  The famous Yang-Yau inequality provides an upper bound for the first
eigenvalue of the Laplacian on an orientable Riemannian surface solely in terms
of its genus $\gamma$ and the area. Its proof relies on the existence of
holomorhic maps to $\mathbb{CP}^1$ of low degree. Very recently, A.~Ros was
able to use certain holomorphic maps to $\mathbb{CP}^2$ in order to give a
quantitative improvement of the Yang-Yau inequality for $\gamma=3$. In the
present paper, we generalize Ros' argument to make use of holomorphic maps to
$\mathbb{CP}^n$ for any $n>0$. As an application, we obtain a quantitative
improvement of the Yang-Yau inequality for all genera $\gamma>3$ except for
$\gamma = 4,6,8,10,14$.

Differential Geometry;Spectral Theory