Bounding the first Dirichlet eigenvalue of a tube around a complex submanifold of CP^n by the degrees of the polynomials defining it

We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model centers of the tube. The bounds are sharp on these models. Moreover, when the models used are $CP^q$ or the complex hyperquadric, these bounds also give gap phenomena and comparison results.