concerned. Related results for Poincare metrics are also established.
In this paper, we give a brute-force proof of the Kastler-Kalau-Walze type theorem for $7$-dimensional manifolds with boundary about Witten deformation, and give a theoritic explaination of the gravitational action for $7$ dimensional manifolds with boundary.
In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. Using the notion of balanced metrics and recent work of Donaldson, Wang, and Phong-Sturm, we show that the statement holds for higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group.
We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature Kähler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a corollary, we recover uniqueness and splitting results for extremal metrics in the polarized case.
On a Hermitian manifold we construct a symmetric $(1,1)$- tensor $H$ using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor $H$ for a harmonic $1$-form to be analytic and for an analytic $1$-form to be harmonic. We prove that if $H$ is positive definite then the first Betti number $b_1 = 0$ and the Hodge number $h^{1,0} = 0$. We obtain an obstruction to the existence of Killing vector fields in terms of the Ricci tensor of the Chern connection: if the Chern form of the Chern connec…
We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi continuity Theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with transversely Riemannian foliations. We use this to prove that the $\partial\bar{\partial}$-lemma and being transversely Kähler are rigid properties under small deformations of the transversely holomorphic structure which preserve the foliation. Finally, we study an example which shows that this is not the case for arbitrary …
A nearly parallel $G_{2}$-manifold $Y$ is a Riemannian 7-manifold whose cone $C(Y) = \mathbb{R}_{>0} \times Y$ has the holonomy group contained in ${\rm Spin(7)}$. In other words, it is a spin 7-manifold with a real Killing spinor. We have a special class of calibrated submanifolds called Cayley submanifolds in $C(Y)$. An associative submanifold in $Y$ is a minimal 3-submanifold whose cone is Cayley. We study its deformations, namely, Cayley cone deformations, explicitly when it is homogeneous in the 7-sphere $S^{7}$.
We study the deformation theory of $\mathrm{G}_2$-instantons on nearly $\mathrm{G}_2$ manifolds. There is a one-to-one correspondence between nearly parallel $\mathrm{G}_2$ structures and real Killing spinors, thus the deformation theory can be formulated in terms of spinors and Dirac operators. We prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to describe the deformation space of the canonical connection on the four normal homoge…
We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new ingredient of independent interest is homotopy density of the subspace of metrics with cylindrical ends.
In this paper, we consider a class of balanced manifolds and provide a proof of a problem proposed by Silva-that compact complex manifolds that are Kahler outside an analytic subset are balanced. Next, as a special case of this theorem, we prove that compact complex manifolds which are Kahler outside a Kahler submanifold are class C. Finally, we prove that a blow up along a submanifold of Hironaka's examples are Kahler, and we establish that Hironaka's examples are examples of this theorem.
We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\infty}$ a priori estimates for a solution of the complex Monge-Ampere equation when the background metric is Hermitian (in complex dimension two) or balanced (in higher dimensions), giving an alternative proof of a theorem of Cherrier. We relate this to recent results of Guan-Li.
In this note we give a simplified proof of a recent result of X.X. Chen, which together with work of G. Szekelyhidi implies that on a sufficiently small deformation of a polarized constant scalar curvature Kahler manifold the K-energy has a lower bound.
The Witten deformation associated to a Morse function on a closed Riemannian manifold, via Rellich-Kato theorem, relates analytically the spectral package of the Riemannian manifold (eigenvalues and eigenforms) to the Morse complex defined by the pair (Morse function, Riemannian metric) coupled with the "multivariable harmonic oscillators" associated to the critical points of the Morse function. We survey this relation and discuss some implications, including the finite subset of the spectral package referred to as the "virtually small spectral package" .
In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which admits deformations to generalized complex manifolds and obtain non-trivial generalized Kahler structures on Fano surfaces and toric Kahler manifolds. In particular, we show that holomorphic Poisson structures on a Kahler manifold induce deformations of generalized Kahler structur…
We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi--Yau geometry and analysis and the Kodaira--Kuranishi--Spencer theory and building up on results of N.Koiso for the case of compact manifolds, we show that under rather general hypotheses any `small' asymptotically cylindrical Ricci-flat deformations of asymptotically cylindrical Ricci-flat Kähler metrics are again Kähler, possibly with respect to a perturbed complex structure. We also find the dimension of the moduli space for these small deformations. In the class o…
We prove that, for some classes of complex nilmanifolds, the Bott-Chern cohomology is completely determined by the Lie algebra associated to the nilmanifold with the induced complex structure. We use these tools to compute the Bott-Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations, completing the computations in arXiv:0709.3528v1 [math.AG] by M. Schweitzer.
