Let $X$, $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$-regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$, we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ can be approximated by $k$-regulous maps in the $\mathcal {C}^k$ topology if and only if $f$ is homotopic to a $k$-regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking $Y=\mathbb {S}^p$ (the unit $p$-dimensional sphere), we obtain several new results on approximation of $\mathcal {C}^{\infty }$ maps from $X$ into $\mathbb {S}^p$ by $k$-regulous maps in the $\mathcal {C}^k$ topology, for $k \geq 0$.

Given a graph G without loops, the pseudograph associahedron PG is a smooth polytope, so there is a projective smooth toric variety XG corresponding to PG. Taking the real locus of XG, we have the projective smooth real toric variety $X^{\mathbb{R}}_G$. The integral cohomology groups of $X^{\mathbb{R}}_G$ can be computed by studying the topology of certain posets of even subgraphs of G; such a poset is neither pure nor shellable in general. We completely characterize the graphs whose posets of even subgraphs are always shellable. It follows that we get a family of projective smooth real toric varieties whose integral cohomology groups are torsion-free or have only 2-torsion.

We compare two partitions of real bitangents to smooth plane quartics into sets of 4: one coming from the closures of connected components of the avoidance locus and another coming from tropical geometry. When both are defined, we use the Tarski principle for real closed fields in combination with the topology of real plane quartics and the tropical geometry of bitangents and theta characteristics to show that they coincide.

We propose two systems of “intrinsic” weights for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class $-2K$, but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.

We give a motivic proof of the fact that for nonsingular real tropical complete intersections, the Euler characteristic of the real part is equal to the signature of the complex part. This was originally proved by Itenberg in the case of surfaces in $\mathbb {C}P^{3}$, and has been successively generalized by Bertrand and by Bihan and Bertrand. Our proof, different from previous approaches, is an application of the motivic nearby fiber of semistable degenerations. In particular, it extends the original result by Itenberg, Bertrand, and Bihan to real analytic families admitting a $\mathbb {Q}$-nonsingular tropical limit.

Let $X$ be a $\text{CW}$ complex with a continuous action of a topological group $G$. We show that if $X$ is equivariantly formal for singular cohomology with coefficients in some field $\Bbbk $, then so are all symmetric products of $X$ and in fact all its $\Gamma $-products. In particular, symmetric products of quasi-projective $\text{M}$-varieties are again $\text{M}$-varieties. This generalizes a result by Biswas and D’Mello about symmetric products of $\text{M}$-curves. We also discuss several related questions.

To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincaré series as a motivic measure. We show Denef–Loeser formulas for the equivariant zeta functions and prove that they are invariants for equivariant blow-Nash equivalence via equivariant blow-Nash isomorphisms. Equivariant blow-Nash equivalence between invariant Nash germs is defined as a generalization involving equivariant data of the blow-Nash equivalence.

Let Y be a compact nonsingular real algebraic variety of positive dimension. Then one can find a compact connected nonsingular real algebraic variety X, which admits a continuous map into Y that is not homotopic to any regular map. It is hard to determine the minimum dimension of such a variety X. In this paper, new upper bounds for dim X are obtained. The main role in the constructions is played by complex algebraic cycles on Y.

Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah–Bott's “Yang–Mills over a Riemann Surface” to compute $\mathbb{Z}/2$–Betti numbers of these spaces.

Let $X$ be a smooth complex projective manifold of dimension $n$ equipped with an ample line bundle $L$ and a rank $k$ holomorphic vector bundle $E$. We assume that $1\leqslant k\leqslant n$, that $X$, $E$ and $L$ are defined over the reals and denote by $\mathbb{R}X$ the real locus of $X$. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in $\mathbb{R}X$ of holomorphic real sections of $E\otimes L^{d}$, where $d$ is a large enough integer. Moreover, given any closed connected codimension $k$ submanifold ${\it\Sigma}$ of $\mathbb{R}^{n}$ with trivial normal bundle, we prove that a real section of $E\otimes L^{d}$ has a positive probability, independent of $d$, of containing around $\sqrt{d}^{n}$ connected components diffeomorphic to ${\it\Sigma}$ in its vanishing locus.

We address the problem of existence and uniqueness of a factorization of a given element of the modular group into a product of two Dehn twists. As a geometric application, we conclude that any maximal real elliptic Lefschetz fibration is algebraic.

In this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.

Let (X, cX) be a convex projective surface equipped with a real structure. The space of stable maps carries different real structures induced by cX and any order two element τ of permutation group Sk acting on marked points. Each corresponding real part ℝτ is a real normal projective variety. As the singular locus is of codimension bigger than two, these spaces thus carry a first Stiefel–Whitney class for which we determine a representative in the case k = c1(X)d − 1 where c1(X) is the first Chern class of X. Namely, we give a homological description of these classes in term of the real part of boundary divisors of the space of stable maps.

The topological classification of smooth real cubic surfaces is recalled and compared to the classification in terms of the number of real lines and of real tritangent planes, as obtained by $\text{L}$. Schläfli in 1858. Using this, explicit examples of surfaces of every possible type are given.

It is proved that each mod 2 homology class of a compact Nash manifold can be represented by a semialgebraic arc-symmetric subset.

Classification of real K3 surfaces $X$ with non-symplectic involution $\tau$ is considered. We define a natural notion of degeneration for them. We show that the connected component of moduli of non-degenerate surfaces of this type is defined by the isomorphism class of the action of $\tau$ and the anti-holomorphic involution $\varphi$ in the homology lattice. (There are very few similar results known.) For their classification we apply invariants of integral lattice involutions with conditions that were developed by the first author in 1983. As a particular case, we describe connected components of moduli of real non-singular curves $A \in | -2 K_V|$ for the classical real surfaces: $V = P^2$, hyperboloid, ellipsoid, $F_1$, $F_4$.

As an application, we describe all real polarized K3 surfaces that are deformations of general real K3 double rational scrolls (the surfaces $V$ above). There are very few exceptions. For example, any non-singular real quartic in $P^3$ can be constructed in this way.

We define invariants of the blow-Nash equivalence of Nash function germs, in a similar way to the motivic zeta functions of Denef and Loeser. As a key ingredient, we extend the virtual Betti numbers, which were known for real algebraic sets, to a generalized Euler characteristic for projective constructible arc-symmetric sets. Actually we prove more: the virtual Betti numbers are not only algebraic invariants, but also Nash invariants of arc-symmetric sets. Our zeta functions enable one to distinguish the blow-Nash equivalence classes of Brieskorn polynomials of two variables. We prove moreover that there are no moduli for the blow-Nash equivalence in the case of an algebraic family with isolated singularities.

Many new universal relations are obtained between the Euler numbers of manifolds of singular supporting hyperplanes of an arbitrary generic smooth closed $k$-dimensional submanifold in ${{\mathbb R}}^n$ where $n\leq 7$ or $k=1$. These relations are applied to Barner-convex curves in an odd-dimensional space ${{\mathbb R}}^n$. A universal (nontrivial) linear relation is established between the numbers of singular supporting hyperplanes of various types but of the same total multiplicity $n$ of tangency with a given generic smooth closed connected Barner-convex curve in ${{\mathbb R}}^n$. The coefficients of this relation are defined by Catalan numbers.

We consider a polynomial $f \,{:}\, \mathbb{R}^n \rightarrow \mathbb{R}$ with isolated critical points and we relate $\chi(f^{-1}(0))$ and $\chi(\{f \ge 0\})-\chi(\{f \le 0\})$ to the topological degrees of polynomial maps defined in terms of $f$.

On a real algebraic variety there may exist an algebraic cycle that is algebraically equivalent to zero and whose cohomology class is non-zero. The group of such cohomology classes can be highly non-trivial. It is interesting since it allows one to detect cohomology classes, in complementary dimension, which cannot be represented by algebraic cycles.