Form–type Calabi–Yau equations

Fu, Jixiang; Wang, Zhizhang; Wu, Damin

doi:10.4310/mrl.2010.v17.n5.a7

Cited by 90 publications

(82 citation statements)

References 12 publications

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“…This is called a "form-type Calabi-Yau equation" in [27]. Regarding the solvability of (4.2) we have the following conjecture: ) for some smooth function u, such that (4.2) holds.…”

Section: Canonical Metrics On Non-kähler Calabi-yau Manifoldsmentioning

confidence: 98%

“…For emphasis, we state this as a conjecture (which is part of the folklore of this subject, see e.g. [25,29,27,62]). There are many examples of such manifolds.…”

Section: Canonical Metrics On Non-kähler Calabi-yau Manifoldsmentioning

confidence: 99%

“…Lastly for (c), we use the recent solution of the Monge-Ampère equation for (n − 1)-plurisubharmonic functions by Weinkove and the author [76, Theorem 1.1] (see also our earlier work [75] for the case when ω is Kähler, as well as [27,28]) to find a unique Hermitian metricω withω n−1…”

Section: The Canonical Bundle Of Non-kähler Calabi-yau Manifoldsmentioning

confidence: 99%

“…When ω is Kähler, (2) is Yau's solution of the Calabi Conjecture [84], and in the non-Kähler case (2) follows from work of Cherrier [16] when n = 2 and of Weinkove and the author [71] in general (see also [41,72]). Part (3) uses the solvability of an equation introduced by Fu-Wang-Wu [27,28], which was recently established by Weinkove and the author [75,76].…”

Section: Introductionmentioning

confidence: 99%

“…A lot of interest in the subject was generated by "Reid's fantasy" [64] that all Calabi-Yau threefolds with trivial canonical bundle should form a connected family provided one allows deformations and singular transitions through non-Kähler manifolds with trivial canonical bundle. The geometry of compact complex manifolds with trivial canonical bundle has been investigated for example by [2,6,12,19,21,25,26,27,30,33,45,52,62,65] and others. In this paper we will consider a more general class of manifolds, that we now define, and argue that they can naturally be considered as "non-Kähler Calabi-Yau" manifolds.…”

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confidence: 99%

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Non-Kähler Calabi-Yau manifolds

Tosatti¹

2015

Analysis, Complex Geometry, and Mathematical Physics

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Abstract. We study the class of compact complex manifolds whose first Chern class vanishes in the Bott-Chern cohomology. This class includes all manifolds with torsion canonical bundle, but it is strictly larger. After making some elementary remarks, we show that a manifold in Fujiki's class C with vanishing first Bott-Chern class has torsion canonical bundle. We also give some examples of non-Kähler CalabiYau manifolds, and discuss the problem of defining and constructing canonical metrics on them. IntroductionIn this paper, Calabi-Yau manifolds are defined to be compact Kähler manifolds M with c 1 (M ) = 0 in H 2 (M, R). Thanks to Yau's theorem [84] these are precisely the compact manifolds that admit Ricci-flat Kähler metrics. Using this, Calabi proved a decomposition theorem [13] which shows that any such manifold has a finite unramified cover which splits as a product of a torus and a Calabi-Yau manifold with vanishing first Betti number. From this one can easily deduce that Calabi-Yau manifolds have holomorphically torsion canonical bundle (see Theorem 1.4).One can ask how much of this theory carries over to the case of non-Kähler Hermitian manifolds. Simple examples, such as a Hopf surface diffeomorphic to S 1 × S 3 , show that the condition that c 1 (M ) = 0 in H 2 (M, R) is too weak in general (see Example 3.3). On the other hand, much interest has been devoted to studying non-Kähler compact complex manifolds with holomorphically trivial (or more generally torsion) canonical bundle, and many examples can be found in [6,15,17,20,23,33,35,36,37,40,43,53,60,78,83,86] and references therein. For example, every compact complex nilmanifold with a left-invariant complex structure has trivial canonical bundle, and it is always non-Kähler unless it is a torus [6]. A lot of interest in the subject was generated by "Reid's fantasy" [64] that all Calabi-Yau threefolds with trivial canonical bundle should form a connected family provided one allows deformations and singular transitions through non-Kähler manifolds with trivial canonical bundle. The geometry of compact complex manifolds with trivial canonical bundle has been investigated for example by [2,6,12,19,21,25,26,27,30,33,45,52,62,65] and others. In this paper we will consider a more general class of manifolds, that we now define, and argue that they can naturally be considered as "non-Kähler Calabi-Yau" manifolds.On any compact complex manifold there is a (finite-dimensional) cohomology theory called Bott-Chern cohomology. We will need only the real (1, 1) Bott-Chern cohomologyThere is a "first Bott-Chern class" map c BC 1 : Pic(M ) → H 1,1 BC (M, R), which can be described as follows. Given any holomorphic line bundle L → M and any Hermitian metric h on the fibers of L, its curvature form R h is locally given by − √ −1∂∂ log h. Then R h is a closed real (1, 1)-form and if we choose a different metricIf g is any Hermitian metric on M , with fundamental 2-form ω, then its first Chern form given locally byWe will call Ric(ω) the Chern-Ricci form of ω.We the...

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“…This is called a "form-type Calabi-Yau equation" in [27]. Regarding the solvability of (4.2) we have the following conjecture: ) for some smooth function u, such that (4.2) holds.…”

Section: Canonical Metrics On Non-kähler Calabi-yau Manifoldsmentioning

confidence: 98%

“…For emphasis, we state this as a conjecture (which is part of the folklore of this subject, see e.g. [25,29,27,62]). There are many examples of such manifolds.…”

Section: Canonical Metrics On Non-kähler Calabi-yau Manifoldsmentioning

confidence: 99%

Section: The Canonical Bundle Of Non-kähler Calabi-yau Manifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Non-Kähler Calabi-Yau manifolds

Tosatti¹

2015

Analysis, Complex Geometry, and Mathematical Physics

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A Survey on Balanced Metrics

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry

2016

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Abstract. We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation on this space. The associated leaves are related to generalized geometry and correspond to moduli spaces of solutions of suitable Killing spinor equations on a Courant algebroid. As an application, we propose a unifying framework for metrics with holonomy SU(3) and solutions of the Strominger system.

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Form–type Calabi–Yau equations

Cited by 90 publications

References 12 publications

Non-Kähler Calabi-Yau manifolds

Non-Kähler Calabi-Yau manifolds

A Survey on Balanced Metrics

Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry

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