An effective Matsusaka big theorem (Q1337588)

We prove the following effective form of Matsusaka's Big Theorem. For an ample line bundle \(L\) over a compact complex manifold \(X\) of complex dimension \(n\) with canonical line bundle \(K_ X\), the line bundle \(mL\) is very ample for \(m\) no less than \[ (2^{3n-1} 5n)^{4^{n-1}} (3(3n- 2)^ nL^ n + K_ X \cdot L^{n - 1})^{4^{n - 1}3n} \over (6(3n - 2)^ n - 2n - 2)^{4^{n - 1} n - {2 \over 3}} (L^ n)^{4^{n-1} 3(n - 1)}. \]