arXiv subjects
Publications and source records attributed to Joaquín Moraga.
We study the birational complexity of log Calabi-Yau $3$-folds. For such a pair $(X,B)$ of index one and coregularity zero, we show that $c_{\rm bir}(X,B)\in \{0,2,3\}$. Further, we prove that $(X,B)$ has a log Calabi-Yau crepant birational model that admits a crepant contraction to $(\mathbb{P}^{3-c},H_0+\dots+H_{3-c})$, where $c=c_{\rm bir}(X,B)$. To prove this, we give a geometric characterization of standard $\mathbb{P}^1$-links.
We introduce the notion of birational complexity of a log Calabi-Yau pair. This invariant measures how far the log Calabi-Yau pair is to being birational to a toric pair. We study fundamental properties of the new invariant, with a particular focus on the geometry of dual complexes.
In this article, we study the geometry of log Calabi-Yau pairs $(X,B)$ of index one and birational complexity zero. Firstly, we propose a conjecture that characterizes such pairs $(X,B)$ in terms of their dual complex and the rationality of their log canonical places. Secondly, we show that for these pairs the open set $X\setminus B$ is divisorially covered by open affine subvarieties which are isomorphic to open subvarieties of algebraic tori. We introduce and study invariants that measure the geometry and the number of these open subvarieties of algebraic tori. Thirdly, we study boundedness properties of log Calabi-Yau pairs of index one and birational complexity zero. For instance, in dimension $2$ we prove that such pairs are affinely bounded.
Let $X$ be a Fano type variety and $(X,Δ)$ be a log Calabi-Yau pair with $Δ$ a Weil divisor. If $(X,Δ)$ admits a polarized endomorphism, then we show that $(X,Δ)$ is a finite quotient of a toric pair. Along the way, we prove that a klt Calabi-Yau pair $(X,Δ)$ with standard coefficients that admits a polarized endomorphism is the quotient of an abelian variety.
We study Fano varieties endowed with a faithful action of a symmetric group, as well as analogous results for Calabi--Yau varieties, and log terminal singularities. We show the existence of a constant $m(n)$, so that every symmetric group $S_k$ acting on an $n$-dimensional Fano variety satisfies $k \leq m(n)$. We prove that $m(n)> n+\sqrt{2n}$ for every $n$. On the other hand, we show that $\lim_{n\to \infty} m(n)/(n+1)^2 \leq 1$. However, this asymptotic upper bound is not expected to be sharp. We obtain sharp bounds for certain classes of varieties. For toric varieties, we show that $m(n)=n+2$ for $n\geq 4$. For Fano quasismooth weighted complete intersections, we prove the asymptotic equality $\lim_{n\to \infty} m(n)/(n+1)=1$. Among the Fano weighted complete intersections, we study the maximally symmetric ones and show that they are closely related to the Fano--Fermat varieties, i.e., Fano complete intersections in $\mathbb P^N$ cut out by Fermat hypersurfaces. Finally, we draw a connection between maximally symmetric Fano varieties and boundedness of Fano varieties. For instance, we show that the class of $S_8$-equivariant Fano $4$-folds forms a bounded family. In contrast, the $S_7$-equivariant Fano $4$-folds are unbounded.
We prove the special termination for log canonical pairs and its generalisation in the context of generalised pairs.
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 that, up to special degenerations, the normalized volume bounds singularities with a complexity-1 torus action. We exhibit a 3-dimensional example which shows that this last statement is optimal.
In this article, we study the fundamental groups of low-dimensional log canonical singularities, i.e., log canonical singularities of dimension at most $4$. In dimension $2$, we show that the fundamental group of an lc singularity is a finite extension of a solvable group of length at most $2$. In dimension $3$, we show that every surface group appears as the fundamental group of a $3$-fold log canonical singularity. In contrast, we show that for $r\geq 2$ the free group $F_r$ is not the fundamental group of a $3$-dimensional lc singularity. In dimension $4$, we show that the fundamental group of any $3$-manifold smoothly embedded in $\mathbb{R}^4$ is the fundamental group of an lc singularity. In particular, every free group is the fundamental group of a log canonical singularity of dimension $4$. In order to prove the existence results, we introduce and study a special kind of polyhedral complexes: the smooth polyhedral complexes. We prove that the fundamental group of a smooth polyhedral complex of dimension $n$ appears as the fundamental group of a log canonical singularity of dimension $n+1$. Given a $3$-manifold $M$ smoothly embedded in $\mathbb{R}^4$, we show the existence of a smooth polyhedral complex of dimension $3$ that is homotopic to $M$. To do so, we start from a complex homotopic to $M$ and perform combinatorial modifications that mimic the resolution of singularities in algebraic geometry.
In this article, we introduce the generalized complexity of a generalized Calabi--Yau pair $(X,B,\textbf{M})$. This invariant compares the dimension of $X$ and Picard rank of $X$ with the sum of the coefficients of $B$ and $\textbf{M}$. It generalizes the complexity introduced by Shokurov. We show that a generalized log Calabi-Yau pair $(X,B,\textbf{M})$ of dimension $2$ with generalized complexity $0$ satisfies that $X$ is toric. This generalizes a result due to Brown, McKernan, Svaldi, and Zhong in the case of surfaces. Furthermore, we show that a generalized klt log Calabi-Yau surface $(X,\textbf{M})$ with generalized complexity $0$ satisfies that $X\simeq \mathbb{P}^2$ or $X\simeq \mathbb{P}^1\times \mathbb{P}^1$. Thus, this invariant interpolates between the characterization of toric varieties and the Kobayashi-Ochiai Theorem. As an application, we show that $3$-fold singularities with generalized complexity $0$ are toric. Furthermore, we show a local version of Kobayashi-Ochiai Theorem in dimension $3$.
We prove the ascending chain condition for log canonical thresholds of bounded coregularity.
In this article, we study the index of log Calabi--Yau pairs $(X,B)$ of coregularity 0. We show that $2λ(K_X+B)\sim 0$, where $λ$ is the Weil index of $(X,B)$. This is in contrast to the case of klt Calabi--Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends to the context of generalized log Calabi--Yau pairs, semi-log canonical pairs, and isolated log canonical singularities of coregularity 0. As a consequence, we show that the index of a variety appearing in the Gross--Siebert program or in the Kontsevich--Soibelman program is at most $2$. Finally, we discuss applications to Calabi--Yau varieties endowed with a finite group action, including holomorphic symplectic varieties endowed with a purely non-symplectic automorphism.
In this article, we study $G$-covers of klt varieties, where $G$ is a reductive group. First, we exhibit an example of a klt singularity admitting a $\mathbb{P}{\rm GL}_n(\mathbb{K})$-cover that is not of klt type. Then, we restrict ourselves to $G$-quasi-torsors, a special class of $G$-covers that behave like $G$-torsors outside closed subsets of codimension two. Given a $G$-quasi-torsor $X\rightarrow Y$, where $G$ is a finite extension of a torus $\mathbb{T}$, we show that $X$ is of klt type if and only if $Y$ is of klt type. We prove a structural theorem for $\mathbb{T}$-quasi-torsors over normal varieties in terms of Cox rings. As an application, we show that every sequence of $\mathbb{T}$-quasi-torsors over a variety with klt type singularities is eventually a sequence of $\mathbb{T}$-torsors. This is the torus version of a result due to Greb-Kebekus-Peternell regarding finite quasi-torsors of varieties with klt type singularities. On the contrary, we show that in any dimension there exists a sequence of finite quasi-torsors and $\mathbb{T}$-quasi-torsors over a klt type variety, such that infinitely many of them are not torsors. We show that every variety with klt type singularities is a quotient of a variety with canonical factorial singularities. We prove that a variety with Zariski locally toric singularities is indeed the quotient of a smooth variety by a solvable group. Finally, motivated by the work of Stibitz, we study the optimal class of singularities for which the previous results hold.
The regularity of a Fano variety, denoted by ${\rm reg}(X)$, is the largest dimension of the dual complex of a log Calabi--Yau structure on $X$. The coregularity is defined to be \[ {\rm coreg}(X):= \dim X - {\rm reg}(X)-1. \] The coregularity is the complementary dimension of the regularity. We expect that the coregularity of a Fano variety governs, to a large extent, the geometry of $X$. In this note, we review the history of Fano varieties, give some examples, survey some important theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties.
Using Altmann-Hausen-Suss description of $\mathbb{T}$-varieties via divisorial fans and Kóvacs-Schwede-Smith characterization of Du Bois singularities, we study Cohen-Macaulay Du Bois $\mathbb{T}$-singularities of complexity one. We exhibit cohomological criteria for a $\mathbb{T}$-variety to be Cohen-Macaulay and Du Bois in terms of polyhedral divisors. We give an example of a Cohen-Macaulay Du Bois singularity of complexity one which does not have rational singularities.
We show the Jordan property for regional fundamental groups of klt singularities of fixed dimension. Furthermore, we prove the existence of effective simultaneous index one covers for $n$-dimensional klt singularities. We give an application to the study of local class groups of klt singularities.
We expand the theory of log canonical $3$-fold complements. We prove that if $X\rightarrow T$ is a $3$-dimensional contraction of log Calabi-Yau type, then we can find $B\geq 0$ on $X$ for which $(X,B)$ is log canonical and $n(K_X+B)\sim_T 0$, where $n$ is an uniform natural number. This means that every $3$-fold of log Calabi-Yau type can be turned into a log Calabi-Yau pair in an effective way.
We study the normalized volume of toric singularities. As it turns out, there is a close relation to the notion of (non-symmetric) Mahler volume from convex geometry. This observation allows us to use standard tools from convex geometry, such as the Blaschke-Santaló inequality and Radon's theorem to prove non-trivial facts about the normalized volume in the toric setting. For example, we prove that for every $ε> 0$ there are only finitely many $\mathbb{Q}$-Gorenstein toric singularities with normalized volume at least $ε$. From this result, it directly follows that there are also only finitely many toric Sasaki-Einstein manifolds of volume at least $ε$ in each dimension. Additionally, we show that the normalized volume of every toric singularity is bounded from above by that of the rational double point of the same dimension. Finally, we discuss certain bounds of the normalized volume in terms of topological invariants of resolutions of the singularity. We establish two upper bounds in terms of the Euler characteristic and of the first Chern class, respectively. We show that a lower bound, which was conjectured earlier by He, Seong, and Yau, is closely related to the non-symmetric Mahler conjecture in convex geometry.
Given a klt singularity $(X,Δ;x)$, we define the iteration of Cox rings of $(X,Δ;x)$. The first result of this article is that the iteration of Cox rings ${\rm Cox}^{(k)}(X,Δ;x)$ of a klt singularity stabilizes for $k$ large enough. The second result is a boundedness one, we prove that for a $n$-dimensional klt singularity $(X,Δ;x)$ the iteration of Cox rings stabilizes for $k\geq c(n)$, where $c(n)$ only depends on $n$. Then, we use Cox rings to establish the existence of a simply connected factorial canonical cover (or scfc cover) of a klt singularity. We prove that the scfc cover dominates any sequence of quasi-étale finite covers and reductive abelian quasi-torsors of the singularity. We characterize when the iteration of Cox rings is smooth and when the scfc cover is smooth. We also characterize when the spectrum of the iteration coincides with the scfc cover. Finally, we give a complete description of the regional fundamental group, the iteration of Cox rings, and the scfc cover of klt singularities of complexity one. Analogous versions of all our theorems are also proved for Fano type morphisms. To extend the results to this setting, we show that the Jordan property holds for the regional fundamental group of Fano type morphisms.