Birational Boundedness Research Articles

On the boundedness of globally F-split varieties

This paper proposes the use of F-split and globally F-regular conditions in the pursuit of BAB type results in positive characteristic. The main technical work comes in the form of a detailed study of threefold Mori fibre spaces over positive dimensional bases. As a consequence we prove the main theorem, which reduces birational boundedness for a large class of varieties to the study of prime Fano varieties.

Geometry of polarised varieties

In this paper, we investigate the geometry of projective varieties polarised by ample and more generally nef and big Weil divisors. First we study birational boundedness of linear systems. We show that if X is a projective variety of dimension d with epsilon -lc singularities for epsilon >0, and if N is a nef and big Weil divisor on X such that N-K_{X} is pseudo-effective, then the linear system |mN| defines a birational map for some natural number m depending only on d,epsilon . This is key to proving various other results. For example, it implies that if N is a big Weil divisor (not necessarily nef) on a klt Calabi-Yau variety of dimension d, then the linear system |mN| defines a birational map for some natural number m depending only on d. It also gives new proofs of some known results, for example, if X is an epsilon -lc Fano variety of dimension d then taking N=-K_{X} we recover birationality of |-mK_{X}| for bounded m.We prove similar birational boundedness results for nef and big Weil divisors N on projective klt varieties X when both K_{X} and N-K_{X} are pseudo-effective (here X is not assumed epsilon -lc).Using the above, we show boundedness of polarised varieties under some natural conditions. We extend these to boundedness of semi-log canonical Calabi-Yau pairs polarised by effective ample Weil divisors not containing lc centres. We will briefly discuss applications to existence of projective coarse moduli spaces of such polarised Calabi-Yau pairs.

Birational boundedness of low-dimensional elliptic Calabi–Yau varieties with a section

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi–Yau manifolds$Y\rightarrow X$with a rational section, provided that$\dim (Y)\leq 5$and$Y$is not of product type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of Kawamata log terminal pairs$(X, \Delta )$with$K_X+\Delta$numerically trivial and not of product type, in dimension at most four.

On birational boundedness of foliated surfaces

Abstract In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function {P:\mathbb{Z}_{\geq 0}\to\mathbb{Z}} , then there exists an integer {N>0} such that if {(X,{\mathcal{F}})} is a canonical or nef model of a foliation of general type with Hilbert polynomial {\chi(X,{\mathcal{O}}_{X}(mK_{\mathcal{F}}))=P(m)} for all {m\in\mathbb{Z}_{\geq 0}} , then {|mK_{\mathcal{F}}|} defines a birational map for all {m\geq N} . On the way, we also prove a Grauert–Riemenschneider-type vanishing theorem for foliated surfaces with canonical singularities.

Birational boundedness of rationally connected Calabi–Yau 3-folds

We prove that rationally connected Calabi–Yau 3-folds with Kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected 3-folds of ϵ-CY type form a birationally bounded family for ϵ>0. Moreover, we show that the set of ϵ-lc log Calabi–Yau pairs (X,B) with coefficients of B bounded away from zero is log bounded modulo flops. As a consequence, we deduce that rationally connected klt Calabi–Yau 3-folds with mld bounded away from 1 are bounded modulo flops.

Fano varieties with large Seshadri constants

We show that the set of Fano varieties (with arbitrary singularities) whose anticanonical divisors have large Seshadri constants satisfies certain weak and birational boundedness. We also classify singular Fano varieties of dimension n whose anticanonical divisors have Seshadri constants at least n, generalizing an earlier result of Liu and the author.

On birational boundedness of Fano fibrations

We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to boundedness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of $\epsilon$-Fano type. Furthermore, we show birational boundedness of $3$-folds of $\epsilon$-Fano type.

Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for K3 surfaces, and the Tate conjecture

We investigate boundedness results for families of holomorphic symplectic varieties up to birational equivalence. We prove the analogue of Zarhin's trick by for $K3$ surfaces by constructing big line bundles of low degree on certain moduli spaces of stable sheaves, and proving birational versions of Matsusaka's big theorem for holomorphic symplectic varieties. As a consequence of these results, we give a new geometric proof of the Tate conjecture for $K3$ surfaces over finite fields of characteristic at least $5$, and a simple proof of the Tate conjecture for $K3$ surfaces with Picard number at least $2$ over arbitrary finite fields -- including characteristic $2$.

On complete intersections with trivial canonical class

We prove birational boundedness results on complete intersections with trivial canonical class of base point free divisors in (some version of) Fano varieties. Our results imply in particular that Batyrev–Borisov toric construction produces only a bounded set of Hodge numbers in any given dimension, even as the codimension is allowed to grow.

Effective log Iitaka fibrations for surfaces and threefolds

We prove an analogue of Fujino and Mori’s “bounding the denominators” (Theorem 3.1 of Fujino and Mori, J Diff Geom 50:167–188, 2000) in the log canonical bundle formula (see also Theorem 8.1 of Prokhorov and Shokurov, J Algebraic Geom 18(1):151–199, 2009) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair (X,Δ) of Kodaira codimension one and dimension at most three such that the coefficients of Δ are in a DCC set \({\mathcal{A}}\), there is a natural number N that depends only on \({\mathcal{A}}\) for which \({\lfloor{N(K_X+\Delta)\rfloor}}\) induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type.

Boundedness of threefolds of Fano type with Mori fibration structures

We show boundedness of $3$-folds of $\epsilon$-Fano type with Mori fibration structures. The proof is based on the birational boundedness result in our previous work arXiv:1509.08722 combining with arguments in Kawamata \cite{K} and Koll\'ar--Miyaoka--Mori--Takagi \cite{KMMT}.