The Estimation Of Lower Bound Of Canonical Volumes Of 3-folds Of General Type

The explicit classification of three-dimensional algebraic varieties is one of the most active fields in the current reasearch of birational geometry. In recent years, there has been a series of new methods and results. Especially, the estimation of lower bound of canonical volumes and the generalization of Noether inequality to three dimension have made significant progress.Jungkai Alfred Chen and Meng Chen in 2008 obtained a lower bound of the canon-ical volumes of irregular smooth projective 3-folds of general type. In 2010, they esti-mated the lower bound of the canonical volumes for arbitrary smooth projective 3-folds of general type, and found the optimal lower bound for smooth projective 3-folds of general type with the Euler-Poincare characteristic less than or equal 1.Noether inequality is an important part of the geography of 3-folds, and forms the boundary in the geography with Miyaoka-Yau inequality together. Many mathemati-cians have concerned on the problem what is the three dimensional version of Noether’s inequality since Reid proposed it in the 1980’s. But it was unknown until 2006 that the optimal inequality of Noether type in three dimension, proved by Catanese, Meng Chen and Deqi Zhang, for smooth minimal 3-folds of general type.In this paper, we considered the lower bound of the canonical volumes, which have close relationship with inequalities of Noether type, for 3-folds of general type. We improved the existing estimation methods and gave a better lower bound for the canonical volumes of irregular smooth 3-folds of generaly type. The lower bound now can be improved from 1/22 to 1/10, and we also provided a better inequality of Noether type in more general category.