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Valentino Tosatti : The Calabi-Yau equation on symplectic four-manifolds

Abstract: The Calabi conjecture, proved by Yau thirty years ago, says that on a compact Kahler manifold one can find a unique Kahler metric in every Kahler class with prescribed volume form. Donaldson recently conjectured that this theorem can be extended to symplectic forms with a compatible almost complex structure in 4 dimensions, and gave possible applications to the symplectic topology of 4-manifolds. I will discuss Donaldson's conjecture and some recents developments (joint work with B. Weinkove and partly with S.-T. Yau).

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