A survey on strong KT structures - SSMR

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$Bull. Math. Soc. Sci. Math. Roumanie Tome 49(97) No. 4 ... - SSMR$

100 Anna Fino and Adriano Tomassinifor any vector field X,Y,Z. Since ∇ B is a Hermitian c<strong>on</strong>necti<strong>on</strong>, its (restricted)hol<strong>on</strong>omy group Hol(∇ B ) is in general c<strong>on</strong>tained in the unitary group U(n).By [20] the c<strong>on</strong>necti<strong>on</strong> ∇ B bel<strong>on</strong>gs to a <strong>on</strong>e-parameter family of can<strong>on</strong>icalHermitian c<strong>on</strong>necti<strong>on</strong>s∇ t = t∇ C + (1 − t)∇ 0 ,where ∇ C and ∇ 0 denote the Chern c<strong>on</strong>necti<strong>on</strong> and the first can<strong>on</strong>ical c<strong>on</strong>necti<strong>on</strong>respectively. For t = −1 <strong>on</strong>e gets the Bismut c<strong>on</strong>necti<strong>on</strong> ∇ B .If c = 0, then the Bismut c<strong>on</strong>necti<strong>on</strong> ∇ B coincides with the Levi-Civita ∇ LCc<strong>on</strong>necti<strong>on</strong> and in this case the Hermitian manifold (M,J,g) is Kähler. In thisnote we are interested <strong>on</strong> n<strong>on</strong>-Kähler geometry and we study Hermitian structureswhose fundamental 2-form F satisfies weaker c<strong>on</strong>diti<strong>on</strong>s than dF = 0. We recallthe following (see [22])Definiti<strong>on</strong> 1.1. Let M be a manifold of real dimensi<strong>on</strong> 2n. A Hermitian structure(J,g) <strong>on</strong> M or a J-Hermitian metric g is said to be str<strong>on</strong>g Kähler withtorsi<strong>on</strong> (str<strong>on</strong>g KT) if dc = 0, or, equivalently, if ∂∂F = 0 or dd c F = 0.The str<strong>on</strong>g KT metrics have been recently studied by many authors and theyhave also applicati<strong>on</strong>s in type II string theory and in 2-dimensi<strong>on</strong>al supersymmetricσ-models [22, 44, 34]. Moreover, they have also relati<strong>on</strong>s with generalizedKähler geometry (see for instance [22, 24, 31, 3, 12, 18]). Indeed, by [24, 3] itfollows that a generalized Kähler structure <strong>on</strong> a 2n-dimensi<strong>on</strong>al manifold M isequivalent to a pair of str<strong>on</strong>g KT structures (J + ,g) and (J − ,g), where J ± aretwo integrable almost complex structures <strong>on</strong> M and g is a Hermitian metric withrespect to J ± , such that the torsi<strong>on</strong> 3-form d c +F + = −d c −F − is closed, where F ±are the two fundamental 2-forms associated with the Hermitian structures (J ± ,g)and d c ± = i(∂ ± − ∂ ± ).By [43] any real compact semisimple Lie group G of even dimensi<strong>on</strong> has anatural str<strong>on</strong>g KT metric and a twisted generalized Kähler structure (see [24]).For the str<strong>on</strong>g KT structure, <strong>on</strong>e may choose as Hermitian structure (J,g), withJ a left-invariant complex structure and g a compatible bi-invariant metric. Theassociated Bismut c<strong>on</strong>necti<strong>on</strong> ∇ B is the flat c<strong>on</strong>necti<strong>on</strong> with skew-symmetrictorsi<strong>on</strong> g(X,[Y,Z]) corresp<strong>on</strong>ding to an invariant 3-form <strong>on</strong> the Lie algebra of G.As far as we know, the <strong>on</strong>ly known solvmanifolds, i.e. compact quotientsof solvable Lie groups by discrete subgroups, admitting a generalized Kählerstructure are the Inoue surface of type S M defined in [33] and the 6-dimensi<strong>on</strong>al<strong>on</strong>e found in [18]. This solvmanifold turns out to be a T 2 -bundle over the Inouesurface of type S M , whose c<strong>on</strong>structi<strong>on</strong> we will review in the last secti<strong>on</strong>.On a complex manifold of complex dimensi<strong>on</strong> n, if the (n − 1,n − 1)-formF n−1 =}F ∧ ...{{∧ F},(n−1)−timesis ∂∂-closed, then, according to [23], the Hermitian metric g is said to be standard.If we denote by δ the coderivative, then a Hermitian metric is standard if and
A <str<strong>on</strong>g>survey</str<strong>on</strong>g> <strong>on</strong> str<strong>on</strong>g KT structures 101<strong>on</strong>ly if the 1-form θ = JδF is co-closed. The 1-form θ is also called the Leeform and a Hermitian manifold is said to be balanced if θ = 0, or equivalently ifdF n−1 = 0 and c<strong>on</strong>formally balanced if θ is exact. C<strong>on</strong>sequently, <strong>on</strong> a complexsurface a standard Hermitian metric coincides with a str<strong>on</strong>g KT metric. If n > 2,then a direct computati<strong>on</strong> shows that a str<strong>on</strong>g KT g is standard if and <strong>on</strong>ly if|dF | 2 = (n − 1) 2 |θ ∧ F | 2 , where | · | denotes the norm <strong>on</strong> forms induced by g.By [23], it follows that any c<strong>on</strong>formal class of a Hermitian metric <strong>on</strong> a compactcomplex manifold can be represented by a standard metric. Thus, in particular,<strong>on</strong> a complex surface <strong>on</strong>e can be found a str<strong>on</strong>g KT metric in the c<strong>on</strong>formal classof any given Hermitian metric. So it is interesting to study str<strong>on</strong>g KT structuresstarting from real dimensi<strong>on</strong> at least six. Compact quotients of solvable (inparticular nilpotent) Lie groups by uniform discrete subgroups provide a naturalclass of examples to be investigated. Indeed, in general, it is well known thata nilmanifold, i.e. a compact quotient of a nilpotent Lie group by a discretesubgroup, cannot admit any Kähler structure unless it is a torus (see for instance[7]). Moreover, in the case of solvmanifolds, Hasegawa proved in [27, 28] thata solvmanifold carries a Kähler metric if and <strong>on</strong>ly if it is covered by a finitequotient of a complex torus, which has the structure of a complex torus bundleover a complex torus.This <str<strong>on</strong>g>survey</str<strong>on</strong>g> c<strong>on</strong>sists essentially of two parts. The first part collects somedefiniti<strong>on</strong>s and results <strong>on</strong> the hol<strong>on</strong>omy of the Bismut c<strong>on</strong>necti<strong>on</strong>, str<strong>on</strong>g KTstructures, in relati<strong>on</strong> also with the blow-up of a complex manifold and withresoluti<strong>on</strong>s of orbifolds. The sec<strong>on</strong>d part reviews the c<strong>on</strong>structi<strong>on</strong> of the examplesfound in [16, 17, 18, 19].Acknowledgements We would like to thank Liviu Ornea for the invitati<strong>on</strong> to writethis paper.2 Hol<strong>on</strong>omy of the Bismut c<strong>on</strong>necti<strong>on</strong> in SU(n)A 2n-dimensi<strong>on</strong>al Hermitian manifold (M,J,g) is said to be Calabi-Yau withtorsi<strong>on</strong>, shortly CY T, if the restricted hol<strong>on</strong>omy group of the Bismut c<strong>on</strong>necti<strong>on</strong>Hol(∇ B ) is c<strong>on</strong>tained in SU(n). In such a case, we have that the first Chern classc 1 (M) of (M,J) is zero.In [25] it was c<strong>on</strong>jectured that any compact complex manifold (M,J) withc 1 (M) = 0 admits a Hermitian metric such that Hol(∇ B ) ⊆ SU(n). A counterexampleto the above c<strong>on</strong>jecture has been found, showing that the c<strong>on</strong>diti<strong>on</strong>Hol(∇ B ) ⊆ SU(n) is also not stable under small deformati<strong>on</strong>s of the complexstructure in [16]. The example was c<strong>on</strong>structed by using the followingTheorem 2.1. [16] Let (M,J) be a compact complex manifold of complex dimensi<strong>on</strong>n with a holomorphic n<strong>on</strong>-vanishing (n,0)-form. If the Ricci tensor ofthe Bismut c<strong>on</strong>necti<strong>on</strong> of some Hermitian metric g vanishes, then (M,J,g) isc<strong>on</strong>formally balanced and in particular admits a balanced metric.The previous theorem can be applied for instance to any nilmanifold Γ\G,
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