Viatcheslav M Kharlamov

University of Strasbourg | UNISTRA · Institut de Recherche Mathématique Avancée

We undertake a study of topological properties of the real Mordell-Weil group $\operatorname{MW}_{\mathbb R}$ of real rational elliptic surfaces $X$ which we accompany by a related study of real lines on $X$ and on the "subordinate" del Pezzo surfaces $Y$ of degree 1. We give an explicit description of isotopy types of real lines on $Y_{\mathbb R}$...

We give an expression for the Smith–Thom deficiency of the Hilbert square of a smooth real algebraic variety in terms of the rank of a suitable Mayer– Vietoris mapping in several situations. As a consequence, we establish a necessary and sufficient condition for the maximality of in the case of projective complete intersections, and show that with...

We give an expression for the Smith-Thom deficiency of the Hilbert square X [2] of a smooth real algebraic variety X in terms of the rank of a suitable Mayer-Vietoris mapping in several situations. As a consequence, we establish a simple necessary and sufficient condition for the maximality of X [2] in the case of projective complete intersections...

We prove that the space of affine, transversal at infinity, nonsingular real cubic surfaces has 15 connected components. We give a topological criterion to distinguish them and show also how these 15 components are adjacent to each other via wall-crossing.

We continue our quest for real enumerative invariants not sensitive to changing the real structure and extend the construction we uncovered previously for counting curves of anti-canonical degree ⩽2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathr...

We explore the maximality of the Hilbert square of maximal real surfaces, and find that in many cases the Hilbert square is maximal if and only if the surface has connected real locus. In particular, the Hilbert square of no maximal K3-surface is maximal. Nevertheless, we exhibit maximal surfaces with disconnected real locus whose Hilbert square is...

We prove that the space of affine, transversal at infinity, real cubic surfaces has 15 connected components. We show also how this division into 15 species and their wall-crossing adjacency are determined by the topology of these affine surfaces.

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 res...

We continue our quest for real enumerative invariants not sensitive to changing the real structure and extend the construction we uncovered previously for counting curves of anti-canonical degree $\leqslant 2$ on del Pezzo surfaces with $K^2=1$ to curves of any anti-canonical degree and on any del Pezzo surfaces of degree $K^2\leqslant 3$.

We show how the real lines on a real del Pezzo surface of degree 1 can be split into two species, elliptic and hyperbolic, via a certain distinguished, intrinsically defined, Pin-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{up...

We propose two systems of "intrinsic" signs 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 result...

We show how the real lines on a real del Pezzo surface of degree 1 can be split into two species, elliptic and hyperbolic, via a certain distinguished, intrinsically defined, Pin-structure on the real locus of the surface. We prove that this splitting is invariant under real automorphisms and real deformations of the surface, and that the differenc...

This is an expanded version of the talk given by the first author at the conference “Topology, Geometry, and Dynamics: Rokhlin – 100”. The purpose of this talk was to explain our current results on the classification of rational symplectic 4-manifolds equipped with an anti-symplectic involution. A detailed exposition will appear elsewhere.

This paper was conceived as an addendum to the note "Rokhlin's signature theorems" (by O.Viro and the authors of this paper). In the main section we give an overview of Rokhlin's proof of his famous theorem on divisibility of signature by 16. In the appendix we retrace some of further developments that show how this theorem became a cornerstone in...

This note is written for a book dedicated to outstanding St-Petersburg mathematicians and timed to the ICM-2022 in St-Petersburg. In accordance with the plan of ICM-organizers, we try to tell about one of the most prominent Rokhlin's achievements in an accessible form and respecting the allowed volume.

This is an expanded version of the talk given be the first author at the conference "Topology, Geometry, and Dynamics: Rokhlin - 100". The purpose of this talk was to explain our current results on classification of rational symplectic 4-manifolds equipped with an anti-symplectic involution. Detailed exposition will appear elsewhere.

In our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of coefficients preserving the hypersurface non-singular. Here, we perform a finer classification giving a full answer to the chirality pr...

We suggest a short proof of O.Benoist and O.Wittenberg theorem (arXiv:1907.10859) which states that for each real non-singular cubic hypersurface $X$ of dimension $\ge 2$ the real lines on $X$ generate the whole group $H_1(X(\Bbb R);\Bbb Z/2)$.

In our previous paper [5] we have elaborated a certain signed count of real lines on real hypersurfaces of degree $2n-1$ in $P^{n+1}$. Contrary to the honest “cardinal” count, it is independent of the choice of a hypersurface and, by this reason, provides a strong lower bound on the honest count. In this count the contribution of a line is its loca...

In our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of coefficients preserving the hypersurface non-singular. Here, we perform a finer classification giving a full answer to the chirality pr...

In our previous paper we have elaborated a certain signed count of real lines on real projective n-dimensional hypersurfaces of degree 2n-1. Contrary to the honest "cardinal" count, it is independent of the choice of a hypersurface, and by this reason provides a strong lower bound on the honest count. In this count the contribution of a line is its...

The surfaces considered are real, rational and have a unique smooth real \((-2)\)-curve. Their canonical class K is strictly negative on any other irreducible curve in the surface and \(K^2>0\). For surfaces satisfying these assumptions, we suggest a certain signed count of real rational curves that belong to a given divisor class and are simply ta...

We prove that the space of pairs $(X,l)$ formed by a real non-singular cubic hypersurface $X\subset P^4$ with a real line $l\subset X$ has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surfa...

We prove that the space of pairs $(X,l)$ formed by a real non-singular cubic hypersurface $X\subset P^4$ with a real line $l\subset X$ has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surfa...

We establish the enumerativity of (original and modified) Welschinger invariants for every real divisor on any real algebraic del Pezzo surface and give an algebro-geometric proof of the invariance of that count both up to variation of the point constraints on a given surface and variation of the complex structure of the surface itself.

We study qualitative aspects of the Welschinger-like $\mathbb Z$-valued count of real rational curves on primitively polarized real $K3$ surfaces. In particular, we prove that with respect to the degree of the polarization, at logarithmic scale, the rate of growth of the number of such real rational curves is, up to a constant factor, the rate o...

We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree ≥ 2. We show that under some conditions, for such a surface X and a real nef and big divisor class D ∈ Pic(X), through any generic collection of −DK X − 1 real points lying on a connected component of the real part RX of X one ca...

The surfaces considered are real, rational and have a unique smooth real $(-2)$-curve. Their canonical class $K$ is strictly negative on any other irreducible curve in the surface and $K^2>0$. For surfaces satisfying these assumptions, we suggest a certain signed count of real rational curves that belong to a given divisor class and are simply tang...

We show that a generic real projective n-dimensional hypersurface of odd degree d, such that 4(n-2)=(d+33), contains “many” real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, d³log d, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as t...

We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree ≥ 2. We show that under some conditions, for such a surface X and a real nef and big divisor class D ∈ Pic(X), through any generic collection of −DK X − 1 real points lying on a connected component of the real part RX of X one ca...

We establish the enumerativity of (original and modified) Welschinger invariants for every real divisor on any real algebraic Del Pezzo surface and give an algebro-geometric proof of the invariance of that count both up to variation of the point constraints on a given surface and variation of the complex structure of the surface itself.

We provide a real analog of the Yau-Zaslow formula counting rational curves on $K3$ surfaces.

In this article, we investigate some properties of cyclic coverings of complex surfaces of general type branched along smooth curves that are numerically equivalent to a multiple of the canonical class. The main results concern coverings of surfaces of general type with p_g=0 and Miyaoka--Yau surfaces; in particular, they provide new examples of mu...

We give a complete deformation classification of real Zariski sextics, that is of generic apparent contours of nonsingular real cubic surfaces. As a by-product, we observe a certain "reversion" duality in the set of deformation classes of these sextics.

We show that a generic real projective n-dimensional hypersurface of degree 2n − 1 contains many real lines, namely not less than (2n − 1)!!, which is approximately the square root of the number of complex lines. This estimate is based on the interpretation of a suitable signed count of the lines as the Euler number of an appropriate bundle.

We introduce and study a semigroup structure on the set of irreducible components of the Hurwitz space of marked coverings of a complex projective curve with given Galois group of the coverings and fixed ramification type. As application, we give new conditions on the ramification type that are sufficient for irreducibility of the Hurwitz spaces, s...

We show that a generic real projective n-dimensional hypersurface of degree 2n-1 contains "many" real lines, namely, not less than (2n-1)!!, which is approximately the square root of the number of complex lines. This estimate is based on the interpretation of a suitable signed count of the lines as the Euler number of an appropriate bundle.

We give a recursive formula for purely real Welschinger invariants of real Del Pezzo surfaces of degree $K^2\ge 3$, where in the case of surfaces of degree 3 with two real components we introduce a certain modification of Welschinger invariants and enumerate exclusively the curves traced on the non-orientable component. As an application, we prove...

We give a recursive formula for purely real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown up at $q$ real and $s \leq 1$ pairs of conjugate imaginary points, where $q+2s\le 5$, and the real quadric blown up at $s \leq 1$ pairs of conjugate imaginary points and having non-empty real part. The formula is s...

A solution of the problem of topological classification of real cubic fourfolds is given. It is proven that the real locus of a real non-singular cubic fourfold is diffeomorphic either to a connected sum ℝP4 #i(S2 × S2)#j(S1 × S3) or to a disjoint union ℝP4 ⊔ S4. La voie la plus courte et la meilleure entre deux vérités de domaine réel passe souve...

Our main results-5pc]Please check the text is ok? concern complete intersections of three real quadrics. We prove that the maximal number B 20(N) of connected components that a regular complete intersection of three real quadrics in ℙ N may have differs at most by one from the maximal number of ovals of the submaximal depth \([(N - 1)/2]\)of a real...

According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and to obtain a pure deformation classification, that is how...

We investigate the automorphism groups of Galois coverings induced by pluricanonical generic coverings of projective spaces. In dimensions one and two, it is shown that such coverings yield sequences of examples where specific actions of the symmetric group S d on curves and surfaces cannot be deformed together with the action of S d into manifolds...

We consider the product of two projective lines equipped with the complex conjugation transforming $(x,y)$ into $(\bar{y},\bar{x})$ and blown up in at most two real, or two complex conjugate, points. For these four surfaces we prove the logarithmic equivalence of Welschinger and Gromov-Witten invariants.

We study real elliptic surfaces and trigonal curves (over a base of an arbitrary genus) and their equivariant deformations. We calculate the real Tate-Shafarevich group and reduce the deformation classification to the combinatorics of a real version of Grothendieck's {\it dessins d'enfants}. As a consequence, we obtain an explicit description of th...

These notes reproduce the content of a short, 50-minutes, survey talk given at the Nice University in September, 2004. We added a few topics that have not been touched on in the lecture by lack of time.

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.

We study real nonsingular projective cubic fourfolds 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 the middle homology. Moreover, we provide a graph whose vertices represent the equivalence classes of such cubics and...

We show that the number of equivariant deformation classes of real structures in a given deformation class of compact hyperkahler manifolds is finite.

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.

Let $P^2_k$ be the projective plane blown up at $k$ points and $D\in H_2(P^2_k,Z)$. In this paper the authors study the asymptotic behavior of the Gromov-Witten invariants ${\rm GW}_{nD}(P^2_k)$. The motivation for this problem comes from the relation between Gromov-Witten invariants and their real analogs---Welschinger invariants, as well as a rel...

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 logarithmic equivalence of sequences we mean the asymptotic equivalence of their log...

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 logarithmic equivalence of sequences we mean the asymptotic equivalence of their log...

The invariance of the Welschinger numbers for real unnodal Del Pezzo surfaces, which we used for the enumeration of real rational curves on real toric Del Pezzo surfaces (see math.AG/0303378 and IMRN 49 (2003), 2639-2653), follows from J.-Y. Welschinger's theorem stated and proved in a symplectic setting (math.AG/0303145 and CRAS, Ser. I, 336 (2003...

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 ex...

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 n...

The number of topologically different plane real algebraic curves of a given degree $d$ has the form $\exp(C d^2 + o(d^2))$. We determine the best available upper bound for the constant $C$. This bound follows from Arnold inequalities on the number of empty ovals. To evaluate its rate we show its equivalence with the rate of growth of the number of...

The work presents some results on the asymptotics of the number of real plane algebraic curves as the degree grows. In particular, we obtain the asymptotics of the number of curves considered up to the isotopy and rigid isotopy, as well as the number of isotopic classes of maximal curves realizable by T-curves. Some results are generalized to hyper...

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 exi...

"These notes reproduce the content of the colloquium talks givenat G"ottingen Univeristy in December, 1999 and at Milan Univeristy in April, 2000. They are devoted essentially to quasi-simplicity and finiteness phenomena in the deformation theory of real algebraic surfaces. An additional result, which was not given in these lectures and was obtaine...

We compare the smooth and deformation equivalence of actions of finite groups on K3-surfaces by holomorphic and anti-holomorphic transformations. We prove that the number of deformation classes is finite and, in a number of cases, establish the expected coincidence of the two equivalence relations. More precisely, in these cases we show that an act...

We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for the important Shapiro and Shapiro conjecture in the real Schubert calculus. We establish restrictions on the n...

Here, we resume and broaden the results concerned which appeared in math.AG/0101098 and math.AG/0104021. We start from summing up our example of a complex algebraic surface which is not deformation equivalent to its complex conjugate and which, moreover, has no homeomorphisms reversing the canonical class. Then, we construct several series of highe...

In this paper we prove that there is an infinite sequence of pairs of plane cuspidal curves, Cm,1 and Cm,2, of degree deg(Cm,1)=deg(Cm,2)→∞, such that the pairs and are diffeomorphic, but Cm,1 and Cm,2 have non-equivalent braid monodromy factorizations. These curves give rise to the negative solutions of “Dif ⇒ Def” and “Dif ⇒ Iso” problems for pla...

We prove that there is an infinite sequence of pairs of plane cuspidal curves $C_{m,1}$ and $C_{m,2}$, such that the pairs $(\Bbb CP^2, C_{m,1})$ and $(\Bbb CP^2, C_{m,2})$ are diffeomorphic, but $C_{m,1}$ and $C_{m,2}$ have non-equivalent braid monodromy factorizations. These curves give rise to the negative solutions of "Dif=Def" and "Dif=Iso" pr...

We show that a real rational (over $\C$) surfaces are quasi-simple, i.e., that such a surface is determined up to deformation in the class of real surfaces by the topological type of its real structure.

We construct examples of rigid surfaces (that is, surfaces whose deformation class consists of a unique surface) with a particular behaviour with respect to real structures. In one example the surface has no real structure. In another it has a unique real structure, which is not maximal with respect to the Smith-Thom inequality. These examples give...

In this paper we prove that there is an infinite sequence of pairs of plane cuspidal curves, Cm,1 and Cm,2, of degree deg(Cm,1)=deg(Cm,2)→∞, such that the pairs (CP2,Cm,1) and (CP2,Cm,2) are diffeomorphic, but Cm,1 and Cm,2 have non-equivalent braid monodromy factorizations. These curves give rise to the negative solutions of “Dif ⇒ Def” and “Dif ⇒...

14J99 14P25 Topology of real algebraic varieties 57S25 Groups acting on specific manifolds

The nonsingular real plane algebraic curves of given degree $d$ are considered either up to isotopy or up to deformation. The asymptotic behavior of the number $I_d$ of isotopy classes and the number $D_d$ of deformation classes are studied. It is shown, in particular, that $log I_d\asypt d^2$. Other related problems and their higher dimensional ge...