Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds

Popovici, Dan

doi:10.24033/bsmf.2704

Cited by 63 publications

(76 citation statements)

References 8 publications

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“…In [STW17], see also [Pop15], by proving the Gauduchon's generalization of the Calabi's conjecture for compact complex manifolds satisfying c BC 1 (X) = 0, it is proved that it is also always possible to find Gauduchon (1)-Chern-Ricci-flat metrics, hence providing existence of non-Kähler special metrics satisfying both curvature and cohomological conditions.…”

Section: First-chern-einstein Metricsmentioning

confidence: 99%

Remarks on Chern–Einstein Hermitian metrics

2019

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Section: First-chern-einstein Metricsmentioning

confidence: 99%

Remarks on Chern–Einstein Hermitian metrics

2019

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“…In this case the Calabi-Yau theorem is replaced by its Hermitian counterpart, proved by Weinkove and the author [41] (see also [2,7,9,17,24,26,27,34,44,42] for earlier results and later developments, [19,20,21,22,30,36,45,46] for other Monge-Ampère type equations on non-Kähler manifolds, and [40] for a very recent Calabi-Yau theorem for Gauduchon metrics on Hermitian manifolds). The key new difficulty is that now in general we have to modify the function F in (1.1) by adding a constant to it, namely we obtain…”

Section: Introductionmentioning

confidence: 99%

The Calabi–Yau theorem and Kähler currents

Tosatti

2016

Advances in Theoretical and Mathematical Physics

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“…ω is Gauduchon), then so isω 3 , and similarly if ∂(ω n−1 ) is ∂-exact (i.e. ω is strongly Gauduchon [63]). …”

Section: Introductionmentioning

confidence: 91%

“…If instead of dω 0 = 0 we assume the astheno-Kähler condition ∂∂(ω .2) falls into the class of equations studied in [76,63] (the only difference is the factor of 2 in front of Re( √ −1∂u∧ ∂(ω n−2 0 ), which does not affect any of the results in [76]). In particular, if M admits astheno-Kähler metrics, then Conjectures 4.1 and 4.2 are reduced to proving a suitable second order estimate for the solution u (see [76,Theorem 1.7]).…”

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

confidence: 99%

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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Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds

Cited by 63 publications

References 8 publications

Remarks on Chern–Einstein Hermitian metrics

Remarks on Chern–Einstein Hermitian metrics

The Calabi–Yau theorem and Kähler currents

Non-Kähler Calabi-Yau manifolds

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