= Riemann-Roch index for a real Cauchy-Riemann operator
{c}
{title2=$\operatorname{ind}_{\mathbb R}D=2n(1-g)+2c_1(E)[\Sigma]$}
A real-linear <Cauchy–Riemann operator> on a complex rank-$n$ <vector bundle> over a closed genus-$g$ <Riemann surface> is a <Fredholm operator> between the usual <Sobolev spaces> and has the displayed index. Zero-order real-linear terms do not change this index, since they form a continuous deformation of <Fredholm operators>. For the bundle pulled back by a sphere, subtracting the six-dimensional reparametrization group gives the <dimension formula for regular J-holomorphic spheres>.
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