Algebraic cuts

In this note we give an algebraic version of a construction called symplectic cutting, which is due to Lerman. Our construction is valid for projective varieties defined over arbitrary fields. Using the equivariant intersection theory developed by the authors, it is a useful tool for studying quotients by torus actions. At the end of the paper, we give an algebraic proof of the Kalkman residue formula and use it to give some formulas for characteristic numbers of quotients.

     

Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kähler quotient.     

Scaling of conformal blocks and generalized theta functions over $$\overline{\mathcal {M}}_{g,n}$$

By way of intersection theory on \(\overline{\mathcal {M}}_{g,n}\), we show that geometric interpretations for conformal blocks, as sections of ample line bundles over projective varieties, do not have to hold at points on the boundary. We show such a translation would imply certain recursion relations for first Chern classes of these bundles. While recursions can fail, geometric interpretations are shown to hold under certain conditions.     

The GIT-Equivalence for G-Line Bundles

Let X be a projective variety with an action of a reductive group G. Each ample G-line bundle L on X defines an open subset X^ss(L) of semi-stable points. Following Dolgachev and Hu, define a GIT-class as the set of algebraic equivalence classes of L's with fixed X^ssL. We show that the GIT-classes are the relative interiors of rational polyhedral convex cones, which form a fan in the G-ample cone. We also study the corresponding variations of quotients X^ss(L)//G. This sharpens results of Thaddeus and Dolgachev-Hu.     

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group G^Mis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ? 2. A Galois cover f :X ? Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X)= G.     

Bubble tree compactification of moduli spaces of vector bundles on surfaces

We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c ₁ = 0, c ₁ = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.     

Toric Kempf-Ness sets

In the theory of algebraic group actions on affine varieties, the concept of a Kempf-Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. Using recent achievements of “toric topology,” we show that an appropriate notion of a Kempf-Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arrangement complements) arising in the Batyrev-Cox “geometric invariant theory” approach to toric varieties. We proceed by studying the cohomology of these “toric” Kempf-Ness sets. In the case of projective nonsingular toric varieties the Kempf-Ness sets can be described as complete intersections of real quadrics in a complex space. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 159–172.     

Open subsets of projective spaces with a good quotient by an action of a reductive group

The aim of the paper is to describe all open subsets of a projective space with an action of a reductive group which admits a good quotient. As in the case of Mumford’s geometric invariant theory (which concerns projective good quotients) the problem can be reduced to the case of an action of a torus. We also show how to distinguish examples of open subsets with a good quotient coming from Mumford’s theory and give examples of open subsets with non-quasi-projective quotients.     

On the number of Frobenius-trivial vector bundles on specific curves

This note investigates, through direct computational methods, the existence of infinitely many isomorphism classes of stable vector bundles which become trivial after being pulled back by the Frobenius morphism. We obtain examples, in characteristic two, where infinitely many such isomorphism classes exist. In characteristic three, however, the computations show that the aforementioned sets are finite.     

Mumford's example and a general construction

We construct a line bundle on a complex projective manifold (a general ruled variety over a curve) which is not ample, but whose restriction to every proper subvariety is ample. This example is of interest in connection with ampleness questions of vector bundles on varieties of dimension greater than one. The method of construction shows that a stable bundle of positive degree on a curve is ample. The example can be used to show that there is no restriction theorem for Bogomolov stability.     

GIT-equivalence and diagonal actions

In the paper, the GIT-equivalence classes of the linearized linear bundles for the diagonal actions of the linear algebraic groups SL(V) and SO(V) on the projective varieties ?(V) ^m ₁ × ?(V*) ^m ₂ and ?(V) ^m , respectively, are described explicitly.     

Parameters of AG codes from vector bundles

We investigate the parameters of the algebraic–geometric codes constructed from vector bundles on a projective variety defined over a finite field. In the case of curves we give a method of constructing weakly stable bundles using restriction of vector bundles on algebraic surfaces and illustrate the result by some examples.     

An algebraic version of Demailly's asymptotic Morse inequalities

We give an elementary algebraic proof of some asymptotic estimates (called by Demailly asymptotic Morse inequalities) for the dimensions of cohomology groups of the difference of two ample line bundles on a smooth complex projective variety of any dimension.

     

Positive polynomials on projective limits of real algebraic varieties

We reveal some important geometric aspects related to non-convex optimization of sparse polynomials. The main result, a Positivstellensatz on the fibre product of real algebraic affine varieties, is iterated to a comprehensive class of projective limits of such varieties. This framework includes as necessary ingredients recent works on the multivariate moment problem, disintegration and projective limits of probability measures and basic techniques of the theory of locally convex vector spaces. A variety of applications illustrate the versatility of this novel geometric approach to polynomial optimization.     

Some Koszul rings from geometry

We give examples of Koszul rings that arise naturally in algebraic geometry. In the first part, we prove a general result on Koszul property associated to an ample line bundle on a projective variety. Specifically, we show how Koszul property of multiples of a base point free ample line bundle depends on its Castelnuovo–Mumford regularity. In the second part, we give examples of Koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type.     

Finite Equivalence Relations on Algebraic Varieties and Hidden Symmetries

This paper can be considered as a continuation of Miyanishi's paper which contains a theorem on existence of a quotient of an affine normal or a projective smooth variety by a finite equivalence relation such that every component of the relation projects onto the variety (we call such an equivalence relation a wide finite equivalence relation). Later papers of Kollar and Keel-Mori shed new light on the subject and can serve as a base for further studies. The results of the present paper are based on the fact that every wide finite equivalence relation on a normal variety V is determined by an action of a finite group on the normalization of V in some Galois extension of k(V). Hence, such an equivalence relation hides some symmetry of a (ramified) cover of V. One may find some analogy of the situation with the concept of a hidden symmetry considered in physics. An important part of the paper is examples described in Section 6 which show that the main result of the paper (Theorem 2.3) is valid neither in the seminormal case, nor under the additional assumptions that there exists a finite morphism whose fibers contain equivalence classes of a given finite relation. In the nonnormal case, identification of some points described by a finite wide equivalence relation may force identification of some other nonequivalent points. This seems to show that the class of normal varieties and wide equivalence relation is a proper frame for considering the general problems of quotients by finite equivalence relations.     

Locally trivial 𝔾a−actions on ℂ5 with singular algebraic quotients

Simple examples are given of proper algebraic actions of the additive group of complex numbers on ?⁵ whose geometric quotients are, respectively, a?ne, strictly quasia?ne, and algebraic spaces which are not schemes. Moreover, a Zariski locally trivial action is given whose ring of invariant regular functions defines a singular factorial a?ne fourfold embedded in ?¹². The geometric quotient for the action embeds as a strictly quasia?ne variety in the smooth locus of the algebraic quotient with complement isomorphic to the normal a?ne surface with the A₂?singularity at the origin.     

Mod 2 cohomology of combinatorial Grassmannians

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles. It defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles. It then gives a transformation from matroid bundles to spherical quasifibrations, by showing that the geometric realization of a matroid bundle is a spherical quasifibration. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. A consequence is that the mod 2 cohomology of the poset of rank k oriented matroids (this poset classifies matroid bundles) contains the free polynomial ring on the first k Stiefel-Whitney classes.     

Codes associated to the zero-schemes of sections of vector bundles

We consider the algebraic geometric codes associated to the zero-schemes of sections of vector bundles on a smooth projective variety. We give lower bounds for the minimum distances of the codes exploiting the Cayley–Bacharach property of zero-dimensional subschemes.