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Strong Moser theorem for Kähler symplectic forms on K3-surfaces

Speaker: 
V. Shevchishin
Institution: 
University of Warmia and Mazury in Olsztyn, Poland
Schedule: 
Wednesday, August 9, 2017 - 14:30
Location: 
A-136
Abstract: 

A classical theorem due to Moser (called Moser's trick)states that a family of symplectic forms \omega_t on a compactmanifold X with a constant cohomology class [\omega_t] is obtained byan isotopy: There exists an isotopy F:X -> X such that\omega_1 =F_*(\omega_0).   Dusa McDuff proved the following generalisation of this theorem("strong Moser theorem"):  If X is a rational or ruled 4-manifold,and \omega_t is a family of symplectic forms on X, such that[\omega_0]=[\omega_1] (equality of cohomology classes  only forthe  endpoints  of the family), then the forms \omega_0 and \omega_1are isotopic: There exists an isotopy F:X -> X such that\omega_1 =F_*(\omega_0).   Also McDuff found a counterexample which shows that the strongMoser theorem does not hold for general symplectic manifolds.   In my talk I give a proof of the strong Moser theorem for the casewhen \omega_t is a family of Kähler symplectic forms on a K3-surface X.The Kählerness means that every \omega_t is a Kähler form for somecomplex structure J_t on X. The proof exploits the Kähler andCalabi-Yau geometries of K3-surfaces. I also explains the meaning of this theorem for the action ofthe diffeomorphism group Diff(X) of a K3-surface X on the spacesof symplectic and Kähler forms.

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