On Canonically Fibred Algebraic 3-folds

This paper mainly estimates the upper bound of geometric invariants of generic irreducible components of general fibers of the canonical map of 3-fold of general type. Let V be a nonsingular projective 3-fold of general type. Assume X is a minimal model of V with at worst Q-factorial terminal singularities. Since the birational behaviors of?|KV| and?|KX|are the same, we can focus our study on X. When X is canonically of fiber type, let F be a birational smooth model of the generic irreducible component in the general fiber of canonical map. We hope to get an optimal upper bound of the birational invariants of F. For technical reasons, we need to assume X is Gorenstein.Chen-Hacon [10,12] once proved the boundedness theorem like g(F)?487 when F is a curve and pg(F)?434 when F is a surface. Besides, they also give examples with g(F)= 5 and pg(F)= 5, which are the biggest values among known examples. In this paper, firstly, we will improve known upper bounds as follows:g(F)?91 when F is a curve and pg(X) is sufficiently large; pg(F)?37 when F is a surface and pg(X) is sufficiently large. Secondly, we will present several new classes of Gorenstein minimal 3-folds with geometric genus as large as 13 when F is a curve and 19 when F is a surface. Finally, as a byproduct of our method in constructing these 3-fold examples, we in fact have found a new class of surfaces which is canonically fibred by curves of genus as large as 13.