Canonical Bundle Research Articles

Canonical torsor bundles of prescribed rational functions on complex curves

The prescribed rational functions constitute a subset of rational functions satisfying certain symmetry and analyticity conditions. Over a smooth complex curve [Formula: see text], we construct explicitly a bundle [Formula: see text] with values in the prescribed rational functions. An intrinsic coordinate-independent formulation (the main result of the paper, Proposition 1) for such bundles is given. The construction presented in this paper is useful for studies of the canonical cosimplicial cohomology of infinite-dimensional Lie algebras on smooth manifolds, as well as for the purposes of conformal field theory and the theory of foliations.

Intersection matrices for the minimal regular model of X0(N)${X}_0(N)$ and applications to the Arakelov canonical sheaf

Intersection matrices for the minimal regular model of X0(N)${X}_0(N)$ and applications to the Arakelov canonical sheaf

On the Canonical Bundle of Complex Solvmanifolds and Applications to Hypercomplex Geometry

On the Canonical Bundle of Complex Solvmanifolds and Applications to Hypercomplex Geometry

Explicit abelian instantons on S1-invariant Kähler Einstein 6-manifolds

We consider the dimensional reduction of the (deformed) Hermitian Yang–Mills condition on S1-invariant Kähler Einstein 6-manifolds. This allows us to reformulate the (deformed) Hermitian Yang–Mills equations in terms of data on the quotient Kähler 4-manifold. In particular, when the gauge group is U(1) we apply this construction to the canonical bundles of CP2 and CP1×CP1 endowed with the Calabi ansatz metric to find abelian instantons. We show that these are determined by a suitable subset of the spectrum of the zero section and are explicitly given in terms of certain hypergeometric functions. As a by-product of our investigation we find a coordinate expression for the holomorphic volume form on OCP2(−3) and use it to construct a new special Lagrangian foliation. We also find 1-parameter families of explicit deformed Hermitian Yang–Mills connections on certain non-compact S1-invariant Kähler Einstein 6-manifolds.

Berndtsson-Lempert-Szőke fields associated to proper holomorphic families of vector bundles

Drawing on work of Berndtsson and of Lempert and Szőke, we define a complex analytic structure for families of (possibly finite-dimensional) Hilbert spaces that might not fit together to form a holomorphic vector bundle but nevertheless have a reasonable definition of curvature that agrees with the curvature of the Chern connection when the family of Hilbert spaces is locally trivial. As one consequence, we obtain a new proof of a theorem of Berndtsson on the curvature of direct images of semi-positively twisted relative canonical bundles (i.e., adjoint bundles), and of its higher-rank generalization by Liu-Yang.

Poincaré duality for smooth Poisson algebras and BV structure on Poisson cohomology

Similar to the modular vector fields in Poisson geometry, modular derivations are defined for smooth Poisson algebras with trivial canonical bundle. By twisting Poisson module with the modular derivation, the Poisson cochain complex with values in any Poisson module is proved to be isomorphic to the Poisson chain complex with values in the corresponding twisted Poisson module. Then a version of twisted Poincaré duality is proved between the Poisson homologies and cohomologies. Furthermore, a notion of pseudo-unimodular Poisson structure is defined. It is proved that the Poisson cohomology as a Gerstenhaber algebra admits a Batalin-Vilkovisky operator inherited from some one of its Poisson cochain complex if and only if the Poisson structure is pseudo-unimodular. This generalizes the geometric version due to P. Xu. The modular derivation and Batalin-Vilkovisky operator are also described by using the dual basis of the Kähler differential module.

On the zero set of the holomorphic sectional curvature

AbstractA notable example due to Heier, Lu, Wong, and Zheng shows that there exist compact complex Kähler manifolds with ample canonical line bundle such that the holomorphic sectional curvature is negative semi‐definite and vanishes along high‐dimensional linear subspaces in every tangent space. The main result of this note is an upper bound for the dimensions of these subspaces. Due to the holomorphic sectional curvature being a real‐valued bihomogeneous polynomial of bidegree (2,2) on every tangent space, the proof is based on making a connection with the work of D'Angelo on complex subvarieties of real algebraic varieties and the decomposition of polynomials into differences of squares. Our bound involves an invariant that we call the holomorphic sectional curvature square decomposition length, and our arguments work as long as the holomorphic sectional curvature is semi‐definite, be it negative or positive.

[formula omitted]-cohomology projective planes from Enriques surfaces in odd characteristic

We give a complete classification of Qℓ-cohomology projective planes with isolated ADE-singularities and numerically trivial canonical bundle in odd characteristic. This leads to a beautiful relation with certain Enriques surfaces which parallels the situation in characteristic zero, yet displays intriguing subtleties.

New slope inequalities for families of complete intersections

We prove f -positivity of \mathcal{O}_X(1) for arbitrary dimensional fibrations over curves f\colon X\to B whose general fibre is a complete intersection. In the special case where the family is a global complete intersection, we prove numerical sufficient and necessary conditions for f -positivity of powers of \mathcal{O}_X(1) and for the relative canonical sheaf. From these results we also derive a Chow instability condition for the fibres of relative complete intersections in the projective bundle of a \mu -unstable bundle.

Tensor triangulated category structures in the derived category of a variety with big (anti)canonical bundle

Tensor triangulated category structures in the derived category of a variety with big (anti)canonical bundle

Cup-length of oriented Grassmann manifolds via Gröbner bases

The aim of this paper is to prove a conjecture made by T. Fukaya in 2008. This conjecture concerns the exact value of the Z2-cup-length of the Grassmann manifold G˜n,3 of oriented 3-planes in Rn. Along the way, we calculate the heights of the Stiefel–Whitney classes of the canonical vector bundle over G˜n,3.

On algebraically coisotropic submanifolds of holomorphic symplectic manifolds

We investigate algebraically coisotropic submanifolds $X$ in a holomorphic symplectic projective manifold $M$. Motivated by our results in the hypersurface case, we raise the following question: when $X$ is not uniruled, is it true that up to a finite \'etale cover, the pair $(X,M)$ is a product $(Z\times Y, N\times Y)$ where $N, Y$ are holomorphic symplectic and $Z\subset N$ is Lagrangian? We prove that this is indeed the case when $M$ is an abelian variety, and give some partial answer when the canonical bundle $K_X$ is semi-ample. In particular, when $K_X$ is nef and big, $X$ is Lagrangian in $M$ (in fact this also holds without nefness assumption). We also remark that Lagrangian submanifolds do not exist on a sufficiently general Abelian variety, in contrast to the case when $M$ is irreducible hyperk\"ahler.

A note on varieties of weak CM-type

CM-type projective varieties X of complex dimension n are characterized by their CM-type rational Hodge structures on the cohomology groups. One may impose such a condition in a weakest form when the canonical bundle of X is trivial; the rational Hodge structure on the level-n subspace of Hn(X;Q) is required to be of CM-type. This brief note addresses the question whether this weak condition implies that the Hodge structure on the entire H⁎(X;Q) is of CM-type. We study in particular abelian varieties when the dimension of the level-n subspace is two or four, and K3×T2. It turns out that the answer is affirmative. Moreover, such an abelian variety is always isogenous to a product of CM-type elliptic curves or abelian surfaces. This extends a result of Shioda and Mitani in 1974.

The moduli space of stable supercurves and its canonical line bundle

Abstract: We prove that the moduli of stable supercurves with punctures is a smooth proper DM stack and study an analog of the Mumford's isomorphism for its canonical line bundle.

Existence of Ulrich bundle on some surfaces of general type

Let X be a smooth projective algebraic surface of Picard rank one with very ample canonical bundle KX. We further assume that q+1≤χ(OX). In this article, we will study the existence of an Ulrich bundle and its stability property of it with respect to KX.

On global ACC for foliated threefolds

In this paper, we prove the rational coefficient case of the global ACC for foliated threefolds. Specifically, we consider any lc foliated log Calabi-Yau triple ( X , F , B ) (X,\mathcal {F},B) of dimension 3 3 whose coefficients belong to a set Γ \Gamma of rational numbers satisfying the descending chain condition, and prove that the coefficients of B B belong to a finite set depending only on Γ \Gamma . To prove our main result, we introduce the concept of generalized foliated quadruples, which is a mixture of foliated triples and Birkar-Zhang’s generalized pairs. With this concept, we establish a canonical bundle formula for foliations in any dimension. As for applications, we extend Shokurov’s global index conjecture in the classical MMP to foliated triples and prove this conjecture for threefolds with nonzero boundaries and for surfaces. Additionally, we introduce the theory of rational polytopes for functional divisors on foliations and prove some miscellaneous results.

Six dimensional homogeneous spaces with holomorphically trivial canonical bundle

We classify all the 6-dimensional unimodular Lie algebras g admitting a complex structure with non-zero closed (3,0)-form. This gives rise to 6-dimensional compact homogeneous spaces M=Γ﹨G, where Γ is a lattice, admitting an invariant complex structure with holomorphically trivial canonical bundle. As an application, in the balanced Hermitian case, we study the instanton condition for any metric connection ∇ε,ρ in the plane generated by the Levi-Civita connection and the Gauduchon line of Hermitian connections. In the setting of the Hull-Strominger system with connection on the tangent bundle being Hermitian-Yang-Mills, we prove that if a compact non-Kähler homogeneous space M=Γ﹨G admits an invariant solution with respect to some non-flat connection ∇ in the family ∇ε,ρ, then M is a nilmanifold with underlying Lie algebra h3, a solvmanifold with underlying algebra g7, or a quotient of the semisimple group SL(2, C). Since it is known that the system can be solved on these spaces, our result implies that they are the unique compact non-Kähler balanced homogeneous spaces admitting such invariant solutions. As another application, on the compact solvmanifold underlying the Nakamura manifold, we construct solutions, on any given balanced Bott-Chern class, to the heterotic equations of motion taking the Chern connection as (flat) instanton.

Effective divisors on projectivized Hodge bundles and modular Forms

AbstractWe construct vector‐valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles over moduli of curves. Cycle relations tell us the weight of these modular forms. In particular, we construct basic modular forms for genus 2 and 3. We also discuss modular forms on the moduli of hyperelliptic curves. In that case, the relative canonical bundle is a pull back of a line bundle on a ‐bundle over the moduli of hyperelliptic curves and we extend that line bundle to a compactification so that its push down is (close to) the Hodge bundle and use this to construct modular forms. In the Appendix, we use our method to calculate divisor classes in the dual projectivized k‐Hodge bundle determined by Gheorghita–Tarasca and by Korotkin–Sauvaget–Zograf.

Polarisation of SKT Calabi-Yau ∂∂̄-manifolds by Aeppli classes

Given a ∂ ∂ ¯ \partial \bar \partial -manifold X X with trivial canonical bundle and carrying a metric ω \omega such that ∂ ∂ ¯ ω = 0 \partial \bar \partial \omega =0 , we introduce the concept of small deformations of X X polarised by the Aeppli cohomology class [ ω ] A [\omega ]_A of a strong Kähler with torsion metric ω \omega . There is a correspondence between the manifolds polarised by [ ω ] A [\omega ]_A in the Kuranishi family of X X and the Bott-Chern classes that are primitive in a sense that we define. We also investigate the existence of a primitive element in an arbitrary Bott-Chern primitive class and compare the metrics on the base space of the subfamily of manifolds polarised by [ ω ] A [\omega ]_A within the Kuranishi family.

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.