Complex Torus Research Articles

HOLOMORPHIC MAPS ONTO KÄHLER MANIFOLDS WITH NON-NEGATIVE KODAIRA DIMENSION

This paper studies the deformation theory of a holomorphic surjective map from a normal compact complex space X to a compact <TEX>$K\"{a}hler$</TEX> manifold Y. We will show that when the target has non-negative Kodaira dimension, all deformations of surjective holomorphic maps <TEX>$X{\rightarrow}Y$</TEX> come from automorphisms of an unramified covering of Y and the underlying reduced varieties of associated components of Hol(X, Y) are complex tori. Under the additional assumption that Y is projective algebraic, this was proved in [7]. The proof in [7] uses the algebraicity in an essential way and cannot be generalized directly to the <TEX>$K\"{a}hler$</TEX> setting. A new ingredient here is a careful study of the infinitesimal deformation of orbits of an action of a complex torus. This study, combined with the result for the algebraic case, gives the proof for the <TEX>$K\"{a}hler$</TEX> setting.

Cohomology of the fundamental groups of toroidal groups

We construct an isomorphism between the $\overline{\partial}$-cohomology and the cohomology of the fundamental groups of toroidal groups, and get a standard form of $p$-cocycles, which was given by Vogt [6] in case of $1$-cocycles. Using differential forms via the above isomorphism enables us to obtain new results in higher dimensional cases. An explicit isomorphism between the Cech cohomology and the cohomology of the fundamental groups of complex tori is given in [5] (p.14). Our results give a generalization of this isomorphism to toroidal groups.

Connections and Higgs fields on a principal bundle

Let M be a compact connected Kahler manifold and G a connected linear algebraic group defined over \({\mathbb{C}}\) . A Higgs field on a holomorphic principal G-bundle eG over M is a holomorphic section θ of \(\text{ad}(\epsilon_{G})\otimes {\Omega}^{1}_{M}\) such that θ∧ θ = 0. Let L(G) be the Levi quotient of G and (eG(L(G)), θl) the Higgs L(G)-bundle associated with (eG, θ). The Higgs bundle (eG, θ) will be called semistable (respectively, stable) if (eG(L(G)), θl) is semistable (respectively, stable). A semistable Higgs G-bundle (eG, θ) will be called pseudostable if the adjoint vector bundle ad(eG(L(G))) admits a filtration by subbundles, compatible with θ, such that the associated graded object is a polystable Higgs vector bundle. We construct an equivalence of categories between the category of flat G-bundles over M and the category of pseudostable Higgs G-bundles over M with vanishing characteristic classes of degree one and degree two. This equivalence is actually constructed in the more general equivariant set-up where a finite group acts on the Kahler manifold. As an application, we give various equivalent conditions for a holomorphic G-bundle over a complex torus to admit a flat holomorphic connection.

Categories of holomorphic line bundles on higher dimensional noncommutative complex tori

We construct explicitly noncommutative deformations of categories of holomorphic line bundles over higher dimensional tori. Our basic tools are Heisenberg modules over noncommutative tori and complex∕holomorphic structures on them introduced by Schwarz [“Theta functions on noncommutative tori,” Lett. Math. Phys. 58, 81–90 (2001)]. We obtain differential graded (DG) categories as full subcategories of curved DG categories of Heisenberg modules over the complex noncommutative tori. Also, we present the explicit composition formula of morphisms, which, in fact, depends on the noncommutativity.

Holomorphic connections on some complex manifolds

Let M be a compact connected complex manifold equipped with a holomorphic submersion to a complex torus such that the fibers are all rationally connected. Then any holomorphic vector bundle over M admitting a holomorphic connection actually admits a flat holomorphic connection. A similar statement is valid for any finite quotient of M. To cite this article: I. Biswas, J.N. Iyer, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

On the connectedness of a centralizer

This letter finds the connected components of certain subgroups of a complex torus that arise in completing the Hamiltonian flows of the full complex Toda lattice.

Non-commutative tori and Fourier–Mukai duality

The classical Fourier-Mukai duality establishes an equivalence of categories between the derived categories of sheaves on dual complex tori. In this article we show that this equivalence extends to an equivalence between two dual objects. Both of these are generalized deformations of the complex tori. In one case, a complex torus is deformed formally in a non-commutative direction specified by a holomorphic Poisson structure. In the other, the dual complex torus is deformed in a B-field direction to a formal gerbe. We show these two deformations are Fourier-Mukai equivalent.

On abelian principal bundles

Let T be a complex torus and E T a holomorphic principal T-bundle over a connected complex manifold M. We prove that the total space of E T admits a Kahler structure if and only if M admits a Kahler structure and E T admits a flat holomorphic connection whose monodromy preserves a Kahler form on T. If E T admits a Kahler structure, then $$ H* (E_T ,\user2{\mathbb{C}}) $$ is isomorphic to $$ H* (M,\user2{\mathbb{C})} \otimes _{\user2{\mathbb{C}}} H* (T,\user2{\mathbb{C}}) $$ .

On splitting of certain Jacobian varieties

We give three examples of non-hyperelliptic curves of genus 4 whose Jacobian varieties are isomorphic to products of four elliptic curves. Two of the examples belong to one-parameter families of curves whose Jacobian varieties are isomorphic to products of two 2-dimensional complex tori. By constructing analogous families, we prove that for each $n>1$, there is a one-parameter family of non-hyperelliptic curves of genus $2n$ whose Jacobian varieties are isomorphic to products of two $n$-dimensional tori.

On Intersection Indices of Subvarieties in Reductive Groups

In this paper, I give an explicit formula for the intersection indices of the Chern classes (defined in [11]) of an arbitrary reductive group with hypersurfaces. This formula has the following applications. First, it allows to compute explicitly the Euler characteristic of complete intersections in reductive groups thus extending the beautiful result by D.Bernstein and Khovanskii, which holds for a complex torus. Second, for any regular compactification of a reductive group, it computes the intersection indices of the Chern classes of the compactification with hypersurfaces. The formula is similar to the Brion– Kazarnovskii formula for the intersection indices of hypersurfaces in reductive groups. The proof uses an algorithm of De Concini and Procesi for computing such intersection indices. In particular, it is shown that this algorithm produces the Brion–Kazarnovskii formula.

A structure theorem of compact complex parallelizable pseudo-Kähler solvmanifolds

In this paper, we prove that the Mostow fibration of a compact complex parallelizable pseudo-Kahler solvmanifold is a complex torus bundle over a complex torus.

Mirror Symmetry in Two Steps: A–I–B

We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A-model). The second theory is an intermediate model, which we call the I-model. The equivalence between the A-model and the I-model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T-duality. On the other hand, the I-model is closely related to the twisted Landau-Ginzburg model (the B-model) that is mirror dual to the A-model. Thus, the mirror symmetry is realized in two steps, via the I-model. In particular, we obtain a natural interpretation of the superpotential of the Landau-Ginzburg model as the sum of terms corresponding to the components of a divisor in the toric variety. We also relate the cohomology of the supercharges of the I-model to the chiral de Rham complex and the quantum cohomology of the underlying toric variety.

A TOPOLOGICAL MIRROR SYMMETRY ON NONCOMMUTATIVE COMPLEX TWO-TORI

In this paper, we study a topological mirror symmetry on noncommutative complex tori. We show that deformation quantization of an elliptic curve is mirror symmetric to an irrational rotation algebra. From this, we conclude that a mirror reflection of a noncommutative complex torus is an elliptic curve equipped with a Kronecker foliation.

On the Levi-flats in Complex Tori of Dimension Two

Compact Levi flat real analytic hypersurfaces in complex tori of dimension two are completely classified. The method is a combination of Siu’s \bar ∂ -regularity theory on weakly pseudoconvex domains, an updated variant of the L^2 estimate for the \bar ∂ operator originated from Berndtsson–Charpentier, and basic arguments in complex analytic geometry.

Existence of Torus bundles associated to cocycles

A Kuga fibre variety is a fibre bundle over a locally symmetric space whose fibre is a polarized Abelian variety. We describe a complex torus bundle associated to a 2-cocycle of a discrete group, which may be regarded as a generalized Kuga fibre variety, and prove the existence of such a bundle.

Finite linear groups, lattices, and products of elliptic curves

Let V be a finite-dimensional complex linear space and G an irreducible finite subgroup of GL ( V ) . For a G-invariant lattice Λ in V of maximal rank, we describe the structure of complex torus V / Λ .

Dirac operator zero-modes on a torus

We study Dirac operator zero-modes on a torus for gauge background with uniform field strengths. Under the basic translations of the torus coordinates the wave functions are subject to twisted periodic conditions. In suitable torus coordinates the zero-mode wave functions can be related to holomorphic functions of the complex torus coordinates. Half of the twisted boundary conditions for the holomorphic part of the zero-mode wave function can be made periodic or anti-periodic. The remaining half is until coordinate dependent but diagonal. We completely solve the twisted boundary conditions and construct the zero-mode wave functions. The chirality and the degeneracy of the zero-modes are uniquely determined by the gauge background and are consistent with the index theorem.

A note on compact solvmanifolds with Kahler structures

In this note we show that a compact solvmanifold admits a Kahler structure if and only if it is a finite quotient of a complex torus which has a structure of a complex torus bundle over a complex torus. We can show in particular that a compact solvmanifold of completely solvable type has a Kahler structure if and only if it is a complex torus, which is known as the Benson-Gordon's conjecture.

Star Product Formula of Theta Functions

As a noncommutative generalization of the addition formula of theta functions, we construct a class of theta functions which are closed with respect to the Moyal star product of a fixed noncommutative parameter. These theta functions can be regarded as bases of the space of holomorphic homomorphisms between holomorphic line bundles over noncommutative complex tori.

Kähler manifolds with numerically effective Ricci class and maximal first Betti number are tori

Let M be a n-dimensional Kähler manifold with numerically effective Ricci class c 1 ( M ) . In this Note we prove that, if the first Betti number b 1 ( M ) is equal to 2 n, then M is biholomorphic to a n-dimensional complex torus. To cite this article: F. Fang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).