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In this paper, we study the volume of algebraically integrable foliations and locally stable families. We show that, for any canonical algebraically integrable foliation, its volume belongs to a discrete set depending only on its rank and the volume of its general leaves. In particular, if the foliation is of general type, then its volume has a positive lower bound depending only on its rank and the volume of its general leaves. This implies some special cases of a question posed by Cascini, Hacon, and Langer. As a consequence, we show that the relative volume of a stable family with a normal generic fiber belongs to a discrete set if the dimension and the volume of its general fibers are bounded. Log versions of the aforementioned theorems are also provided and proved., Comment: 24 pages

We study the structures of $\epsilon$-lc Calabi--Yau pairs. We show that such pairs possess a finite cover that is isomorphic in codimension 1 with a fiber product of a Calabi--Yau variety and a rationally connected variety. Moreover, the degree of the cover depends only on $\epsilon$ and the dimension. As a corollary, we show that the index of 4-dimensional non-canonical Calabi--Yau variety is bounded. We also prove a boundedness result on polarized $\epsilon$-lc Calabi--Yau pairs., Comment: 32 pages, comments welcome!

We study the birational boundedness of special fibers of log Calabi-Yau fibrations and Fano fibrations. We show that for a locally stable family of Fano varieties or polarised log Calabi-Yau pairs over a curve, if the general fiber satisfies some natural boundedness conditions, then every irreducible component of the special fiber is birationally bounded., Comment: 15 pages, comments welcome

We study the behavior of generalized lc pairs with $\mathrm{\textbf b}$-log abundant nef part, a meticulously designed structure on algebraic varieties. We show that this structure is preserved under the canonical bundle formula and sub-adjunction formulas, and is also compatible with the non-vanishing conjecture and the abundance conjecture in the classical minimal model program., Comment: 20 pages, v2. Reference update

In this paper, we investigate the boundedness of good minimal models with intermediate Kodaira dimensions, which has a natural Iitaka fibration to the canonical models. We prove that good minimal models are bounded modulo crepant birational equivalence when the base (canonical models) are bounded and the general fibers of the Iitaka fibration are in a bounded family of polarized Calabi-Yau pairs., Comment: 23 pages; revised argument in sections 3, 4, and 6, results unchanged

It is conjectured that the canonical models of varieties (not of general type) are bounded when the Iitaka volume is fixed. We confirm this conjecture when the general fibers of the corresponding Iitaka fibration are in a fixed bounded family of polarized Calabi-Yau pairs., Comment: 26 pages; Comments welcome

In this paper, we generalize the finiteness of models theorem in [BCHM06] to Kawamata log terminal pairs with fixed Kodaira dimension. As a consequence, we prove that a Kawamata log terminal pair with $\mathbb{R}-$boundary has a canonical model, and can be approximated by log pairs with $\mathbb{Q}-$boundary and the same canonical model.