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1
vote
Why do we study biholomorphically invariant pseudodistances/metrics
Just as in Riemannian (and Finsler) geometry, we can often distinguish domains by metric information, such as determining the behaviour of geodesics, or finding some analogue of curvature to compute, …
18
votes
Accepted
Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology
On differential forms, take complex conjugate to turn $\partial$ into $\bar\partial$, and holomorphic functions into conjugate holomorphic.
All of the proofs about differential forms then go through …