Let Y be a normal crossing divisor in the smooth projective algebraic variety X (defined over ${\mathbb C}$) and let U be a tubular neighbourhood of Y in X. We construct homological cycles generating $H_*(A,B)$, where (A,B) is one of the following pairs $(Y,\emptyset)$, (X,Y), (X,X-Y), $(X-Y,\emptyset)$ and $(\partial U,\emptyset)$. The construction is compatible with the weights in $H_*(A,B,{\mathbb Q})$ of Deligne's mixed Hodge structure.
In this paper we study certain homogeneous spaces, which we call generalised affine Stiefel algebraic varieties. The main aim is to characterise the canonical divisor of generalised affine Stiefel algebraic varieties in terms of group representations. Affine Stiefel algebraic varieties and in particular $S^{n}$ are two special cases of the generalised affine Stiefel algebraic varieties.
We study Minkowski sums and Hadamard products of algebraic varieties. Specifically we explore when these are varieties and examine their properties in terms of those of the original varieties.
We generalize classical results about the topology of toric varieties to the case of projective Q-factorial T-varieties of complexity one using the language of divisorial fans. We describe the Hodge-Deligne polynomial in the smooth case, the cohomology ring and the Chow ring in the contraction-free case.
We give sufficient cohomological criteria for the classes of given varieties over a field $k$ to be algebraically independent in the Grothendieck ring of varieties over $k$ and construct some examples.
We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small deformation of the coefficients of the original equations. This method is based on the properties of Zariski equisingular families of varieties. Moreover we construct an algorithm, that, given a system of equations defining a variety $V$, produces a system of equations with algebraic coefficients of a variety homeomorphic to $V$
We develop a moduli theory of algebraic varieties and pairs of non-negative Kodaira dimension. We define stable minimal models and construct their projective coarse moduli spaces under certain natural conditions. This can be applied to a wide range of moduli problems in algebraic geometry.
We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in th…
In this paper, we establish the following version at infinity of Whitney's theorem [6, 7]: Geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of geometric tangent cones at infinity and using the global Łojasiewicz inequality with explicit exponents for complex algebraic varieties. We also show that tangent cones at infinity of complex algebraic varieties can be computed using Gröbner bases.
This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern--Schwartz--MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity (EIDS). The formula is given in terms of the local Euler obstruction and Gaffney's $m_{d}$ multiplicity.
We extend the Altmann-Hausen presentation of normal affine algebraic C-varieties endowed with effective torus actions to the real setting. In particular, we focus on actions of quasi-split real tori, in which case we obtain a simpler presentation.
This is a report on some of the main developments in birational geometry in the last few years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.
This article proposes a generalization of tautological rings introduced by Beauville and Moonen for Jacobians. The main result is that, under certain hypotheses, the special subvarieties of Prym varieties are algebraically equivalent and their classes belong to the tautological ring.
We discuss relations between certain invariants of varieties in positive characteristic, like the a-number and the height of the Artin-Mazur formal group. We calculate the a-number for Fermat surfaces
We define the class of admissible linear embeddings of flag varieties. The definition is given in the general language of algebraic geometry. We then prove that an admissible linear embedding of flag varieties has a certain explicit form in terms of linear algebra. This result enables us to show that any direct limit of admissible embeddings of flag varieties is isomorphic to an ind-variety of generalized flags as defined in [DP]. These latter ind-varieties have been introduced in terms of the ind-group SL(\infty) (respectively, O(\infty) or Sp(\infty) for isotropic generalized flags), and the…
We give a short appreciation of Mumford's work on the moduli of varieties by putting it into historical context. By reviewing earlier works we highlight the innovations introduced by Mumford. Then we discuss recent developments whose origins can be traced back to Mumford's ideas.
We prove that a quasi-finite endomorphism of an algebraic variety over an algebraically closed field of characteristic zero, that is injective on the complement of a closed subvariety, is an automorphism. We also prove that an endomorphism of complex algebraic variety that is injective on the complement of a closed subvariety of codimension at least $2$, is an automorphism.
The weak factorization theorem for birational maps is used to prove that for all nonnegative i the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a "virtual Betti number" beta_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then beta_i(X) = beta_i(X\Y) + beta_i(Y).
This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants.
Lee and Pandharipande studied a "double point" algebraic cobordism theory of varieties equipped with vector bundles, and speculated that some features of that story might extend to the case of varieties with principal G-bundles. This note shows that this expectation holds rationally, and more generally after inverting the torsion index of the group, for reductive G. We show that (after inverting the torsion index) the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism, that the theory for a point is dual to Omega^*(BG), and describe how the theory f…
Connected components of real algebraic varieties invariant under the $CB_{n}$-Coxeter group are investigated. In particular, we consider their maximal number and their geometric and topological properties. This provides a decomposition for the space of $CB_{n}$-algebraic varieties. We construct $CB_{n}$-polynomials using Young-posets and partitions of integers. Our results establish bounds on the number of connected components for a given set of coefficients. It turns out that this number can achieve an upper bound of $2^{n}+1$ for specific coefficients. We introduce a new method to characteri…
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.
We extend Kollar's technique to look for an explicit function h(n) with phi_m birational onto its image for all integers $m\geq h(n)$ and for all n-dimensional nonsingular projective varieties of general type.
In this paper we study a notion of volume for Cartier divisors on arbitrary blow-ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study a volume for normal isolated singularities, generalizing work on surfaces done by Wahl. We also compare this volume of isolated singularities to a different generalization due to Boucksom, de Fernex and Favre.
We associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on classical (compactly supported) homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties. This complements our previous definition of the weight filtration of Borel-Moore homology.
Let $k$ be a field of characteristic $p>0$ not necessarily perfect. Using Berthelot's theory of arithmetic $\mathcal{D}$-modules, we construct a $p$-adic formalism of Grothendieck's six operations for realizable $k$-schemes of finite type.
We extend Kolchin's results on linear dependence over projective varieties in the constants, to linear dependence over arbitrary complete differential varieties. We show that in this more general setting, the notion of linear dependence still has necessary and sufficient conditions given by the vanishing of a certain system of differential polynomials equations. We also discuss some conjectural questions around completeness and the catenary problem.
We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than $d$, and for the size of the sum of $\mod 2$ Betti numbers for the real form of complex manifolds of complex degree less than $d$.
We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.
We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety, obtained by pinching a smooth variety along a finite subscheme. In contrast, no Witt group of dimension at least 3 over a perfect field of prime characteristic is isogenous to the Picard variety obtained by this construction.
This paper has been withdrawn by the author. It is not true that plt blow-up of toric singularity is toric (in dimension 3 too) and the first main theorem is incorrect also. The error is in deformation argument. I am grateful to Ivan Cheltsov for discussions.
For a smooth manifold $X$ equipped with a volume form, let $\dL$ be the Lie algebra of volume preserving smooth vector fields on $X$. A. Lichnerowicz proved that the abelianization of $\dL$ is a finite-dimensional vector space, and that its dimension depends only on the topology of $X$. In this paper we provide analogous results for some classical examples of non-singular complex affine algebraic varieties that admit a nowhere-zero algebraic form of top degree (which plays the role of a volume form).
This note is about cycle-theoretic properties of the Fano variety of lines on a smooth cubic fivefold. The arguments are based on the fact that this Fano variety has finite-dimensional motive. We also present some results concerning Chow groups of Fano varieties of lines on certain cubics in other dimensions.
We present a modern proof of a theorem of Rosenlicht, asserting that every variety as in the title is isomorphic to a product of affine lines and punctured affine lines.
We discuss the basic properties of various versions of two variable elliptic genus with special attention to the equivariant elliptic genus. The main applications are to the elliptic genera attached to non-compact GITs, including the elliptic genera of Witten's phases on $N=2$ theories.
