Wing Keung To, National U. Singapore, Singapore

Effective isometric embeddings and Lojasiewicz inequality

In this talk, I will discuss some joint work with Sai-Kee Yeung on effective estimates on certain modifications needed to turn  positive bihomogeneous polynomials on complex Euclidean spaces into squares of norms of vector-valued holomorphic polynomials.  Such results can be interpreted as effective estimates on powers of associated Hermitian holomorphic line bundles needed for isometric projective embedding.  We will also discuss generalizations to the case of bihomogeneous polynomials positive on a homogeneous affine hypersurfaces.  In the higher codimensional case of arithmetically defined homogeneous affine varieties, we also obtain an effective Lojasiewicz inequality for such varieties, which is needed for the generalization.