Algebraic Subvarieties Research Articles

Transience of algebraic varieties in linear groups - applications to generic Zariski density

We study the transience of algebraic varieties in linear groups. In particular, we show that a “non elementary” random walk in SL 2 (ℝ) escapes exponentially fast from every proper algebraic subvariety. We also treat the case where the random walk takes place in the real points of a semisimple split algebraic group and show such a result for a wide family of random walks.

Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces

We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the $d$-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for the Abel-Jacobi map, which has played an important role in the recent study of D-branes (by Morrison and Walcher). Using the variation formalism, we prove that the relative periods of toric B-branes on Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature (by Alim, Hecht, Mayr, and Mertens), and discuss the relations between several works in the recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.

Solving algebraic equations in roots of unity

This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called maximal torsion cosets. We obtain new explicit upper bounds for the number of maximal torsion cosets on an algebraic subvariety of the complex algebraic -torus . In contrast to earlier work that gives the bounds of polynomial growth in the maximum total degree of defining polynomials, the proofs of our results are constructive. This allows us to obtain a new algorithm for determining maximal torsion cosets on an algebraic subvariety of .

Residual $${\bar\partial}$$ -cohomology and the complex Radon transform on subvarieties of $${\mathbb C P^n}$$

We show that the complex Radon transform realizes an isomorphism between the quotient-space of residual \({\bar\partial}\) -cohomologies of a locally complete intersection algebraic subvariety in a linearly concave domain of \({{{\mathbb C}}P^n}\) and the space of holomorphic solutions of the associated homogeneous system of differential equations with constant coefficients in the dual domain in \({({{\mathbb C}}P^n)^*}\) .

Nilpotent algebras and affinely homogeneous surfaces

To every nilpotent commutative algebra \({\mathcal{N}}\) of finite dimension over an arbitrary base field of characteristic zero a smooth algebraic subvariety \({S\subset\mathcal{N}}\) can be associated in a canonical way whose degree is the nil-index and whose codimension is the dimension of the annihilator \({\mathcal{A}}\) of \({\mathcal{N}}\). In case \({\mathcal{N}}\) admits a grading, the surface S is affinely homogeneous. More can be said if \({\mathcal{A}}\) has dimension 1, that is, if \({\mathcal{N}}\) is the maximal ideal of a Gorenstein algebra. In this case two such algebras \({\mathcal{N}}\), \({\tilde{\mathcal{N}}}\) are isomorphic if and only if the associated hypersurfaces S, \({\tilde S}\) are affinely equivalent. If one of S, \({\tilde S}\) even is affinely homogeneous, ‘affinely equivalent’ can be replaced by ‘linearly equivalent’. In case the nil-index of \({\mathcal{N}}\) does not exceed 4 the hypersurface S is always affinely homogeneous. Contrary to the expectation, in case nil-index 5 there exists an example (in dimension 23) where S is not affinely homogeneous.

On Certain Quasivarieties of Quasi-MV Algebras

Quasi-MV algebras are generalisations of MV algebras arising in quantum computational logic. Although a reasonably complete description of the lattice of subvarieties of quasi-MV algebras has already been provided, the problem of extending this description to the setting of quasivarieties has so far remained open. Given its apparent logical repercussions, we tackle the issue in the present paper. We especially focus on quasivarieties whose generators either are subalgebras of the standard square quasi-MV algebra S, or can be obtained therefrom through the addition of some fixpoints for the inverse.

Splitting fields of elements in arithmetic groups

We prove that the number of unimodular integral n × n matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group Sn is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases. More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field F . Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes developed by Jouve, Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice point counting result, and a non-concentration phenomenon for lattice points in algebraic subvarieties of the group variety, both established previously by the authors. The results answer a question posed by Rivin and by Jouve, Kowalski and Zywina, who have considered Galois groups of random products of elements in algebraic groups.

Global classification of the planar Lotka–Volterra differential systems according to their configurations of invariant straight lines

In this article, we study the Lotka–Volterra planar quadratic differential systems. We denote by LV systems all systems which can be brought to a Lotka–Volterra system by an affine transformation and time homotheties. All these systems possess invariant straight lines. We classify the family of LV systems according to their geometric properties encoded in the configurations of invariant straight lines which these systems possess. We obtain a total of 65 such configurations which are distinguished, roughly speaking, by the multiplicity of their invariant lines and by the multiplicities of the singularities of the systems located on these lines. We determine an algebraic subvariety of \({\mathbb{R}^{12}}\) which contains all these systems and we find the bifurcation diagram of the configurations of LV systems within this algebraic subvariety, in terms of polynomial invariants with respect to the group action of affine transformations and time homotheties. This geometric classification will serve as a basis for the full topological classification of LV systems.

Characteristic foliation on a hypersurface of general type in a projective symplectic manifold

AbstractA foliation on a non-singular projective variety is algebraically integrable if all leaves are algebraic subvarieties. A non-singular hypersurface X in a non-singular projective variety M equipped with a symplectic form has a naturally defined foliation, called the characteristic foliation on X. We show that if X is of general type and dim M≥4, then the characteristic foliation on X cannot be algebraically integrable. This is a consequence of a more general result on Iitaka dimensions of certain invertible sheaves associated with algebraically integrable foliations by curves. The latter is proved using the positivity of direct image sheaves associated to families of curves.

Effective algebraic degeneracy

We show that for every smooth projective hypersurface X⊂ℙn+1 of degree d and of arbitrary dimension n ≥2, if X is generic, then there exists a proper algebraic subvariety Y ⊊ X such that every nonconstant entire holomorphic curve f :ℂ→X has image f(ℂ) which lies in Y, as soon as its degree satisfies the effective lower bound $d\geqslant 2^{n^{5}}$ .

A Liouville theorem for plurisubharmonic currents

The goal of this paper is to extend the concepts of algebraic and Liouville currents, previously defined for positive closed currents by M. Blel, S. Mimouni and G. Raby, to psh currents on ℂ n . Thus, we study the growth of the projective mass of positive currents on ℂ n whose support is contained in a tubular neighborhood of an algebraic subvariety. We also give a sufficient condition guaranteeing that a negative psh current is Liouville. Moreover, we prove that every negative psh algebraic current is Liouville. For the particular case of closed currents, under adequate support conditions, we obtain a structure theorem.

Sums of hermitian squares on pseudoconvex boundaries

We give an abstract characterization of all real algebraic subvarieties of complex ane space on which every positive polynomial is a sum of hermitian squares, and we find obstructions to this phenomenon. As a consequence we construct a strictly pseudoconvex domain with smooth algebraic boundary on which there exists a degree two positive polynomial which is not a sum of hermitian squares, answering thus in the negative a question of John D'Angelo.

Singularities of algebraic subvarieties and problems of birational geometry

We consider the connection between the problem of estimating the multiplicity of an algebraic subvariety at a given singular point and the problem of describing birational maps of rationally connected varieties. We describe the method of hypertangent divisors which makes it possible to give bounds for the multiplicities of singular points. The concept of birational rigidity of algebraic varieties is discussed.

Transcendental manifolds in real projective space and Stiefel-Whitney classes

It is shown that the Stiefel-Whitney classes of a smooth manifold can give obstructions to realizing this manifold as the set of real points of a nonsingular real algebraic subvariety of projective space of a given dimension, even when the manifold can be embedded as an algebraic subset of real projective space of that dimension (meaning that the corresponding real algebraic variety must have complex singularities outside the real points). This strengthens earlier results by Akbulut and King. The result is an application of more technical results concerning algebraic cycles on real varieties combined with the Barth-Larsen Theorem in complex geometry. Mathematics Subject Classification (2000): 14P05 (primary); 14C25, 57R20 (secondary).

Manifolds with a unique embedding

Let X be a smooth, compact manifold of dimension k: We show that any two smooth embeddings X !R n are tamely equivalent provided n ‚ 2k + 2: Moreover, if additionally X is a real analytic manifold, then we show that any two real analytic embeddings X !R n are real analytically equivalent. We extend this result to some interesting sub-categories of the category of real smooth manifold. In particular we will prove that if X;Y are Nash kidimensional submanifolds of R n (where n ‚ 2k + 2) and ` : X ! Y is a difieomorphism (Nash isomorphism, real analytic isomorphism), thencan be extended to a tame difieomorphism (tame Nash isomorphism, tame real analytic isomorphism) ' :R n !R n : We prove also that if X;Y are kidimensional smooth algebraic subvarieties of C n (where n ‚ 2k + 2), and ` : X ! Y is a biholomorphism (or polynomial isomorphism), thencan be extended to a global tame biholomorphism (polynomial automorphism) ' :C n !C n : Finally we prove: if X;Y are kidimensional closed semi-algebraic subsets of R n (where n ‚ 2k + 2), and ` : X ! Y is a semi-algebraic homeomorphism, thatcan be extended to a global tame semi-algebraic homeomorphism ' :R n !R n : This last result is a semi-algebraic counterpart of a classical result of Herman Gluck (6), on extension of homeomorphisms of compact polyhedrons.

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.

Extension of Fréchet valued real analytic functions from subvarieties of ℝd

It is shown that for an algebraic subvariety X of ℝd every Fréchet valued real analytic function on X can be extended to a real analytic function on ℝd if and only if X is of type (PL), i.e. all of its singularities are of a certain type. Necessity of this condition is shown for any real analytic variety. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Polynomial symplectomorphisms

Let 𝕂 be the field of real or complex numbers. Let (X ≅ 𝕂2n, ω) be a symplectic affine space. We study the group of polynomial symplectomorphisms of X. We show that for an arbitrary k ∈ ℕ the group of polynomial symplectomorphisms acts k-transitively on X. Moreover, if 2 ⩽ l ⩽ 2n − 2 then elements of this group can be characterized by polynomial automorphisms which preserve the symplectic type of all algebraic l-dimensional subvarieties of X.

The probability that a slightly perturbed numerical analysis problem is difficult

We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of \epsilon-tubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a disk of radius \sigma. Besides \epsilon and \sigma, this bound depends only the dimension of the sphere and on the degree of the defining equations.

Approximation of holomorphic maps with a lower bound on the rank

Let K be a closed polydisc or ball in C, and let Y be a quasi-projective algebraic manifold which is Zariski locally equivalent to C P , or a complement of an algebraic subvariety of codimension ≥ 2 in such a manifold. If r is an integer satisfying (n - r + 1)(p - r + 1) ≥ 2, then every holomorphic map from a neighborhood of K to Y with rank > r at every point of K can be approximated uniformly on K by entire maps C n → Y with rank > r at every point of C.