Let $E\rightarrow \mathbb{P}^1$ be a complex vector bundle and let $a_{(0,0)},a_{(1,0)},a_{(0,1)},a_{(1,1)}$ be differential forms such that $a_{(i,j)}\in\Omega^{i,j}(\mathbb{P}^1,End(E))$. I would like to compute the degree (1,1) part of the form $$Tr(\exp(a_{(0,0)}+a_{(1,0)}+a_{(0,1)}+a_{(1,1)}))\in\Omega^{*,*}(\mathbb{P}^1,\mathbb{C})$$ Any idea please?
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1$\begingroup$ I am not clear how to define the exponential, because they don't commute. $\endgroup$– Ben McKayCommented Jun 8, 2020 at 13:54
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$\begingroup$ @BenMcKay this is why it is harder to compute, but since you can multiply them the exponential is define $\endgroup$– BinAckerCommented Jun 8, 2020 at 14:09
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