Summary: In this paper, we investigate compact complex curves and compact totally real minimal surfaces in a complex projective space. Denote by \({|A|^2}\) the squared norm of the second fundamental form of the compact complex curve \(\Sigma\). We prove that if \(\int_\Sigma {|A|^2}{\mathrm{d}}\mu < \frac{1}{378\pi}\), then \(\Sigma\) is totally geodesic. For totally real minimal surfaces, we also obtain a similar theorem. More generally, we prove a global pinching theorem for minimal surfaces in a complex projective space.