Complex Torus Research Articles

Dynamics of automorphisms on projective complex manifolds

We show that the dynamics of automorphisms on all projective complex manifolds X (of dimension 3, or of any dimension but assuming the Good Minimal Model Program or Mori's Program) are canonically built up from the dynamics on just three types of projective complex manifolds: complex tori, weak Calabi-Yau manifolds and rationally connected manifolds. As a by-product, we confirm the conjecture of Guedj for automorphisms on 3-dimensional projective manifolds, and also determine pi_1(X).

Killing fields of holomorphic Cartan geometries

We study local automorphisms of holomorphic Cartan geometries. This leads to classification results for compact complex manifolds admitting holomorphic Cartan geometries. We prove that a compact Kähler Calabi–Yau manifold bearing a holomorphic Cartan geometry of algebraic type admits a finite unramified cover which is a complex torus.

Gluing construction of compact complex surfaces with trivial canonical bundle

We obtain a new construction of compact complex surfaces with trivial canonical bundle. In our construction we glue together two compact complex surfaces with an anticanonical divisor under suitable conditions. Then we show that the resulting compact manifold admits a complex structure with trivial canonical bundle by solving an elliptic partial differential equation. We generalize this result to cases where we have other than two components to glue together. With this generalization, we construct examples of complex tori, Kodaira surfaces and K3 surfaces. Lastly we deal with the smoothing problem of a normal crossing complex surface X with at most double curves. We prove that we still have a family of smoothings of X in a weak sense even when X is not Kahlerian or H 1 ( X , O X ) ≠ 0 , in which cases the smoothability result of Friedman [Fr] is not applicable.

On the classification of complex tori arising from real Abelian surfaces

Let A′ be an Abelian surface over ℝ and denote by A its complexification. We define an intrinsic volume vol(A) of A and show that there are seven possibilities with respect to the rank of End(A) and if vol(A) is rational or not. We prove that each possibility determines the Picard number and the endomorphism algebra of A′ and A respectively.

Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers

The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasi-projective case results of Green–Lazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers h q , 0 of congruence covers in any dimension, generalizing results of E. Hironaka and Sakuma. We extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems. In particular, we obtain a generalization of the polynomial periodicity of Betti numbers of unbranched congruence covers due to Sarnak–Adams. We derive a geometric characterization of finite abelian covers, which recovers the classic one and the one of Pardini. We use this, for example, to prove a conjecture of Libgober about Hodge numbers of abelian covers.

Positive toric fibrations

A principal toric bundle M is a complex manifold equipped with a free holomorphic action of a compact complex torus T. Such a manifold is fibred over M/T, with fibre T. We discuss the notion of positivity in fibre bundles and define positive toric bundles. Given an irreducible complex subvariety X ⊂ M of a positive principal toric bundle, we show that either X is T-invariant, or it lies in an orbit of a T-action. For principal elliptic bundles, this theorem is known from Verbitsky [Math. Res. Lett. 12 (2005) 251–264]. As follows from the Borel–Remmert–Tits theorem, any simply connected compact homogeneous complex manifold is a principal toric bundle. We show that compact Lie groups with left-invariant complex structure I are positive toric bundles, if I is generic. Other examples of positive toric bundles are discussed.

Connecting certain rigid birational non-homeomorphic Calabi-Yau threefolds via Hilbert scheme

A Calabi–Yau manifold is a compact Kahler simply connected manifold with a nowhere vanishing global n-form but no global i-form with 0 < i < n = dimX. By Kodaira’s criterion, it is projective if the dimension n ≥ 3. As well known, Calabi–Yau manifolds, hyperkahler manifolds and complex tori form the building blocks of compact Kahler manifolds with vanishing first Chern class [2, 6]. A famous theorem of Huybrechts states that two bimeromorphic hyperkahler manifolds are equivalent under smooth deformation [8,9]. In particular, they are homeomorphic to each other, having the same Betti numbers and Hodge numbers. Clearly, the same holds true for complex tori. Another theorem, originally due to Batyrev and Kontsevich, says that two birational Calabi–Yau manifolds have the same Betti numbers and Hodge numbers [1, 4, 10, 19, 21]. However, there are rigid birational non-isomorphic Calabi–Yau manifolds (cf. Theorem 0.1). Obviously, they are not equivalent under any smooth deformation. The aim of this paper is to remark that there nevertheless exist birational Calabi–Yau threefolds which are rigid, non-homeomorphic, but are connected by (necessarily non-smooth) projective flat deformation:

Conjecture of Tits type for complex varieties and Theorem of Lie-Kolchin type for a cone

First, we formulate and prove Theorem of Lie-Kolchin type for a cone and derive some algebro-geometric consequences. Next, inspired by a recent result of Dinh and Sibony we pose a conjecture of Tits type for a group of automorphisms of a complex variety and verify its weaker version. Finally, applying Theorem of Lie-Kolchin type for a cone, we confirm the conjecture of Tits type for complex tori, hyperk\ahler manifolds, surfaces, and minimal threefolds.

Torus configuration and materials selection on a fusion DEMO reactor, SlimCS

SlimCS is the conceptual design of a compact fusion DEMO plant assuming technologies foreseeable in 2020–2030s. For continuity of blanket technology from the Japanese ITER-TBM, the prime option of blanket is water-cooled solid breeder with Li 2TiO 3 (or Li 4SiO 4) and Be. A reduced-activation ferritic–martensitic steel (RAFM) and subcritical water are chosen as the structural material and coolant, respectively. The reactor has somewhat complex torus configuration with a sector-wide conducting plate slipped in between the replaceable (front) and permanent (back) blanket. In order to allow flexibility of maintenance in such a complex configuration, sector transport hot cell maintenance scheme is adopted. This paper describes characteristics of SlimCS with a focus on materials selection.

Mirror duality and noncommutative tori

In this paper, we study a mirror duality on a generalized complex torus and a noncommutative complex torus. First, we derive a symplectic version of Riemann conditions using mirror duality on ordinary complex tori. Based on this, we will find a mirror correspondence on generalized complex tori and generalize the mirror duality on complex tori to the case of noncommutative complex tori.

Classification des tissus exceptionnels quasilinéaires complètement décomposables

We present two results on the classification of exceptional CDQL webs on P 2 and on 2-dimensional complex tori. These are the exceptional k-webs formed by k global foliations, k − 1 of which are linear. To cite this article: J.V. Pereira, L. Pirio, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

HOLOMORPHIC PRINCIPAL BUNDLES WITH AN ELLIPTIC CURVE AS THE STRUCTURE GROUP

Let Λ ⊂ ℂ be the ℤ-module generated by 1 and [Formula: see text], where τ is a positive real number. Let Z := ℂ/Λ be the corresponding complex torus of dimension one. Our aim here is to give a general construction of holomorphic principal Z-bundles over a complex manifold X. Let θ1 and θ2 be two C∞ real closed two-forms on X such that the Hodge type (0, 2) component of the form [Formula: see text] vanishes, and the elements in H2(X, ℂ) represented by θ1 and θ2 are contained in the image of H2(X, ℤ). For such a pair we construct a holomorphic principal Z-bundle over X. Conversely, given any holomorphic principal Z-bundle EZ over X, we construct a pair of closed differential forms on X of the above type.

On the unramified spectrum of spherical varieties over p-adic fields

AbstractThe description of irreducible representations of a group G can be seen as a problem in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G×G by left and right multiplication. For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the ‘Langlands dual’ group. We generalize this description to an arbitrary spherical variety X of G as follows. Irreducible unramified quotients of the space $C_c^\infty (X)$ are in natural ‘almost bijection’ with a number of copies of AX*/WX, the quotient of a complex torus by the ‘little Weyl group’ of X. This leads to a description of the Hecke module of unramified vectors (a weak analog of geometric results of Gaitsgory and Nadler), and an understanding of the phenomenon that representations ‘distinguished’ by certain subgroups are functorial lifts. In the course of the proof, rationality properties of spherical varieties are examined and a new interpretation is given for the action, defined by Knop, of the Weyl group on the set of Borel orbits.

Rational points in periodic analytic sets and the Manin–Mumford conjecture

We present a new proof of the Manin–Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri–Pila–Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves contained in an analytic variety supposed invariant for translations by a full lattice, which is a topic with some independent motivation.

On a desingularization of the moduli space of noncommutative tori

It is shown that the moduli space of the noncommutative tori A θ {\mathbb A}_{\theta } admits a natural desingularization by the group E x t ( A θ , A θ ) Ext~ ({\mathbb A}_{\theta },{\mathbb A}_{\theta }) . Namely, we prove that the moduli space of pairs ( A θ , E x t ( A θ , A θ ) ) ({\mathbb A}_{\theta }, Ext~ ({\mathbb A}_{\theta },{\mathbb A}_{\theta })) is homeomorphic to a punctured two-dimensional sphere. The proof is based on a correspondence (a covariant functor) between the complex and noncommutative tori.

Compact Manifolds Covered by a Torus

Let X be a compact complex manifold which is the image of a complex torus by a holomorphic surjective map A→X. We prove that X is Kähler and that up to a finite étale cover, X is a product of projective spaces by a torus.

Holomorphic vector bundles on non-algebraic tori of dimension 2

We describe the Chern classes of holomorphic vector bundles on non-algebraic complex torus of dimension 2.

P-ADIC PERIODS OF MODULAR ELLIPTIC CURVES AND THE LEVEL-LOWERING THEOREM

An elliptic curve defined over the field of rational numbers can be considered as a complex torus. We can describe its complex periods in terms of integration of the weight-2 cusp form corresponding to the elliptic curve. In this paper, we will study an analogous description of the p-adic periods of the elliptic curve, considering the elliptic curve as a p-adic torus. An essential tool for the proof of such a description is the level-lowering theorem of Ribet, which is one of the main ingredients used in the proof of Fermat's Last Theorem.

A generalization of Matsushima’s embedding theorem

Let $M$ be a compact complex manifold and let $L \rightarrow M$ be a homorphic line bundle whose curvature form is everywhere of signature $(s_+, s_-)$. Under some conditions on the curvature form of $L$, it is show that $K \otimes L^{\otimes m}$ admits, for some $K$ and sufficiently large $m \in \mathbb{N}$, $C^{\infty}$ sections $t_0, \ldots, t_N$ such that the ratio $(t_0 : \cdots : t_N)$ embeds $M$ holomorphically in $s_+$ variables and antiholomorphically in $s_-$ variables. The result extends the Kodaira's embedding theorem as well as aresult of Matsushima for complex tori.

Symmetries of the hypergeometric function \phantom{}_{𝑚}𝐹_{𝑚-1}

In this paper, we show that the generalized hypergeometric function m F m − 1 \phantom {}_mF_{m-1} has a one parameter group of local symmetries, which is a conjugation of a flow of a rational Calogero-Mozer system. We use the symmetry to construct fermionic fields on a complex torus, which have linear-algebraic properties similar to those of the local solutions of the generalized hypergeometric equation. The fields admit a nontrivial action of the quaternions based on the above symmetry. We use the similarity between the linear-algebraic structures to introduce the quaternionic action on the direct sum of the space of solutions of the generalized hypergeometric equation and its dual. As a side product, we construct a “good” basis for the monodromy operators of the generalized hypergeometric equation inspired by the study of multiple flag varieties with finitely many orbits of the diagonal action of the general linear group by Magyar, Weyman, and Zelevinsky. As an example of computational effectiveness of the basis, we give a proof of the existence of the monodromy invariant hermitian form on the space of solutions of the generalized hypergeometric equation (in the case of real local exponents) different from the proofs of Beukers and Heckman and of Haraoka. As another side product, we prove an elliptic generalization of Cauchy identity.