Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry

We first build the moduli spaces of real rational pseudo-holomorphic curves in a given real symplectic 4-manifold. Then, following the approach of Gromov and Witten [3, 19, 11], we define invariants under deformation of real symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of points. Mathematics Subject Classification (2000) 14N10, 14P25, 53D05, 53D45     

Welschinger invariants of toric Del Pezzo surfaces with nonstandard real structures

The Welschinger invariants of real rational algebraic surfaces are natural analogs of the Gromov-Witten invariants, and they estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces equipped with a nonstandard real structure. Such a formula for real toric Del Pezzo surfaces with a standard real structure (i.e., naturally compatible with the toric structure) was established by Mikhalkin and the author. As a consequence we prove that for any real ample divisor D on a surface Σ under consideration, through any generic configuration of c ₁(Σ)D − 1 generic real points, there passes a real rational curve belonging to the linear system |D|. To Vladimir Igorevich Arnold on the occasion of his 70th birthday     

Simple components of group algebras Q[G] central over real quadratic fields

Let k be a real quadratic field, and $\text{U}$ a central division quaternion algebra over k. In this paper sufficient conditions are given to insure that $\text{U}$ appears in a simple component of the group algebra Q[G] of some finite group G over the rational field Q. In particular, when k is assumed to be Q(√2) or Q(√5), the necessary and sufficient conditions for $\text{U}$ to appear in some Q[G] are given.     

Automorphisms of real rational surfaces and weighted blow-up singularities

Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e = [e ₁, . . . , e _? ] be a partition of n. Denote by X ^e the set of ?-tuples (P ₁, . . . , P _? ) of disjoint nonsingular curvilinear subschemes of X of orders (e ₁, . . . , e _? ). We show that the group Aut(X) acts transitively on X ^e . This statement generalizes earlier work where the case of the trivial partition e = [1, . . . , 1] was treated under the supplementary condition that X is nonsingular. As an application we classify singular real rational surfaces obtained from nonsingular surfaces by performing weighted blow-ups.     

Inertia and eigenvalue relations between symmetrized and symmetrizing matrices for the real and the general field case

Starting from a theorem of Frobenius that every n×n matrix is the product of two symmetric ones, we study relations between the similarity invariants of a square matrix and the congruence invariants of its symmetric factors. Section 1 treats the real case, Sec. 2 the arbitrary field case, and Sec. 3 the indefinite inner product case for Krein spaces. The proofs are obtained from the real canonical pair form in Secs. 1 and 3 and from the recently found rational canonical pair form in Sec. 2, each time via combinatorial type arguments on weighted partitions of n. The resulting theorems typically give bounds for the elementary divisor structure of A in terms of the index or signature of one or both of its symmetric factors (or vice versa). Our results greatly extend and generalize the classic results of Klein, Loewy, Taussky, et al., and in Sec. 2 put new light on Waterhouse's recent characterization of hereditarily euclidean fields. A short survey on the history of the subject from the early 1800s on completes the paper.     

Towards relative invariants of real symplectic four-manifolds

Let (X, ω, c_X) be a real symplectic four-manifold with real part . Let be a smooth curve such that We construct invariants under deformation of the quadruple (X, ω, c_X, L) by counting the number of real rational J-holomorphic curves which realize a given homology class d, pass through an appropriate number of points and are tangent to L. As an application, we prove a relation between the count of real rational J-holomorphic curves done in [W2] and the count of reducible real rational curves done in [W3]. Finally, we show how these techniques also allow us to extract an integer valued invariant from a classical problem of real enumerative geometry, namely about counting the number of real plane conics tangent to five given generic real conics. Received: March 2005; Revision: September 2005; Accepted: September 2005     

Special values of L-functions of elliptic curves over Q and their base change to real quadratic fields

For an elliptic curve E over Q, and a real quadratic extension F of Q, satisfying suitable hypotheses, we study the algebraic part of certain twisted L-values for E/F. The Birch and Swinnerton-Dyer conjecture predicts that these L-values are squares of rational numbers. We show that this question is related to the ratio of Petersson inner products of a quaternionic form on a definite quaternion algebra over Q and its base change to F.     

Uniqueness of the group operation on the lattice of order-automorphisms of the real line

A(R) is the lattice-ordered group (l-group) of all order-automorphisms of the real lineR, with the usual pointwise order and “of course” with composition as the group operation. In fact, what other choices are there for a group operation having the same identity that would give anl-group? Composition in the reverse order would work. But there are no other choices — the group operation can be recognized in the lattice. Several classes of abelianl-groups having a unique group operation have been found by Conrad and Darnel, but this is the first non-abelian example having the minimum of two group operations. “Conversely”, Holland has shown that for the groupA(R) under composition, the only lattice orderings yielding anl-group are the pointwise order and its dual. These results also hold for the rational lineQ.     

Real cubic surfaces and real hyperbolic geometry

The moduli space of stable real cubic surfaces is the quotient of real hyperbolic four-space by a discrete, nonarithmetic group. The volume of the moduli space is 37π²/1080 in the metric of constant curvature ?1. Each of the five connected components of the moduli space can be described as the quotient of real hyperbolic four-space by a specific arithmetic group. We compute the volumes of these components. To cite this article: D. Allcock et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

The main invariants of a complex finsler space

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwald frame. The geometry of such manifolds is controlled by three real invariants which live on T'M: two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular interest. Complex Berwald spaces coincide with Kähler spaces, in the two – dimensional case. We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the Kähler purely Hermitian spaces by the fact K = W = constant and I = 0. For the class of complex Berwald spaces we have K = W = 0. Finally, a classification of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.     

Invariants of formally real fields and quadratic forms

Let F be a formally real field. Connections between the invariants I(F), N(F), ud(F), ũd(F), β_F(i) (i ϵ [1, l(F) - 1]) are studied, using a new invariant C(F). The possibilities for β_F are described when l(F) ≤ 5 or ud(F) ≤ 4, without considering existence.     

Lichnerowicz-Obata theorem for Kohn Laplacian on the real ellipsoid

We give the sharp lower bound for Ricci curvature on the real ellipsoid in Cⁿ⁺¹, and prove the Lichnerowicz-Obata theorem for Kohn Laplacian.     

Virtual Betti numbers of real algebraic varieties

We show that for all i?0 the i-th mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti numberβ_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X then β_i(X)=β_i(X?Y)+β_i(Y). We show by example that there is no natural weight filtration on the $\text{Z}{}_2$-cohomology of real algebraic varieties with compact supports such that the virtual Betti numbers are the weighted Euler characteristics. To cite this article: C. McCrory, A. Parusiński, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Rational versus real cohomology algebras of low-dimensional toric varieties

We show that the real cohomology algebra of a compact toric variety of complex dimension  is determined, up to isomorphism, by the combinatorial data of its defining fan. Surprisingly enough, this is no longer the case when taking rational coefficients. Moreover, we show that neither the rational nor the real or complex cohomology algebras of compact quasi-smooth toric varieties are combinatorial invariants in general.

     

Computation of the Iwasawa invariants of certain real abelian fields

Let p be a prime number and k a finite extension of . It is conjectured that the Iwasawa invariants λ_p(k) and μ_p(k) vanish for all p and totally real number fields k. Some methods to verify the conjecture for each real abelian field k are known, in which cyclotomic units and a set of auxiliary prime numbers are used. We give an effective method, based on the previous one, to compute the exact value of the other Iwasawa invariant ν_p(k) by using Gauss sums and another set of auxiliary prime numbers. As numerical examples, we compute the Iwasawa invariants associated to in the range 1<f<200 and 5?p<10000.     

Second order homological obstructions on real algebraic manifolds

Let Y be a compact nonsingular real algebraic set whose homology classes (over Z/2) are represented by Zariski closed subsets. It is well known that every smooth map from a compact smooth manifold to Y is unoriented bordant to a regular map. In this paper, we show how to construct smooth maps from compact nonsingular real algebraic sets to Y not homotopic to any regular map starting from a nonzero homology class of Y of positive degree. We use these maps to obtain obstructions to the existence of local algebraic tubular neighborhoods of algebraic submanifolds of Rn and to study some algebro-homological properties of rational real algebraic manifolds.     

On the Kolmogorov complexity of continuous real functions

Kolmogorov complexity was originally defined for finitely-representable objects. Later, the definition was extended to real numbers based on the asymptotic behaviour of the sequence of the Kolmogorov complexities of the finitely-representable objects—such as rational numbers—used to approximate them.This idea will be taken further here by extending the definition to continuous functions over real numbers, based on the fact that every continuous real function can be represented as the limit of a sequence of finitely-representable enclosures, such as polynomials with rational coefficients.Based on this definition, we will prove that for any growth rate imaginable, there are real functions whose Kolmogorov complexities have higher growth rates. In fact, using the concept of prevalence, we will prove that ‘almost every’ continuous real function has such a high-growth Kolmogorov complexity. An asymptotic bound on the Kolmogorov complexities of total single-valued computable real functions will be presented as well.     

Counting curves via lattice paths in polygons

This Note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface is $\text{P}{}^2$ or $\text{P}{}^1\text{×}\text{P}{}^1$ then the invariants under consideration coincide with the Gromov–Witten invariants. The formula gives a new count even in these cases, where other computational techniques are available. To cite this article: G. Mikhalkin, C. R. Acad. Sci. Paris, Ser. I 336 (2003).