 Almost complex structure induced by a complex atlas

ID: almost-complex-structure-induced-by-a-complex-atlas

In a holomorphic chart

z^{j} = x^{j} + i y^{j}

J \frac{\partial}{\partial x ^{j}} = \frac{\partial}{\partial y ^{j}}, J \frac{\partial}{\partial y ^{j}} = - \frac{\partial}{\partial x ^{j}} .

The derivative of every holomorphic transition map is complex linear and therefore commutes with this

J

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