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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeApr 30th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI just ran across this: <http://mathoverflow.net/questions/163377/what-is-the-complex-third-derivative>. The answer points out that the real Hessian is not invariant under coordinate changes, which is exactly the problem of "Cauchy's invariant formula" that's solved by including $\mathrm{d}^2 x$ terms in $\mathrm{d}^2 f$. We can consider complex-valued cogerm differential forms, in particular on $\mathbb{C}^n$ or a more general complex manifold. (Although should the domains of the germs still be intervals in $\mathbb{R}$, or should they be regions in $\mathbb{C}$?) Are there "Dolbeault" versions of the commutative cogerm differential for which the "complex Hessian" can be written as $i \partial \bar{\partial} f$? If so, this might partially answer the second question asked at the link.

   I just ran across this: http://mathoverflow.net/questions/163377/what-is-the-complex-third-derivative. The answer points out that the real Hessian is not invariant under coordinate changes, which is exactly the problem of “Cauchy’s invariant formula” that’s solved by including $\mathrm{d}^2 x$ terms in $\mathrm{d}^2 f$.

   We can consider complex-valued cogerm differential forms, in particular on $\mathbb{C}^n$ or a more general complex manifold. (Although should the domains of the germs still be intervals in $\mathbb{R}$, or should they be regions in $\mathbb{C}$?) Are there “Dolbeault” versions of the commutative cogerm differential for which the “complex Hessian” can be written as $i \partial \bar{\partial} f$? If so, this might partially answer the second question asked at the link.