In this paper we investigate boundedness and volume of generalised pairs and give applications to usual algebraic varieties and pairs, especially to a class of pairs that we call stable minimal models.
We prove a result about descent of nef divisors to bounded families. This is the key to proving various other results in this paper and elsewhere.
As an application, fixing the dimension and a DCC set controlling coefficients, we will show that the set of volumes of all projective generalised lc pairs $(X,B+M)$, under the given data, satisfies the descending chain condition (DCC). Futhermore, we will show that in the klt case, the set of such pairs with ample $K_X+B+M$ and fixed volume forms a bounded family.
We will then apply the above to study projective lc pairs $(X,B)$ with abundant $K_X+B$ of arbitrary Kodaira dimension. In particular, we show that the set of Iitaka volumes of such pairs satisfies DCC under some natural boundedness assumptions on the fibres of the Iitaka fibration.
We define strongly stable minimal models. Fixing appropriate invariants, we show that such models form a bounded family. This and other results of this paper are crucial ingredients of the construction of moduli spaces of stable minimal models in a sequel work.
In this paper we investigate various properties of generalised pairs in families, especially boundedness of several kinds. We show that many statements for usual pairs do not hold for generalised pairs. In particular, we construct an unexpected counter-example to boundedness of generalised lc models with fixed appropriate invariants. We also show that the DCC of Iitaka volumes and existence of nef reduction maps fail in families of generalised pairs. In a positive direction we show boundedness of bases of log Calabi-Yau fibrations with their induced generalised pair structure under natural ass…
We prove Weak Borisov--Alexeev--Borisov Conjecture in dimension three which states that the anti-canonical volume of an $ε$-klt log Fano pair of dimension three is bounded from above.
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 $Φ(\mathscr{R})$ associated to a fixed finite set $\mathscr{R}$ form a bounded family. We also show $α(X,B)^{d-1}\mathrm{vol}(-(K_X+B))$ are bounded from above for all klt weak Fano pairs $(X,B)$ of a fixed dimension $d$.
In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on singularities with a torus action of complexity 1, threefold singularities, and hypersurface singularities. Given a real value v>0, we prove that the class of K-semistable threefold singularities with normalized volume at least v forms a bounded family. Analogous statements are proved in the case of n-dimensional complexity-1 and n-dimensional hypersurface singularities for arbitary n. In the general case of klt singularities, i.e. without the assumption on K-semistability, we show th…
In this article we prove the following version of the Weak-BAB conjecture for $3$-folds in char $p>5$: Fix a DCC set $I\subset [0, 1)$ and an algebraically closed field $k$ of characteristic $p>5$. Let $\mathfrak{D}$ be a collection of klt pairs $(X, Δ)$ satisfying the following properties: (1) $X$ is a projective $3$-fold, (2) $Δ$ is an $\mathbb{R}$-divisor with coefficients in $I$, (3) $K_X+Δ\equiv 0$, and (4) $-K_X$ is ample. Then the set $\{\mbox{vol}_X(-K_X) \ | \ (X, Δ)\in\mathfrak{D}\mbox{ for some }Δ\}$ is bounded from above.
It is conjectured that the base varieties of the Iitaka fibrations are bounded when the Iitaka volumes are bounded above. We confirm this conjecture for Iitaka $ε$-lc Fano type fibrations.
In this paper, we study the behavior of the sets of volumes of the form $\mathrm{vol}(X,K_X+B+M)$, where $(X,B)$ is a log canonical pair, and $M$ is a nef $\mathbb{R}$-divisor. After a first analysis of some general properties, we focus on the case when $M$ is $\mathbb{Q}$-Cartier with given Cartier index, and $B$ has coefficients in a given DCC set. First, we show that such sets of volumes satisfy the DCC property in the case of surfaces. Once this is established, we show that surface pairs with given volume and for which $K_X+B+M$ is ample form a log bounded family. These generalize results …
We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective klt pair (X,D) with $K_X+D$ big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of D and the volume of $K_X+D$. We further show that all n-dimensional projective klt pairs (X,D), such that $K_X+D$ is big and nef of fixed volume and such that the coefficients of D are contained in a given DCC…
In this paper we study the question of whether on smooth projective surfaces the denominators in the volumes of big line bundles are bounded. In particular we investigate how this condition is related to bounded negativity (i.e., the boundedness of self-intersections of irreducible curves). Our first result shows that boundedness of volume denominators is equivalent to \emph{primitive bounded negativity}, which in turn is implied by bounded negativity. We connect this result to the study of semi-effective orders of divisors: Our second result shows that negative classes exist that become effec…
Given $ε>0$, we show that over an algebraically closed field of characteristic $p>5$, the anticanonical volume of a Fano threefold $X$ (with arbitrary singularities) whose anticanonical divisor has Seshadri constant $ε(-K_X,x)>2+ε$ at some smooth point $x\in X$ is bounded from above.
The set of volumes of stable surfaces does have accumulation points. In this paper, we study this phenomenon for surfaces with one cyclic quotient singularity, towards answering the question under which conditions we can still have boundedness. Effective bounds allow listing singularities that might appear on a stable surface after fixing its invariants. We find optimal inequalities for stable surfaces with one cyclic quotient singularity, which can be used to prove boundedness under certain conditions. We also introduce the notion of generalized T-singularity, which is a natural generalizatio…
In this note, we survey some recent developments in birational geometry concerning the boundedness of algebraic varieties. We delineate a strategy to extend some of these results to the case of generalized pairs, first introduced by Birkar and Zhang, when the associated log canonical divisor is ample, and the volume is fixed. In this context, we show a version of deformation invariance of plurigenera for generalized pairs. We conclude by discussing an application to the boundedness of varieties of Kodaira dimension $κ(X)=\dim(X)-1$.
The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $x\in (X,Δ)$ satisfies the ACC if the coefficients of $Δ$ belong to a DCC set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $δ$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hol…
We construct klt projective varieties with ample canonical class and the smallest known volume. We also find exceptional klt Fano varieties with the smallest known anticanonical volume. We conjecture that our examples have the smallest volume in every dimension, and we give low-dimensional evidence for that. In order to improve on earlier examples, we are forced to consider weighted hypersurfaces that are not quasi-smooth. We show that our Fano varieties are exceptional by computing their global log canonical threshold (or $α$-invariant) exactly; it is extremely large, roughly $2^{2^n}$ in dim…
We investigate the class of compact complex Hermitian-symplectic manifolds $X$. For each Hermitian-symplectic metric $ω$ on $X$, we introduce a functional acting on the metrics in the Aeppli cohomology class of $ω$ and prove that its critical points (if any) must be Kähler when $X$ is $3$-dimensional. We go on to exhibit these critical points as maximisers of the volume of the metric in its Aeppli class and propose a Monge-Ampère-type equation to study their existence. Our functional is further utilised to define a numerical invariant for any Aeppli cohomology class of Hermitian-symplectic met…
