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On any smooth manifold $M$, the de Rham complex of smooth differential forms can be complexified pointwise. A smooth involution of the tangent bundle (called an almost complex structure $J$) gives a splitting of this complex into an almost double complex, which becomes an honest double complex (called the Dolbeault complex) when $J$ comes from a complex structure on $M$. There are two types of indecomposible bounded double complexes, zigzags and squares, into which every bounded double complex can be decomposed. While squares are inert, zigzags determine the dimensions of various cohomology spaces, and exhibit geometric properties of the complex structure on $M$. This talk will give a brief review of relevant differential topology background, introduce the Dolbeault complex of a complex manifold, and describe several special cases in which the zigzag decomposition is somewhat easier to calculate.

Please RSVP (optional, but encouraged!) at: https://docs.google.com/spreadsheets/d/1wyOmPJvaqsSngBuIcjnN3vLAWeRFTyru47HFaT0wlMc/edit#gid=1069057501