For the <formal adjoints> $d^*$, $\partial^*$ and $\bar\partial^*$ defined by the <Kähler metric>,
$$
\boxed{\Delta_d=dd^*+d^*d,}
$$
and
$$
\boxed{\Delta_{\bar\partial}=\bar\partial\bar\partial^*+\bar\partial^*\bar\partial,\qquad
\Delta_\partial=\partial\partial^*+\partial^*\partial.}
$$
These are respectively the <Hodge Laplacian> and the two Dolbeault Laplacians.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 5 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-118/5/a/solution