A reduction of canonical stability index of 4 and 5 dimensional projective varieties with large volume

Algebraic threefolds of general type with small volume

“…It is known that the Noether inequality exists in any dimension: there exist a n , b n > 0 such that K n X ≥ a n p g (X) − b n for every minimal n-fold X of general type [11]. However, to obtain the optimal a n and b n for n ≥ 3 is quite challenging (see [20,2,5] for n = 3, as well as [8,30] for general n).…”

Boundedness of fibers for pluricanonical maps of varieties of general type

“…Somewhat surprisingly, if we only consider threefolds of large volume, Todorov [Tod07] has shown that ϕ rK X is birational already for r r 2 = 5. Similarly, in [CJ17] Chen and Jiang proved that for fourfolds of large volume ϕ rK X is birational already for r r 3 . Moreover, they conjectured that if X is a variety of large volume and dimension n, then ϕ rK X is birational for r r n−1 (see [CJ17,Question 6.1]).…”

“…Similarly, in [CJ17] Chen and Jiang proved that for fourfolds of large volume ϕ rK X is birational already for r r 3 . Moreover, they conjectured that if X is a variety of large volume and dimension n, then ϕ rK X is birational for r r n−1 (see [CJ17,Question 6.1]). In this direction we prove: Theorem 1.3.…”

Varieties of general type with doubly exponential asymptotics

“…Noether's inequality for surfaces of general type says that vol(X) ≥ 2p g − 4, where the geometric genus p g means h 0 (X, K X ). More generally, M. Chen and Z. Jiang showed that for every positive integer n there are positive constants a n and b n such that vol(X) ≥ a n p g (X) − b n for every smooth projective n-fold X of general type [13,Corollary 5.1]. Strengthening earlier results, J. Chen, M. Chen, and C. Jiang recently proved a Noether inequality for 3-folds of general type, with optimal constants: we have vol(X) ≥ (4/3)p g (X) − 10/3 if p g (X) ≥ 11 [11].…”