A survey on strong KT structures - SSMR

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

106 Anna Fino and Adriano Tomassinithat a compact complex manifold has a str<strong>on</strong>g KT metric if and <strong>on</strong>ly if there isno n<strong>on</strong>-zero positive current of bi-dimensi<strong>on</strong> (1,1) which is dd c -exact.Miyaoka proved in [41] that, if a compact complex manifold has a Kählermetric in the complement of a point, then it admits itself a Kähler metric.By using a deep extensi<strong>on</strong> and regularity result for positive or negative plurisubharm<strong>on</strong>iccurrents by Alessandrini and Bassanelli (see [2]) we get the followingTheorem 3.1. [19] If M 2n \ {p}, n ≥ 2, admits a str<strong>on</strong>g KT metric, then thereexists a str<strong>on</strong>g KT metric <strong>on</strong> M 2n .This theorem is a generalizati<strong>on</strong> of Miyaoka’s extensi<strong>on</strong> result (see [41]).A classical result by Blanchard (see [9]) states that the blow-up of a Kählermanifold at a point or al<strong>on</strong>g a compact complex submanifold is still Kähler. Forthe str<strong>on</strong>g KT case we have the same result (see [19]), namelyTheorem 3.2. The blow-up of a str<strong>on</strong>g KT manifold (M,J,g) at a point or al<strong>on</strong>ga compact complex submanifold of M is still str<strong>on</strong>g KT.Then, as an applicati<strong>on</strong> of the previous theorem we have that, if M is acomplex manifold and ˜M p is the blow-up of M at a point p ∈ M, then ˜M p has astr<strong>on</strong>g KT metric if and <strong>on</strong>ly if M admits a str<strong>on</strong>g KT metric.Theorem 3.1 can be applied to complex orbifolds endowed with a str<strong>on</strong>g KTmetric in a similar way as for the symplectic orbifolds (see [11]).We start with recalling that a complex orbifold is a singular complex manifold Mof dimensi<strong>on</strong> n such that each singularity p is locally isomorphic to U/G, whereU is an open set of C n , G is a finite subgroup of GL(n, C) acting linearly <strong>on</strong> Uwith the <strong>on</strong>ly <strong>on</strong>e fixed point p. Moreover, the set S of singular points of M ofthe orbifold M has real codimensi<strong>on</strong> at least two.Therefore, for instance, the quotient of a complex manifold by a holomorphicacti<strong>on</strong> of a finite group G with n<strong>on</strong>-identity fixed point sets of real codimensi<strong>on</strong>at least two <strong>on</strong>e is a complex orbifold.The noti<strong>on</strong>s of smooth r-forms and (p,q)-forms make also sense <strong>on</strong> complexorbifolds and the differential d splits as usual as d = ∂ + ∂. A Hermitian metricg <strong>on</strong> a complex orbifold (M,J) is a J-Hermitian metric in the usual sense <strong>on</strong>the n<strong>on</strong>-singular part of (M,J) and G-invariant in any chart U/G. In this case,for any chart U/G, we have G ⊂ U(n). The Hermitian metric <strong>on</strong> the complexorbifold (M,J) is str<strong>on</strong>g KT if ∂∂F = 0, where F is the fundamental 2-formassociated to (J,g).We recall that in general a resoluti<strong>on</strong> ( ˜M,π) of a singular complex variety Mis a normal, n<strong>on</strong>singular complex variety ˜M with a proper surjective birati<strong>on</strong>almorphism π : ˜M → M. In [19] we studied the resoluti<strong>on</strong> of singularities of acomplex orbifold endowed with a str<strong>on</strong>g KT metric in order to obtain a smoothcomplex manifold admitting a str<strong>on</strong>g KT metric. We set the followingDefiniti<strong>on</strong> 3.3. Let (M,J,g) be a complex orbifold endowed with a str<strong>on</strong>g KTmetric g . A str<strong>on</strong>g KT resoluti<strong>on</strong> of a str<strong>on</strong>g KT orbifold (M,J,g) is the datum
A <str<strong>on</strong>g>survey</str<strong>on</strong>g> <strong>on</strong> str<strong>on</strong>g KT structures 107of a smooth complex manifold ( ˜M, ˜J) endowed with a ˜J-Hermitian str<strong>on</strong>g KTmetric ˜g and of a map π : ˜M → M, such thati) π : ˜M \ E → M \ S is a biholomorphism, where S is the singular set of Mand E = π −1 (S) is the excepti<strong>on</strong>al set;ii) ˜g = π ∗ g <strong>on</strong> the complement of a neighborhood of E.By applying Hir<strong>on</strong>aka Resoluti<strong>on</strong> of Singularities Theorem [32] to resolve thesingularities of a complex algebraic variety by a finite number of blow-ups, in [19]we proved the followingTheorem 3.4. Let (M,J) be a complex orbifold of complex dimensi<strong>on</strong> n endowedwith a J-Hermitian str<strong>on</strong>g KT metric g. Then there exists a str<strong>on</strong>g KT resoluti<strong>on</strong>.4 Simply-c<strong>on</strong>nected examplesIn [19] we applied the previous theorem to a quotient of a torus by the finite groupZ 2 . More precisely let T 6 = R 6 /Z 6 be the standard torus and let (x 1 ,...,x 6 ) beglobal coordinates <strong>on</strong> R 2n .C<strong>on</strong>sider <strong>on</strong> T 6 the involuti<strong>on</strong> σ induced byσ ( (x 1 ,...,x 6 ) ) = (−x 1 ,...,−x 6 ).and the complex structure J defined by{η 1 = dx 1 + i ( f dx 3 + dx 4) ,η j = dx j + idx 3+j , j = 2,3,where f = f(x 3 ,x 6 ) is a C ∞ , Z 6 -periodic and even functi<strong>on</strong>.Then, as a c<strong>on</strong>sequence of Theorem 3.4, we can prove the followingTheorem 4.1. [19] The quotient (T 6 /〈σ〉,J) is a complex orbifold with singularpoint setS ={x + Z 6 | x ∈ 1 }2 Z6 .The J-Hermitian metric g = 1 2∑ nj=1(η j ⊗ η j + η j ⊗ η j) is str<strong>on</strong>g KT. Moreover,(T 6 /〈σ〉,J) admits a str<strong>on</strong>g KT resoluti<strong>on</strong> which is simply-c<strong>on</strong>nected.Very interesting examples of 6-dimensi<strong>on</strong>al simply-c<strong>on</strong>nected str<strong>on</strong>g KT manifoldswere found in [21] by using torus bundles. They proved that for everypositive integer k ≥ 1, the manifold (k −1)(S 2 ×S 4 )♯k(S 3 ×S 3 ) admits a str<strong>on</strong>g
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