Effective bounds for very ample line bundles

Abstract.

Let $L$ be an ample line bundle on a non singular projective$n$ -fold $X$. It is first shown that$2K_X+mL$ is very ample for$m\ge 2+{3n+1\choose n}$ . The proof develops an original idea of Y.T. Siu and is based on a combination of the Riemann-Roch theorem together with an improved Noetherian induction technique for the Nadel multiplier ideal sheaves. In the second part, an effective version of the big Matsusaka theorem is obtained, refining an earlier version of Y.T. Siu: there is an explicit polynomial bound$m_0=m_0(L^n,L^{n-1}\cdot K_X)$ of degree${}\le n3^n$ in the arguments, such that $mL$ is very ample for$m\ge m_0$ . The refinement is obtained through a new sharp upper bound for the dualizing sheaves of algebraic varieties embedded in projective space.