arXiv subjects

José Ignacio Yáñez

Publications and source records attributed to José Ignacio Yáñez.

10 recordsLinked to original sources

arXiv2026

Complexity one varieties are cluster type

The complexity of a Calabi-Yau pair $(X,B)$ is an invariant that relates the dimension of $X$, the rank of the group of divisors, and the coefficients of $B$. If the complexity is less than one, then $X$ is a toric variety. We prove that if the complexity is less than two, then $X$ is a Fano type variety. Furthermore, if the complexity is less than 3/2, then $X$ admits a Calabi-Yau structure of complexity one and index at most two, and it admits a finite cover $Y \to X$ of degree at most 2, where $Y$ is a cluster type variety. In particular, if the complexity is one and the index is one, $(X,B)$ is cluster type. Finally, we establish a connection with the theory of $T$-varieties. We prove that a variety of $T$-complexity one admits a similar finite cover from a cluster type variety.

math.AG↗

arXiv2026

Computing Cox rings via the cone conjecture

We initiate a program to study the Cox ring of Calabi-Yau varieties, employing the notion of Morrison-Kawamata dream spaces. In this setting, we establish an analogue of the Hu-Keel GIT constructions for Mori dream spaces. More precisely, for a Morrison-Kawamata dream space $X$, we establish a correspondence between the small $\mathbb{Q}$-factorial modifications of $X$ and the GIT quotients of $\operatorname{Spec}\operatorname{Cox}(X)$. We further show that the Cox ring of a Morrison-Kawamata dream space is a filtered direct limit of subalgebras, each of which is an inverse limit of finitely generated $\mathrm{Cl}(X)$-graded $\mathbb{K}$-algebras. As an application, we give an explicit presentation of the Cox ring of a very general hypersurface of multidegree $(2,\dots,2,n+1)$ in $(\mathbb{P}^1)^m\times \mathbb{P}^n$. Furthermore, we prove that the Cox ring of such a hypersurface is of dense $F$-pure type.

math.AG↗

arXiv2026

MMP for Generalized Pairs on Kähler 3-folds

In this article we define generalized pairs $(X, B+\boldsymbolβ)$ where $X$ is an analytic variety and $\boldsymbolβ$ is a b-(1,1) current. We then prove that almost all standard results of the MMP hold in this generality for compact Kähler varieties of dim $X\leq 3$. More specifically, we prove the cone theorem, existence of flips, existence of log terminal models, log canonical models and Mori fiber spaces, the geography of log canonical and log terminal models, etc.

math.AG↗

arXiv2025

Calabi-Yau pairs of complexity two

A Calabi-Yau pair of index one and complexity zero is toric. Furthermore, a Calabi-Yau pair of index one and complexity one is of cluster type. In this article, we study Calabi-Yau pairs of index one and complexity two. We develop machinery to decide whether a Calabi-Yau of complexity two is of cluster type. This approach reduces the problem to studying del Pezzo fibrations over toric varieties. We apply this to the setting of Gorenstein del Pezzo surfaces of Picard rank one. We prove that such a surface $X$ is cluster type if and only if $X$ has only $A$-type singularities and either $\mathrm{vol}(X)>1$ or $|X^{\rm sing}|\leq 3$.

math.AG↗

arXiv2025

Polarized endomorphisms of log Calabi-Yau pairs

Let $(X,Δ)$ be a dlt log Calabi-Yau pair admitting a polarized endomorphism. We show that $(X,Δ)$ is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety. We provide an example which shows that the previous statement does not hold if we drop the dlt condition of $(X,Δ)$ even if $X$ is a smooth variety. Given a klt type variety $X$ and a log Calabi-Yau pair $(X,Δ)$ admitting a polarized endomorphism, we show that a suitable birational modification of $(X,Δ)$ is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety.

math.AG↗

arXiv2025

Bott-Chern complexity of Kähler pairs

We introduce the Bott-Chern complexity of a compact Kähler pair $(X,B)$. This invariant compares $\dim(X)$, $\dim H^{1,1}_{\rm BC}(X)$ and the sum of the coefficients of $B$. When $(X,B)$ is Calabi-Yau, we show that its Bott-Chern complexity is non-negative. We prove that the Bott-Chern complexity of a Calabi-Yau compact Kähler pair $(X,B)$ is at least three whenever $X$ is not projective. Furthermore, we show this value is optimal and is achieved by certain singular non-projective K3 surfaces.

math.AG↗

arXiv2024

Polarized endomorphisms of Fano varieties with complements

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.

math.AG↗

arXiv2022

Movable cones of complete intersections of multidegree one on products of projective spaces

We study Calabi-Yau manifolds which are complete intersections of hypersurfaces of multidegree $1$ in an $m$-fold product of $n$-dimensional projective spaces. Using the theory of Coxeter groups, we show that the birational automorphism group of such a Calabi-Yau manifold $X$ is infinite and a free product of copies of $\mathbb{Z}$ . Moreover, we give an explicit description of the boundary of the movable cone $\overline{\operatorname{Mov}}(X)$. In the end, we consider examples for the general and non-general case and picture the movable cone and the fundamental domain for the action of $\operatorname{Bir}(X)$.

math.AG↗

arXiv2021

Birational automorphism groups and the movable cone theorem for Calabi-Yau complete intersections of products of projective spaces

For a Calabi-Yau manifold $X$, the Kawamata - Morrison movable cone conjecture connects the convex geometry of the movable cone $\overline{\mathrm{Mov}}(X)$ to the birational automorphism group. Using the theory of Coxeter groups, Cantat and Oguiso proved that the conjecture is true for general varieties of Wehler type, and they described explicitly $\mathrm{Bir}(X)$. We generalize their argument to prove the conjecture and describe $\mathrm{Bir}(X)$ for general complete intersections of ample divisors in arbitrary products of projective spaces. Then, under a certain condition, we give a description of the boundary of $\overline{\mathrm{Mov}}(X)$ and an application connected to the numerical dimension of divisors.

math.AG↗

arXiv2016

Characterization of Kollár surfaces

Kollár introduced in [Ko08] the surfaces $$(x_1^{a_1}x_2+x_2^{a_2}x_3+x_3^{a_3}x_4+x_4^{a_4}x_1=0)\subset \mathbb{P}(w_1,w_2,w_3,w_4)$$ where $w_i=W_i/w^*$, $W_i=a_{i+1}a_{i+2}a_{i+3}-a_{i+2}a_{i+3}+a_{i+3}-1$, and $w^*=$gcd$(W_1,\ldots,W_4)$. The aim was to give many interesting examples of $\mathbb{Q}$-homology projective planes. They occur when $w^*=1$. For that case, we prove that Kollár surfaces are Hwang-Keum [HK12] surfaces. For $w^*>1$, we construct a geometrically explicit birational map between Kollár surfaces and cyclic covers $z^{w^*}=l_1^{a_2 a_3 a_4} l_2^{-a_3 a_4} l_3^{a_4} l_4^{-1}$, where $\{l_1,l_2,l_3,l_4\}$ are four general lines in $\mathbb{P}^2$. In addition, by using various properties on classical Dedekind sums, we prove that: (a) For any $w^*>1$, we have $p_g=0$ iff the Kollár surface is rational. This happens when $a_{i+1} \equiv 1$ or $a_{i}a_{i+1} \equiv -1 ($mod $w^*)$ for some $i$. (b) For any $w^*>1$, we have $p_g=1$ iff the Kollár surface is birational to a K3 surface. We classify this situation. (c) For $w^*>>0$, we have that the smooth minimal model $S$ of a generic Kollár surface is of general type with $K_{S}^2/e(S) \to 1$.

math.AG↗