Invariance of Plurigenera in Smooth Projective Families: An Algebraic Approach

This article presents an algebraic proof of the invariance of plurigenera for families of smooth projective varieties under deformations. While Siu’s original proof relied on analytic tools such as multiplier ideal sheaves and L 2 -extension theorems, our approach reformulates these techniques within the framework of algebraic geometry, emphasizing multiplier ideals, Castelnuovo-Mumford regularity, and Nadel vanishing theorem. Key steps include establishing the surjectivity of restriction maps for pluricanonical sections via careful analysis of base ideals and asymptotic multiplier ideals. This work aligns with recent efforts to translate Siu’s results into algebraic settings and provides a foundation for extending the invariance theorem to singular varieties.