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We obtain upper bounds for the first Dirichlet eigenvalue of a tube around acomplex 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 somemodel centers of the tube. The bounds are sharp on these models. Moreover, whenthe models used are $CP^q$ or the complex hyperquadric, these bounds also givegap phenomena and comparison results.

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