Higher-dimensional log Calabi-Yau pairs

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2. Log rational and cluster type pairs

Definition: Two log pairs

$(X,B)$ and

$(X',B')$ are said to be crepant birational equivalent if the following two conditions are satisfied:

(i) there are two projective birational maps $p\colon Y\rightarrow X$ and $q\colon Y\rightarrow X'$ , and
(ii) the equality of divisors $p^*(K_X+B)=q^*(K_{X'}+B')$ holds.

In the previous setting the birational map

$q\circ p^{-1}\colon (X,B)\dashrightarrow (X',B')$ is said to be a crepant birational map. In simple words, a crepant birational map is a birational map that respects the boundary divisor. Birational equivalent pairs share a lot of common properties such as the coregularity, the property of being log Calabi–Yau, and log discrepancies.

Definition: We say that a log Calabi–Yau pair

$(X,B)$ is log rational if it is crepant birationale quivalent to

$(\mathbb{P}^n,H_0+\dots+H_n)$ where the

$H_i$ are the hyperplane coordinates of the projective space and

$n=\dim X$ .

Definition: We say that a log Calabi–Yau pair

$(X,B)$ is of cluster type if there is a crepant birational map

$\phi\colon (\mathbb{P}^n,H_0+\dots+H_n)\dashrightarrow (X,B)$ for which

${\rm codim}_{\mathbb{P}^n}(\overline{{\rm Ex}(\phi)\cap \mathbb{G}_m^n})\geq 2.$ In other words, the crepant birational map from

$(\mathbb{P}^n,H_0+\dots+H_n)$ does not contract divisors from the algebraic torus

$\mathbb{G}_m^n$ .

While the concepts of rationality and toricness are well-studied in algebraic geometry, these two new notions are novel and less explored. By definition, every toric pair is cluster type, every cluster type pair is log rational, and every log rational pair is rational. However, the converse implications are not true. Below we compile some problems about log rational and cluster type pairs.

By the work of Gross, Hacking, and Keel, we know that a log Calabi–Yau pair $(X,B)$ of dimension 2, index one (i.e., $K_X+B\sim 0$ ), and coregularity zero is log rational. The previous statement was also proved independently by Shokurov. Ducat proved that a log Calabi–Yau pair of the form $(\mathbb{P}^3,B)$ of index one (i.e., $B$ is a quartic surface) and coregularity zero is also log rational. The following seems to be the more natural generalization of Ducat’s work.

Add remark

Problem 2.1.
Let $T$ be a toric $3$ -fold and $B\sim |-K_T|$ be an slc hypersurface of coregularity zero. Show that $(T,B)$ is log rational.
The most ambitious version of the previous problem was proposed by Ducat in dimension 3 and then by Enwright, Figueroa, and Moraga in arbitrary dimension. It is known as the log rationality conjecture.

Add remark

Problem 2.2.
Let $(X,B)$ be a log Calabi–Yau pair of dimension $n$ . Let $(Y,B_Y)\rightarrow (X,B)$ be a dlt modification. Assume that every strata of $(Y,B_Y)$ , inlcuding $Y$ itself, is a rational variety. Furthermore, assume that $\mathcal{D}(Y,B_Y)\simeq_{\rm PL} \mathbf{S}^{n-1}$ . Prove that $(X,B)$ is a log rational pair.
1. Remark. [Joaquin Moraga] Examples in the literature due to Kaloghiros and Mauri-Moraga show that both conditions: the rationality of the strata and the sphericity of the dual complex are indeed necessary in the previous problem.
By the work of Loginov, Moraga, and Vasilkov, we know that given a general (i.e., general in its deformation family) rational smooth Fano $3$ -fold $X$ , we can find $B\in |-K_X|$ for which $(X,B)$ is log rational. Thus, general rational smooth Fano threefolds are indeed log rational. We do not yet understand what happens for special smooth rational Fano $3$ -folds.

Add remark

Problem 2.3.
Classify log rational smooth Fano $3$ -folds.
Smooth quadrics are examples of cluster type varieties. Indeed, the pair $(\mathbb{Q}^n,H_1+\dots+H_n)$ induced by cutting with hyerplanes is a cluster type pair. In the previous pair, the number of of components of the boundary equals $\dim(Q)+\rho(Q)-1$ . We expect that this phenomena holds in a more general setting.

Add remark

Problem 2.4.
Let $(X,B)$ be a log Calabi–Yau pair of dimension $n$ , index one, and Picard rank $\rho$ . Assume that the number of components of $B$ equals $n+\rho-1$ . Show that $(X,B)$ is of cluster type.
By the work of Loginov together with Gross, Hacking, and Keel, it is known that a general del Pezzo surface is of cluster type. Recall that any smooth del Pezzo surface is a blow-up of $\mathbb{P}^2$ . In the case of smooth Fano $3$ -folds, it is unclear which ones are of cluster type yet and this seems to be a very hard problem. In this direction, the most reasonable first step is the following:

Add remark

Problem 2.5.
Classify which smooth Fano blow-ups of $\mathbb{P}^3$ are of cluster type.
Cluster type varieties are closely related to toric varieties. In characteristic zero, if $X$ is a $n$ -dimensional Fano variety and $(X,B)$ is a cluster type pair, then we have an embedding $\mathbb{G}_m^n\hookrightarrow X\setminus B$ . For this reason, it is expected that cluster type varieties defined over a field of characteristic $p>1$ lift.

Add remark

Problem 2.6.
Let $X$ be a cluster type variety defined over an algebraically closed field $k$ of characteristic $p>1$ . Show that $X$ is liftable over $\mathbb{W}_2(k)$ .

Cite this as: AimPL: Higher-dimensional log Calabi-Yau pairs, available at http://aimpl.org/higherdimlogcy.

Higher-dimensional log Calabi-Yau pairs

2. Log rational and cluster type pairs

Problem 2.1.

Problem 2.2.

Problem 2.3.

Problem 2.4.

Problem 2.5.

Problem 2.6.