Complex Torus Research Articles

On a B-field transform of generalized complex structures over complex tori

Let (Xn,Xˇn) be a mirror pair of an n-dimensional complex torus Xn and its mirror partner Xˇn. Then, by SYZ transform, we can construct a holomorphic line bundle with an integrable connection from each pair of a Lagrangian section of Xˇn→Rn/Zn and a unitary local system along it, and those holomorphic line bundles with integrable connections form a dg-category DGXn. In this paper, we focus on a certain B-field transform of the generalized complex structure induced from the complex structure on Xn, and interpret it as the deformation XGn of Xn by a flat gerbe G. Moreover, we construct the deformation of DGXn associated to the deformation from Xn to XGn, and also discuss the homological mirror symmetry between XGn and its mirror partner on the object level.

An invariant for affine maximal type equations

Let y:M→Rn+1 be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space Rn+1, given as the graph of a smooth, strictly convex function xn+1=f(x1,...,xn) defined on a domain Ω⊂Rn. Considering the α-relative normalization of the graph of the convex function f, we will prove a Bernstein theorem for a class of nonlinear, fourth order partial differential equations of affine maximal type. As applications, we define an invariant of the equations and prove a rigidity result of the complete Tn-invariant Kähler metric on complex torus (C⁎)n with vanishing scalar curvature for n≤5.

On neighborhoods of embedded complex tori

On neighborhoods of embedded complex tori

Representations of complex tori and GL(2, C)

Groups and their representations have been studied for a long time. One can extend the notion of a group by asking the group axioms to hold in other categories. A group in the category of smooth manifolds is a Lie group, and a group in the category of algebraic varieties is an algebraic group. In this paper, we discuss the representation theory of algebraic groups, in particular complex tori and GL(2,C).

A Classification of Compact Cohomogeneity One Locally Conformal Kähler Manifolds

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification GC of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case.

A classification of automorphic Lie algebras on complex tori

Abstract We classify the automorphic Lie algebras of equivariant maps from a complex torus to $\mathfrak{sl}_2(\mathbb{C})$ . For each case, we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of $\mathrm{PSL}_2({\mathbb{Z}})$ , apart from four cases, which are all isomorphic to Onsager’s algebra.

Almost complex torus manifolds - a problem of Petrie type

The Petrie conjecture asserts that if a homotopy C P n \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of C P n \mathbb {CP}^n . Petrie proved this conjecture in the case where the manifold admits a T n T^n -action. An almost complex torus manifold is a 2 n 2n -dimensional compact connected almost complex manifold equipped with an effective T n T^n -action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2 n 2n -dimensional almost complex torus manifold M M only shares the Euler number with the complex projective space C P n \mathbb {CP}^n , the graph of M M agrees with the graph of a linear T n T^n -action on C P n \mathbb {CP}^n . Consequently, M M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch χ y \chi _y -genus, Todd genus, and signature as C P n \mathbb {CP}^n , endowed with the standard linear action. Furthermore, if M M is equivariantly formal, the equivariant cohomology and the Chern classes of M M and C P n \mathbb {CP}^n also agree.

A gerby deformation of complex tori and the homological mirror symmetry

A gerby deformation of complex tori and the homological mirror symmetry

Quasi-projectivity of images of mixed period maps

Abstract We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasi-projective, with a natural ample bundle. Specifically, we consider the map from the image of the mixed period map to the image of the period map of the associated graded. On the one hand, we show in a precise manner that the parts of this map parametrizing extension data of non-adjacent-weight pure Hodge structures are quasi-affine. On the other hand, extensions of adjacent-weight pure polarized Hodge structures are parametrized by a compact complex torus (the intermediate Jacobian) equipped with a natural theta bundle which is ample in Griffiths transverse directions. Our proof makes heavy use of o-minimality, and recent work with B. Klingler associating an ℝ an , exp {\mathbb{R}_{\mathrm{an},\exp}} -definable structure to mixed period domains and admissible mixed period maps.

Gaussian Gabor frames, Seshadri constants and generalized Buser–Sarnak invariants

We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander’s {overline{partial }}-L^2 estimate with singular weight, Demailly’s Calabi–Yau method for Kähler currents and a Kähler-variant generalization of the symplectic embedding theorem of McDuff–Polterovich for ellipsoids. Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice. In particular, we obtain an effective Gaussian Gabor frame criterion in terms of the covolume for almost all lattices, which is the first general covolume criterion in multivariate Gaussian Gabor frame theory. The recent Berndtsson–Lempert method and the Ohsawa–Takegoshi extension theorem also allow us to give explicit estimates for the frame bounds in terms of certain Robin constant. In the one-dimensional case we obtain a sharp estimate of the Robin constant using Faltings’ theta metric formula for the Arakelov Green functions.

Differential geometric global smoothings of simple normal crossing complex surfaces with trivial canonical bundle

AbstractLetXXbe a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that ifXXisdd-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus ofXX, and the first author’s existence result of holomorphic volume forms on global smoothings ofXX. In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples ofdd-semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces, andK3K3surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable toK3K3surfaces.

Intersection numbers of twisted homology and cohomology groups associated to the Riemann–Wirtinger integral

The Riemann–Wirtinger integral is an analogue of the hypergeometric integral, which is defined on a one-dimensional complex torus. We study the intersection forms on the twisted homology and cohomology groups associated with the Riemann–Wirtinger integral. We derive explicit formulas of some intersection numbers, and apply them to studies of the monodromy representation, connection problems, and contiguity relations.

On the Curvature of the Bismut Connection: Bismut–Yamabe Problem and Calabi–Yau with Torsion Metrics

We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a conformal Hermitian structure on a compact complex manifold, there exists a metric with constant Bismut scalar curvature in that class when the expected constant scalar curvature is non-negative. A similar result is given in the general case of Gauduchon connections. We then study an Einstein-type condition for the Bismut Ricci curvature tensor on principal bundles over Hermitian manifolds with complex tori as fibers. Thanks to this analysis, we construct explicit examples of Calabi–Yau with torsion Hermitian structures and prove a uniqueness result for them.

Dynamical Tropicalisation

We analyse the dynamics of the pullback of the map z longmapsto z^m on the complex tori and toric varieties. We will observe that tropical objects naturally appear in the limit, and review several theorems in tropical geometry.

Automorphism groups of $\mathbb{P}^1$-bundles over a non-uniruled base

In this survey we discuss holomorphic $\mathbb{P}^1$-bundles $p\colon X \to Y$ over a non-uniruled complex compact Kähler manifold $Y$, paying a special attention to the case when $Y$ is a complex torus. We consider the groups $\operatorname{Aut}(X)$ and $\operatorname{Bim}(X)$ of its biholomorphic and bimeromorphic automorphisms, respectively, and discuss when these groups are bounded, Jordan, strongly Jordan, or very Jordan. Bibliography: 88 titles.

Automorphism groups of $\mathbb{P}^1$-bundles over a non-uniruled base

In this survey we discuss holomorphic $\mathbb{P}^1$-bundles $p\colon X \to Y$ over a non-uniruled complex compact Kähler manifold $Y$, paying a special attention to the case when $Y$ is a complex torus. We consider the groups $\operatorname{Aut}(X)$ and $\operatorname{Bim}(X)$ of its biholomorphic and bimeromorphic automorphisms, respectively, and discuss when these groups are bounded, Jordan, strongly Jordan, or very Jordan. Bibliography: 88 titles.

Numerical characterization of complex torus quotients

This article gives a characterization of quotients of complex tori by finite groups acting freely in codimension two in terms of a numerical vanishing condition on the first and second Chern class. This generalizes results previously obtained by Greb–Kebekus–Peternell in the projective setting, and by Kirschner and the second author in dimension three. As a key ingredient to the proof, we obtain a version of the Bogomolov–Gieseker inequality for stable sheaves on singular spaces, including a discussion of the case of equality.

Alexander modules, Mellin transformation and variations of mixed Hodge structures

To any complex algebraic variety endowed with a morphism to a complex affine torus we associate multivariable cohomological Alexander modules, and define natural mixed Hodge structures on their maximal Artinian submodules. The key ingredients of our construction are Gabber-Loeser's Mellin transformation and Hain-Zucker's work on unipotent variations of mixed Hodge structures. As applications, we prove the quasi-unipotence of monodromy, we obtain upper bounds on the sizes of the Jordan blocks of monodromy, and we explore the change in the Alexander modules after removing fibers of the map. We also give an example of a variety whose Alexander module has non-semisimple torsion.

The bijectivity of mirror functors on tori

By the SYZ construction, a mirror pair (X,Xˇ) of a complex torus X and a mirror partner Xˇ of the complex torus X is described as the special Lagrangian torus fibrations X→B and Xˇ→B on the same base space B. Then, by the SYZ transform, we can construct a simple projectively flat bundle on X from each affine Lagrangian multisection of Xˇ→B with a unitary local system along it. However, there are ambiguities of the choices of transition functions of it, and this causes difficulties when we try to construct a functor between the symplectic geometric category and the complex geometric category. In this paper, we prove that there exists a bijection between the set of the isomorphism classes of their objects by solving this problem.

Almost Complex Torus Manifolds—Graphs and Hirzebruch Genera

Abstract Let a $k$-dimensional torus $T^k$ act on a $2n$-dimensional compact connected almost complex manifold $M$ with isolated fixed points. As in the case of circle actions, we show that there exists a (directed labeled) multigraph that contains information on weights at the fixed points and isotropy submanifolds of $M$. This includes the notion of a GKM (Goresky-Kottwitz-MacPherson) graph as a special case that weights at each fixed point are pairwise linearly independent. If in addition $k=n$, that is, $M$ is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. We show that the Hirzebruch $\chi _y$-genus $\chi _y(M)=\sum _{i=0}^n a_i(M) \cdot (-y)^i$ of an almost complex torus manifold $M$ satisfies $a_i(M)> 0$ for $0 \leq i \leq n$. In particular, the Todd genus of $M$ is positive and there are at least $n+1$ fixed points.