Calabi-Yau Type Research Articles

Abundance for uniruled pairs which are not rationally connected

One of the central aims of the Minimal Model Program is to show that a projective log canonical pair (X,\Delta) with K_X+\nobreak\Delta pseudoeffective has a good model, i.e. a minimal model (Y,\Delta_Y) such that K_Y+\Delta_Y is semiample. The goal of this paper is to show that this holds if X is uniruled but not rationally connected, assuming the Minimal Model Program in dimension \dim X-\nobreak1 . Moreover, if X is rationally connected, then we show that the existence of a good minimal model for (X,\Delta) follows from a nonexistence conjecture for a very specific class of rationally connected pairs of Calabi–Yau type.

On Effective Log Iitaka Fibrations and Existence of Complements

Abstract We study the relationship between Iitaka fibrations and the conjecture on the existence of complements, assuming the good minimal model conjecture. In one direction, we show that the conjecture on the existence of complements implies the effective log Iitaka fibration conjecture. As a consequence, the effective log Iitaka fibration conjecture holds in dimension $3$. In the other direction, for any Calabi-Yau type variety $X$ such that $-K_{X}$ is nef, we show that $X$ has an $n$-complement for some universal constant $n$ depending only on the dimension of $X$ and two natural invariants of a general fiber of an Iitaka fibration of $-K_{X}$. We also formulate the decomposable Iitaka fibration conjecture, a variation of the effective log Iitaka fibration conjecture which is closely related to the structure of ample models of pairs with non-rational coefficients, and study its relationship with the forestated conjectures.

On the Alesker-Verbitsky Conjecture on HyperKähler Manifolds

We solve the quaternionic Monge–Ampère equation on hyperKähler manifolds. In this way we prove the ansatz for the conjecture raised by Alesker and Verbitsky claiming that this equation should be solvable on any hyperKähler with torsion manifold, at least when the canonical bundle is trivial holomorphically. The novelty in our approach is that we do not assume any flatness of the underlying hypercomplex structure which was the case in all the approaches for the higher order a priori estimates so far. The resulting Calabi–Yau type theorem for HKT metrics is discussed.

Solvability of a class of fully nonlinear elliptic equations on tori

We study the solvability of a class of fully nonlinear equations on the flat torus. The equations arise in the study of some Calabi–Yau type problems in torus bundles.

Boundedness of Complements for Log Calabi–Yau Threefolds

In this paper, we study the theory of complements, introduced by Shokurov, for Calabi–Yau type varieties with the coefficient set [0, 1]. We show that there exists a finite set of positive integers N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}$$\\end{document}, such that if a threefold pair (X/Z∋z,B)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X/Z\ i z,B)$$\\end{document} has an R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}-complement which is klt over a neighborhood of z, then it has an n-complement for some n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\in \\mathcal {N}$$\\end{document}. We also show the boundedness of complements for R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}-complementary surface pairs.

Log Calabi–Yau Structure of Projective Threefolds Admitting Polarized Endomorphisms

Abstract Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, that is, $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. It was conjectured by Broustet and Gongyo that $X$ is of Calabi–Yau type, that is, $(X,\Delta )$ is lc for some effective ${\mathbb {Q}}$-divisor such that $K_X+\Delta \sim _{{\mathbb {Q}}} 0$. In this paper, we establish a general guideline based on the equivariant minimal model program and the canonical bundle formula. In this way, we prove the conjecture when $X$ is a smooth projective threefold.

Strictly nef divisors on K-trivial fourfolds

In this paper, we prove the ampleness conjecture and Serrano’s conjecture for strictly nef divisors on K-trivial fourfolds, particularly, hyperkähler fourfolds. Specifically, for a smooth projective fourfold X of Calabi–Yau type with vanishing irregularity, we show that strictly nef divisors on X are ample.

On quartic double fivefolds and the matrix factorizations of exceptional quaternionic representations

On étudie les quartiques doubles de dimension cinq du point de vue des variétés Fano de type Calabi–Yau et des représentations quaternioniques exceptionnelles. On démontre tout d’abord qu’une quartique double de dimension cinq générique peut être représentée comme un recouvrement double de ℙ 5 ramifié le long d’une section linéaire de la quartique Spin 12 -invariante dans ℙ 31 . Le nombre de telles représentations est fini. Ensuite, en utilisant la géométrie des décompositions de Vinberg de type II de certaines représentations quaternioniques exceptionnelles et en se basant sur des calculs effectués grâce au logiciel Macaulay2, on démontre l’existence d’un fibré vectoriel sphérique de rang 6 sur de telles quartiques doubles. On déduit finalement de l’existence de ces fibrés que l’unité homologique des catégories Calabi–Yau de dimension trois associées par Kuznetsov aux quartiques doubles de dimension cinq est ℂ⊕ℂ[3].

The [formula omitted] orbifold, mirror symmetry, and Hodge structures of Calabi–Yau type

Starting from the Kähler moduli space of the rigid orbifold Z=E3∕Z3 one would expect for the cohomology of the generalized mirror to be a Hodge structure of Calabi–Yau type (1,9,9,1). We show that such a structure arises in a natural way from rational Hodge structures on Λ3Q[ω]6, where ω is a primitive third root of unity. We do not try to identify an underlying mirror geometry, but we show how special geometry arises in our abstract construction. We also show how such Hodge structure can be recovered as a polarized substructure of a bigger Hodge structure given by the third cohomology group of a six-dimensional abelian variety of Weil-type. Moreover, we recover a result of Zheng Zhang on the associates variation of Hodge structure.

On Motivic Realizations of the Canonical Hermitian Variations of Hodge Structure of Calabi–Yau Type over type Domains

AbstractLet${\mathcal{D}}$be the irreducible Hermitian symmetric domain of type$D_{2n}^{\mathbb{H}}$. There exists a canonical Hermitian variation of real Hodge structure${\mathcal{V}}_{\mathbb{R}}$of Calabi–Yau type over ${\mathcal{D}}$. This short note concerns the problem of giving motivic realizations for ${\mathcal{V}}_{\mathbb{R}}$. Namely, we specify a descent of${\mathcal{V}}_{\mathbb{R}}$from$\mathbb{R}$to$\mathbb{Q}$and ask whether the$\mathbb{Q}$-descent of${\mathcal{V}}_{\mathbb{R}}$can be realized as sub-variation of rational Hodge structure of those coming from families of algebraic varieties. When$n=2$, we give a motivic realization for ${\mathcal{V}}_{\mathbb{R}}$. When$n\geqslant 3$, we show that the unique irreducible factor of Calabi–Yau type in$\text{Sym}^{2}{\mathcal{V}}_{\mathbb{R}}$can be realized motivically.

Lower bounds for Lyapunov exponents of flat bundles on curves

Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non-expanding cusp monodromies. As an application we obtain the large genus limits of individual Lyapunov exponents in hyperelliptic strata of Abelian differentials. Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, e.g. for Calabi-Yau type families. We conjecture that equality of the sum of Lyapunov exponents and the degree is related to the monodromy group being a thin subgroup of its Zariski closure.

Characterization of Calabi–Yau variations of Hodge structure over tube domains by characteristic forms

Sheng and Zuo’s characteristic forms are invariants of a variation of Hodge structure. We show that they characterize Gross’s canonical variations of Hodge structure of Calabi–Yau type over (Hermitian symmetric) tube domains.

Remarks on Log Calabi-Yau Structure of Varieties Admitting Polarized Endomorphisms

We discuss the Calabi-Yau type structure of normal projective surfaces and Mori dream spaces admitting non-trivial polarized endomorphisms.

Liouville and Calabi-Yau type theorems for complex Hessian equations

We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex Hessian equation on a compact K\ahler manifold. This terminates the program, initiated by Hou, Ma and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right hand side in some $L^p, $ with sharp bound on $p$.

Some Fourth Order CY-type Operators with Non-symplectically Rigid Monodromy

ABSTRACTWe study tuples of matrices with rigidity index two in , which are potentially induced by differential operators of Calabi–Yau type. The constructions of those monodromy tuples via algebraic operations and middle convolutions and the related constructions on the level of differential operators lead to previously known and new examples.

Reduction of triangulated categories and maximal modification algebras for cAn singularities

Abstract In this paper we define and study triangulated categories in which the Hom-spaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi–Yau type property. If 𝒞 \mathcal{C} is such a category, we say that 𝒞 \mathcal{C} is Calabi–Yau with dim ⁡ 𝒞 ≤ 1 \dim\mathcal{C}\leq 1 . We extend the notion of Calabi–Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categories CM ¯ R \mathop{\underline{\textup{CM}}}R of Cohen–Macaulay modules. We explain the connection between Calabi–Yau reduction of CM ¯ R \mathop{\underline{\textup{CM}}}R and both partial crepant resolutions and ℚ \mathbb{Q} -factorial terminalizations of Spec ⁡ R \operatorname{Spec}R , and we show under quite general assumptions that Calabi–Yau reductions exist. In the remainder of the paper we focus on complete local c ⁢ A n cA_{n} singularities R. By using a purely algebraic argument based on Calabi–Yau reduction of CM ¯ R \mathop{\underline{\textup{CM}}}R , we give a complete classification of maximal modifying modules in terms of the symmetric group, generalizing and strengthening results in [I. Burban, O. Iyama, B. Keller and I. Reiten, Cluster tilting for one-dimensional hypersurface singularities, Adv. Math. 217 2008, 6, 2443–2484], [H. Dao and C. Huneke, Vanishing of Ext, cluster tilting and finite global dimension of endomorphism rings, Amer. J. Math. 135 2013, 2, 561–578], where we do not need any restriction on the ground field. We also describe the mutation of modifying modules at an arbitrary (not necessarily indecomposable) direct summand. As a corollary when k = ℂ k=\mathbb{C} we obtain many autoequivalences of the derived category of the ℚ \mathbb{Q} -factorial terminalizations of Spec ⁡ R \operatorname{Spec}R .

Surfaces of globally F-regular and F-split type

We prove that normal projective surfaces of dense globally F-split type (respectively, globally F-regular type) are of Calabi–Yau type (respectively, Fano type).

Wall-crossing in genus zero Landau–Ginzburg theory

Abstract We study a family of moduli spaces and corresponding quantum invariants introduced recently by Fan–Jarvis–Ruan. The family has a wall-and-chamber structure relative to a positive rational parameter ϵ. For a Fermat quasi-homogeneous polynomial W (not necessarily of Calabi–Yau type), we study natural generating functions packaging the invariants. Our wall-crossing formula relates the generating functions by showing that they all lie on the Lagrangian cone associated to the Fan–Jarvis–Ruan–Witten theory of W. For arbitrarily small parameter ϵ, a specialization of our generating function is a hypergeometric series called the big I-function which determines the entire Lagrangian cone. As a special case of our wall-crossing, we obtain a new geometric interpretation of the Landau–Ginzburg mirror theorem.

Finiteness results for 3-folds with semiample anticanonical bundle

The purpose of this paper is to give some evidence for the Morrison–Kawamata cone conjecture for klt pairs. Roughly speaking, the cone conjecture predicts that in appropriate ‘Calabi–Yau-type’ situations, the groups of automorphisms and pseudo-automorphisms of a projective variety act with rational polyhedral fundamental domain on the nef and movable cones of the variety. (See Section 1 for the precise statement.) In this paper we prove some statements in this direction, in the case of a mildly singular 3-fold with semiample anticanonical bundle of positive Iitaka dimension. Let us say a bit about how such a 3-fold looks geometrically. The anticanonical bundle defines a contraction morphism X→S to a positive-dimensional base; by adjunction all smooth fibres are varieties whose canonical bundle is torsion. So the generic fibre is a point, an elliptic curve, or a Calabi–Yau surface, according as the Iitaka dimension is 3, 2, or 1. (Here a Calabi–Yau surface means an abelian, K3, Enriques or hyperelliptic surface.) If the contraction morphism is equidimensional, classification results due to Kodaira and Miranda (for fibre dimension 1) and Kulikov and Crauder–Morrison (for fibre dimension 2) give information about the singular fibres. (See [5] for details of these classification results.) We will see that all 3-folds of this kind fall inside the scope of the Morrison–Kawamata cone conjecture. Our main result is the following finiteness theorem, which can be regarded as a weak form of the conjecture for these varieties. (All varieties are assumed projective, over an algebraically closed field of characteristic zero.)

A realization for a $\mathbb{Q}$-Hermitian variation of Hodge structure of Calabi-Yau type with real multiplication

We show that the $\mathbb{Q}$-descents of the canonical $\mathbb{R}$-variation of Hodge structure of Calabi-Yau type over a tube domain of type $A$ can be realized as sub-variations of Hodge structure of certain $\mathbb{Q}$-variations of Hodge structure which are naturally associated to abelian varieties of (generalized) Weil type.