The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V. I. Arnold and V. A. Rokhlin. We make an attempt to systematize the principal results in the subject. After an exposition of general tools and results, special attention is paid to surfaces and curves on surfaces.
To the memory of Andrey Bolibruch, a lively man of creative mind and open soul Abstract The Welschinger numbers, a kind of a real analog of the Gromov-Witten numbers which count the complex rational curves through a given generic collection of points, bound from below the number of real rational curves for any real generic collection of points. By the… (More)
We study real nonsingular cubic hypersurfaces X ⊂ P 5 up to deformation equivalence combined with projective equivalence and prove that they are classified by the conjugacy classes of involutions induced by the complex conjugation in H 4 (X). Moreover, we provide a graph Γ K4 whose vertices represent the equivalence classes of such cubics and edges… (More)
We introduce and develop a language of semigroups over the braid groups for a study of braid monodromy factorizations (bmf's) of plane algebraic curves and other related objects. As an application we give a new proof of Orevkov's theorem on realization of a bmf over a disc by algebraic curves and show that the complexity of such a realization can not be… (More)
We complete the classiication of the topological types of real En-riques surfaces started by V. Nikulin. The resulting list contains 87 topological types.
Le Yi Jing n'est pas un livre, un texte qu'on lit du débutà la fin, mais un ouvrage que l'on consulte quand on en a besoin. Lorsqu'on hésite sur une voiè a suivre, une attitudè a prendre, un choixà faire, un dilemmè a résoudre, on peut alors s'en servir pour ce qu'il est dans la pratique : un manuel d'aidè a la décision. Abstract. We study real elliptic… (More)
We study real Campedelli surfaces up to real deformations and exhibit a number of such surfaces which are equivariantly diffeomorphic but not real deformation equivalent.
Welschinger's invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using Mikhalkin's approach which deals with a corresponding count of tropical curves. In particular, our estimate implies that, for any positive integer d, there exists a real… (More)
We begin the topological study of real rational plane curves all of whose inflection points are real. The existence of such curves is implied by the results of real Schubert calculus, and their study has consequences for the important Shapiro and Shapiro conjecture in real Schubert calculus. We establish restrictions on the number of real nodes of such… (More)
We define a series of relative tropical Welschinger-type invariants of real toric surfaces. In the Del Pezzo case, these invariants can be seen as real tropical analogs of relative Gromov-Witten invariants, and are subject to a recursive formula. As application we obtain new formulas for Welschinger invariants of real toric Del Pezzo surfaces.