Related papers: Rational differential forms on the variety of flex…

For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.

We give a vanishing and classification result for holomorphic differential forms on smooth projective models of the moduli spaces of pointed K3 surfaces. We prove that there is no nonzero holomorphic k-form for 0<k<10 and for even k>19. In…

Let $X$ be the variety of flexes of plane cubics. We prove that (1) $X$ is an irreducible rational algebraic variety endowed with a faithful algebraic action of ${\rm PSL}_3$; (2) $X$ is ${\rm PSL}_3$-equivariantly birationally isomorphic…

It is proved that on a smooth algebraic variety, fibered into cubic surfaces over the projective line and sufficiently ``twisted'' over the base, there is only one pencil of rational surfaces -- that is, this very pencil of cubics. In…

We construct smooth rational real algebraic varieties of every dimension $\ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.

In this paper we prove that no complex surface of general type is diffeomorphic to a rational surface, thereby completing the smooth classification of rational surfaces and the proof of the Van de Ven conjecture on the smooth invariance of…

We prove that a generic holomorphic foliation on a weighted projective plane has no algebraic solutions when the degree is big enough. We also prove an analogous result for foliations on Hirzebruch surfaces.

We show, among other things, that for each integer $n \ge 3$, there is a smooth complex projective rational variety of dimension $n$, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic…

We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long standing open question if a smooth complex projective rational surface has…

For each nonnegative integer m we show that any closed, oriented topological four-manifold with fundamental group Z_{4m+2} and odd intersection form, with possibly seven exceptions, either admits no smooth structure or admits infinitely…

We prove that nonsingular retract rational algebraic varieties over any infinite field are uniformly retract rational. As a consequence, every rational, projective, nonsingular complex variety is algebraically elliptic.

Let $n=2,3,4,5$ and let $X$ be a smooth complex projective hypersurface of $\mathbb P^{n+1}$. In this paper we find an effective lower bound for the degree of $X$, such that every holomorphic entire curve in $X$ must satisfy an algebraic…

We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $\mathbb{S}^1$ up to equivariant isomorphisms. As an application, we show that every…

We prove that there are no rigid complex filiform Lie algebras in the variety of (filiform) Lie algebras of dimension less than or equal to 11. More precisely we show that in any Euclidean neighborhood of a filiform Lie bracket (of low…

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real…

We construct several rigid (i.e., unique in their deformation class) surfaces which have particular behavior with respect to real structures: in one example the surface has no any real structure, in the other one it has a unique real…

In this paper, we prove the following differential analog of the Noether normalization lemma: for every $d$-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map…

We study ruled real hypersurfaces whose shape operators have constant squared norm in nonflat complex space forms. In particular, we prove the nonexistence of such hypersurfaces in the projective case. We also show that biharmonic ruled…

In this article, we construct differential modular forms for compact Shimura curves over totally real fields bigger than rational of non-zero integral weights that is not classical (of order zero) generalizing the construction of Buium [8].

In this work, we construct the algebra of differential forms with the cube of exterior differential equal to zero on one-dimensional space. We prove that this algebra is a graded q-differential algebra where q is a cubic root of unity.…