Related papers: Strong $(\delta,n)$-complements for semi-stable mo…

We prove a canonical bundle formula for generically finite morphisms in the setting of generalized pairs (with $\mathbb{R}$-coefficients). This complements Filipazzi's canonical bundle formula for morphisms with connected fibres. It is then…

We show the existence of $(\epsilon,n)$-complements for $(\epsilon,\Rr)$-complementary projective generalized pairs of Fano type $(X,B+M)$ when either the coefficients of $B$ and $\mu_j$ belong to a finite set or the coefficients of $B$…

For Fano fibrations with $\epsilon$-lc singularities of a fixed dimension, we show the existence of bounded relative-global complements. If the base of the fibration is of dimension one, we even show the existence of bounded relative-global…

We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from $0$ and the coefficients of the boundaries belonging to a fixed DCC set $S$ form a bounded family. Moreover, such pairs admit a strong…

We prove the boundedness of $n$-complements for surface pairs in a generalized case without restrictions on multiplicities or the Fano type assumption.

In this paper, we generalise the theory of complements to log canonical log fano varieties and prove boundedness of complements for them in dimension less than or equal to 3. We also prove some boundedness results for the canonical index of…

In this paper, we prove the canonical bundle formula for Fano type fibrations and Shokurov's conjecture on boundedness of complements for Fano type threefold pairs $(X,B)$ with fibration structures in large characteristics. In particular,…

Let $(X,\Delta)$ be a projective log canonical pair such that $\Delta \geq A$ where $A \geq 0$ is an ample $\mathbb{R}$-divisor. We prove that either $(X,\Delta)$ has a good minimal model or a Mori fibre space. Moreover, if $X$ is…

Let $f: X \to Z$ be a fibration from a normal projective variety $X$ of dimension $n$ onto a normal curve $Z$ over a perfect field of characteristic $p>2$. Let $(X, B)$ be a dlt pair such that the induced pair on a general fibre is log…

Given a Fano type log Calabi-Yau fibration $(X,B)\to Z$ with $(X,B)$ being $\epsilon$-lc, the first author in \cite{Bi23} proved that the generalised pair $(Z,B_Z+M_Z)$ given by the canonical bundle formula is generalised $\delta$-lc where…

We prove an analogue of Fujino and Mori's ``bounding the denominators'' in the log canonical bundle formula (see also Prokhorov and Shokurov) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a…

We show that a weak version of the canonical bundle formula holds for fibrations of relative dimension one. We provide various applications thereof, for instance, using the recent result of Xu and Zhang, we prove the log non-vanishing…

1) Assuming log Minimal Model Conjecture, we give a construction of a complete moduli space of stable log pairs of arbitrary dimension generalizing directly the space M_{g,n} of pointed stable curves. Each stable pair has semi log canonical…

The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the…

We show the existence of $n$-complements for generalized pairs with additional Diophantine approximation properties when the coefficients of boundaries belong to a DCC set.

We show that the finiteness of the fundamental groups of the smooth locus of lower dimensional log Fano pairs would imply the finiteness of the local fundamental group of klt singularities. As an application, we verify that the local…

We prove the boundedness of complements for Fano type generalized pairs (with the boundary coefficient set $[0,1]$) after Shokurov.

We establish the generalized canonical bundle formula for generalized lc-trivial fibrations with irrational coefficients over non-compact bases in the complex analytic setting, and we show that the discriminant b-divisor and moduli…

We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from $0$ and the coefficients of the boundaries belong to the set of hyperstandard multiplicities $\Phi(\mathscr{R})$ associated to a fixed…

We prove that the anti-canonical divisors of weak Fano 3-folds with log canonical singularities are semiample. Moreover, we consider semiampleness of the anti-log canonical divisor of any weak log Fano pair with log canonical singularities.…