A Gauss-Bonnet-Chern theorem for complex Finsler manifolds

Using Chern’s method of transgression and the currents, we establish a Gauss-Bonnet-Chern theorem for general closed complex Finsler manifolds (M, F). This result extends the classical Gauss-Bonnet-Chern theorem for Hermitian manifolds. Furthermore, a simplified version of the Gauss-Bonnet-Chern theorem is obtained in the case of complex Berwald 1-manifolds.