Complex Affine Space Research Articles

Bounded composition operators on functional quasi-Banach spaces and stability of dynamical systems

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of a composition operator strongly limits the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.

Positively hyperbolic varieties, tropicalization, and positroids

A variety of codimension c in complex affine space is positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimension c. Positively hyperbolic hypersurfaces are defined by stable polynomials. We characterize these varieties using sign variations, and show that they are equivalently defined by being hyperbolic with respect to the positive part of the Grassmannian, in the sense of Shamovich and Vinnikov. Positively hyperbolic projective varieties have tropicalizations that are locally subfans of the type A hyperplane arrangement defined by xi=xj, in which the maximal cones satisfy a non-crossing condition. This gives new proofs of results of Choe–Oxley–Sokal–Wagner and Brändén on Newton polytopes and tropicalizations of stable polynomials. We settle the question of which tropical varieties can be obtained as tropicalizations of positively hyperbolic varieties in the case of tropical toric varieties, constant-coefficient tropical curves, and Bergman fans. Along the way, we give a new characterization of positroids in terms of a non-crossing condition on their Bergman fans.

Affine hyperplane arrangements and Jordan classes

We study the geometry of the stratification induced by an affine hyperplane arrangement H on the quotient of a complex affine space by the action of a discrete group preserving H. We give conditions ensuring normality or normality in codimension 1 of strata. As an application, we provide the list of those categorical quotients of closures of Jordan classes and of sheets in all complex simple algebraic groups that are normal. In the simply connected case, we show that normality of such a quotient is equivalent to its smoothness.

Frameworks for Two-Dimensional Keller Maps

The classical Jacobian Conjecture asserts that every locally invertible polynomial self-map of the complex affine space is globally invertible. A Keller map is a (hypothetical) counterexample to the Jacobian Conjecture. In dimension two every such map, if exists, leads to a map between the Picard groups of suitable compactifications of the affine plane, that satisfy a complicated set of conditions. This is essentially a combinatorial problem. Several solutions to it ("frameworks") are described in detail. Each framework corresponds to a large system of equations, whose solution would lead to a Keller map.

On first order deformations of homogeneous foliations

We study analytic deformations of holomorphic foliations given by homogeneous integrable one-forms in the complex affine space Cn. The deformation is supposed to be of first order (order one in the parameter). We also assume that the deformation is given by homogeneous polynomial one-forms. The deformation takes place in the affine space since we are not assuming that the foliations descent to the projective space. We describe the space of such deformations in three main situations: (1) the foliation is given by the level hypersurfaces of a homogeneous polynomial. (2) the foliation is rational, i.e., has a first integral of type Pr/Qs for some homogeneous polynomials P,Q. (3) the foliation is logarithmic of a generic type. We prove that, for each class mentioned above, the first order homogeneous deformations of same degree are in the very same class. We also investigate the existence of such deformations with different degrees.

Logarithmic Cartan geometry on complex manifolds

We pursue the study of holomorphic Cartan geometry with singularities. We introduce the notion of logarithmic Cartan geometry on a complex manifold, with polar part supported on a normal crossing divisor. In particular, we show that the push-forward of a Cartan geometry constructed using a finite Galois ramified covering is a logarithmic Cartan geometry (the polar part is supported on the ramification locus). We also study the specific case of the logarithmic Cartan geometry with the model being the complex affine space.

Is the Affine Space Determined by Its Automorphism Group?

Abstract In this note we study the problem of characterizing the complex affine space ${\mathbb{A}}^n$ via its automorphism group. We prove the following. Let $X$ be an irreducible quasi-projective $n$-dimensional variety such that $\operatorname{Aut}(X)$ and $\operatorname{Aut}({\mathbb{A}}^n)$ are isomorphic as abstract groups. If $X$ is either quasi-affine and toric or $X$ is smooth with Euler characteristic $\chi (X) \neq 0$ and finite Picard group $\operatorname{Pic}(X)$, then $X$ is isomorphic to ${\mathbb{A}}^n$. The main ingredient is the following result. Let $X$ be a smooth irreducible quasi-projective variety of dimension $n$ with finite $\operatorname{Pic}(X)$. If $X$ admits a faithful $({\mathbb{Z}} / p {\mathbb{Z}})^n$-action for a prime $p$ and $\chi (X)$ is not divisible by $p$, then the identity component of the centralizer $\operatorname{Cent}_{\operatorname{Aut}(X)}( ({\mathbb{Z}} / p {\mathbb{Z}})^n)$ is a torus.

Christoffel transformations for multivariate orthogonal polynomials

Polynomial perturbations of real multivariate measures are discussed and corresponding Christoffel type formulas are found. The 1D Christoffel formula is extended to the multidimensional realm: multivariate orthogonal polynomials are expressed in terms of last quasi-determinants and sample matrices. The coefficients of these matrices are the original orthogonal polynomials evaluated at a set of nodes, which is supposed to be poised. A discussion for the existence of poised sets is given in terms of algebraic hypersurfaces in the complex affine space. Two examples of irreducible perturbations of total degree 1 and 2, for the bivariate product Legendre orthogonal polynomials, are discussed in detail.

Relative cohomology of bi-arrangements

A bi-arrangement of hyperplanes in a complex affine space is data of two sets of hyperplanes along with a coloring information on strata. To such a bi-arrangement, one naturally associates a relative cohomology group, that we call its motive. The motivation for studying such relative cohomology groups comes from notion of motivic period. More generally, we suggest systematic study of motive of a bi-arrangement of hypersurfaces in a complex manifold. We provide combinatorial and cohomological tools to compute structure of these motives. Our main object is Orlik-Solomon bi-complex of a bi-arrangement, which generalizes Orlik-Solomon algebra of an arrangement. Loosely speaking, our main result states that the motive of an exact bi-arrangement is computed by its Orlik-Solomon bi-complex, which generalizes classical facts involving Orlik-Solomon algebra of an arrangement. We show how this formalism allows us to explicitly compute motives arising from study of multiple zeta values and sketch a more general application to periods of mixed Tate motives.

Complex Curves as Lines of Geometries

We investigate Hjelmslev geometries \({\mathcal{H}}\) having a representation in a complex affine space \({\mathbb{C}^n}\) the lines of which are given by entire functions. If \({\mathcal{H}}\) has dimension 2 and the entire functions satisfy some injectivity conditions, then \({\mathcal{H}}\) is a substructure of the complex Laguerre plane. If the lines are geodesics with respect to a natural connection \({\nabla^{\circ}}\), then a detailed classification of them as well as of the corresponding geometries is obtained. Generalizations of complex Grunwald planes play a main role in the classification. Since in the considered geometries the set of lines is invariant under the translation group of \({\mathbb{C}^n}\), we classify all complex curves C in \({\mathbb{C}^n}\) given by entire functions as well as the connections \({\nabla^\circ}\) such that all images of C under the translation group of \({\mathbb C^n}\) consist of geodesics with respect to \({\nabla^{\circ}}\).

English

English

Oka properties of groups of holomorphic and algebraic automorphisms of complex affine space

We show that the group of all holomorphic automorphisms of complex affine space $\mathbb C^n$, $n>1$, and several of its subgroups satisfy the parametric Oka property with approximation and with interpolation on discrete sets.

STRATIFICATIONS ASSOCIATED TO REDUCTIVE GROUP ACTIONS ON AFFINE SPACES

For a complex reductive group G acting linearly on a complex affine space V with respect to a character ρ, we show two stratifications of V associated to this action (and a choice of invariant inner product on the Lie algebra of the maximal compact subgroup of G) coincide. The first is Hesselink’s stratification by adapted 1-parameter subgroups and the second is the Morse theoretic stratification associated to the norm square of the moment map. We also give a proof of a version of the Kempf–Ness theorem, which states that the geometric invariant theory quotient is homeomorphic to the symplectic reduction (both taken with respect to ρ). Finally, for the space of representations of a quiver of fixed dimension, we show that the Morse theoretic stratification and Hesselink’s stratification coincide with the stratification by Harder–Narasimhan types.

Average linking with complex subspaces

We generalize Banchoff–Pohlʼs isoperimetric inequality to complex affine space.

Milnor fibers over singular toric varieties and nearby cycle sheaves

We apply sheaf-theoretical methods to monodromy zeta functions of Milnor fibrations. Classical formulas due to Kushnirenko, Varchenko and Oka, etc. on polynomials over the complex affine space will be generalized to polynomial functions over any toric variety. Moreover our results enable us to calculate the monodromy zeta functions of any constructible sheaf.

Quiver D-modules and homology of local systems over an arrangement of hyperplanes

Let $C$ be an arrangement of affine hyperplanes in a complex affine space $X$, $D$ the ring of algebraic differential operators on $X$. We define a category of quivers associated with $C$. A quiver is a collection of vector spaces, attached to strata of the arrangement, and suitable linear maps between the vector spaces . To a quiver we assign a $D$-module on $X$, called the quiver $D$-module. We describe basic operations for $D$-modules in terms of linear algebra of quivers. We give an explicit construction of a free resolution of a quive $D$-module and use the construction to describe the associated perverse sheaf. As an application, we calculate the cohomology of $X$ with coefficients in the quiver perverse sheaf (under certain assumptions).

The full orbifold K-theory of abelian symplectic quotients

AbstractIn their 2007 paper, Jarvis, Kaufmann, and Kimura defined the full orbifoldK-theory of an orbifold , analogous to the Chen-Ruan orbifold cohomology of in that it uses the obstruction bundle as a quantum correction to the multiplicative structure. We give an explicit algorithm for the computation of this orbifold invariant in the case when arises as an abelian symplectic quotient. To this end, we introduce the inertial K-theory associated to a T -action on a stably complex manifold M, where T is a compact abelian Lie group. Our methods are integral K-theoretic analogues of those used in the orbifold cohomology case by Goldin, Holm, and Knutson in 2005. We rely on the K-theoretic Kirwan surjectivity methods developed by Harada and Landweber. As a worked class of examples, we compute the full orbifold K-theory of weighted projective spaces that occur as a symplectic quotient of a complex affine space by a circle. Our computations hold over the integers, and in the particular case of these weighted projective spaces, we show that the associated invariant is torsion-free.

A NON-EXISTENCE THEOREM FOR MORSE TYPE HOLOMORPHIC FOLIATIONS OF CODIMENSION ONE TRANSVERSE TO SPHERES

We prove that a Morse type codimension one holomorphic foliation is not transverse to a sphere in the complex affine space. Also we characterize the variety of contacts of a linear foliation with concentric spheres.

Siciak–Zahariuta extremal functions, analytic discs and polynomial hulls

We prove two disc formulas for the Siciak–Zahariuta extremal function of an arbitrary open subset of complex affine space. We use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in terms of analytic discs. Similar results in previous work of ours required the subsets to be connected.

Geometry of Puiseux expansions

We consider the space $\mathrm{Curv}$ of complex affine lines $% t\mapsto(x,y)=(\phi(t),\psi(t))$ with monic polynomials $\phi$, $\psi$ of fixed degrees and a map $\mathrm{Expan}$ from $\mathrm{Curv}$ to a complex affine space $\mathrm{Puis}$ with $\dim\m