TY - JOUR

T1 - Erratum

T2 - Unobstructed symplectic packing for tori and hyper-Kähler manifolds (Journal of Topology and Analysis (2016) 8:4 (589-626) DOI: 10.1142/s1793525316500229)

AU - Entov, Michael

AU - Verbitsky, Misha

N1 - Publisher Copyright:
© 2019 World Scientific Publishing Company.

PY - 2019/3/1

Y1 - 2019/3/1

N2 - In the proof of Theorem 9.2 in Section 9.2 of the paper [1] it was erroneously claimed without proof that the Teichmüller space of Kähler formsa on T 2n is connected (meaning that any two cohomologous Kähler forms on T 2n are isotopic as Kähler forms). While we do not know whether this claim is true, the proof of Theorem 9.2 in [1] in the case of T 2n (the only place in [1] where the claim was used) can be fixed in the following way (see [2] for more details). Below we use the notations of [1]. Consider the subgroup DiffH ? Diff+, consisting of all diffeomorphisms of T 2n acting by identity on H(T 2n). The action of DiffH on the Teichmüller space Teichs of Kähler forms (that is, the space of the Kähler forms on T 2n, factorized by the action of Diff0) preserves the fibers of the period map Per : Teichs → H2(M,R) (that is, the map associating to an element of Teichs the cohomology class of any Kähler form representing that element). We claim that this action is transitive on each fiber.b Indeed, by Proposition 6.1 of [1], any Kähler form on T 2n can be identified with a linear symplectic form by a diffeomorphism of T 2n. Moreover, with an appropriate choice of the map F appearing in the proof of that proposition, the latter diffeomorphism can be chosen to belong to DiffH. Since two linear cohomologous forms on T 2n = R2n/Z2n coincide, we get that any two cohomologous Kähler forms on T 2n - that is, representatives of two points of Teichs lying in the same fiber of Per - can be mapped into each other by a diffeomorphism from DiffH, which proves the claim. The transitivity of the action of DiffH on the fibers of Per and the fact that Per is a surjective local diffeomorphism imply that for any Kähler form w on T 2n the orbit of the image of w in Teichs under the action of Diff+/Diff0 is dense in Teichs if and only if the orbit of the cohomology class [w] under the action of Diff+/Diff0on H2(T 2n,R) is dense in the image of Per. The rest of the proof of Theorem 9.2 proceeds as in [1].

AB - In the proof of Theorem 9.2 in Section 9.2 of the paper [1] it was erroneously claimed without proof that the Teichmüller space of Kähler formsa on T 2n is connected (meaning that any two cohomologous Kähler forms on T 2n are isotopic as Kähler forms). While we do not know whether this claim is true, the proof of Theorem 9.2 in [1] in the case of T 2n (the only place in [1] where the claim was used) can be fixed in the following way (see [2] for more details). Below we use the notations of [1]. Consider the subgroup DiffH ? Diff+, consisting of all diffeomorphisms of T 2n acting by identity on H(T 2n). The action of DiffH on the Teichmüller space Teichs of Kähler forms (that is, the space of the Kähler forms on T 2n, factorized by the action of Diff0) preserves the fibers of the period map Per : Teichs → H2(M,R) (that is, the map associating to an element of Teichs the cohomology class of any Kähler form representing that element). We claim that this action is transitive on each fiber.b Indeed, by Proposition 6.1 of [1], any Kähler form on T 2n can be identified with a linear symplectic form by a diffeomorphism of T 2n. Moreover, with an appropriate choice of the map F appearing in the proof of that proposition, the latter diffeomorphism can be chosen to belong to DiffH. Since two linear cohomologous forms on T 2n = R2n/Z2n coincide, we get that any two cohomologous Kähler forms on T 2n - that is, representatives of two points of Teichs lying in the same fiber of Per - can be mapped into each other by a diffeomorphism from DiffH, which proves the claim. The transitivity of the action of DiffH on the fibers of Per and the fact that Per is a surjective local diffeomorphism imply that for any Kähler form w on T 2n the orbit of the image of w in Teichs under the action of Diff+/Diff0 is dense in Teichs if and only if the orbit of the cohomology class [w] under the action of Diff+/Diff0on H2(T 2n,R) is dense in the image of Per. The rest of the proof of Theorem 9.2 proceeds as in [1].

UR - http://www.scopus.com/inward/record.url?scp=85056515165&partnerID=8YFLogxK

U2 - 10.1142/S1793525318920012

DO - 10.1142/S1793525318920012

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AN - SCOPUS:85056515165

SN - 1793-5253

VL - 11

SP - 249

EP - 250

JO - Journal of Topology and Analysis

JF - Journal of Topology and Analysis

IS - 1

ER -