The Fubini-Study 2-form $\alpha$ on complex projective space is a *calibrating* form in the sense that it evaluates to 1 on every complex direction, and is less than 1 on all other real 2-dimensional directions.  Therefore integration of $\alpha$ over any complex curve will give precisely the area of the curve, whereas for any embedded Riemann surface, the integral will provide a lower bound for the area.  This technique is applied very often in Riemannian geometry, following Marcel Berger and Mikhael Gromov after him.