Algebraic Varieties Research Articles

Transfer and the spectrum-level Siegel–Sullivan KO-orientation for singular spaces

Integrally oriented normally nonsingular maps between singular spaces have associated transfer homomorphisms on KO-homology at odd primes. We prove that such transfers preserve Siegel–Sullivan orientations, defined when the singular spaces are compact pseudomanifolds satisfying the Witt condition, for example pure-dimensional compact complex algebraic varieties. This holds for bundle transfers associated to block bundles with manifold fibers as well as for Gysin restrictions associated to normally nonsingular inclusions. Our method is based on constructing a lift of the Siegel–Sullivan orientation to a morphism of highly structured ring spectra which factors through L-theory.

Images of dominant endomorphisms of affine space

A basic problem in algebraic morphisms is which sets can be the image of an endomorphism of affine space. This paper extends the results previously obtained by the first author on the question of existence of surjective maps F : A n → A n ∖ Z , where Z is an algebraic subvariety of A n of codimension at least 2. In particular, we show that for any (affine) algebraic variety Z of dimension at most n − 2 , there is an algebraic variety W ⊂ A n birational to Z and a surjective algebraic morphism A n → A n ∖ W . We also propose a conjectural approach toward resolving unknown cases.

Symplectic Grassmannians and cyclic quivers

AbstractThe goal of this paper is to extend the quiver Grassmannian description of certain degenerations of Grassmann varieties to the symplectic case. We introduce a symplectic version of quiver Grassmannians studied in our previous papers and prove a number of results on these projective algebraic varieties. First, we construct a cellular decomposition of the symplectic quiver Grassmannians in question and develop combinatorics needed to compute Euler characteristics and Poincaré polynomials. Second, we show that the number of irreducible components of our varieties coincides with the Euler characteristic of the classical symplectic Grassmannians. Third, we describe the automorphism groups of the underlying symplectic quiver representations and show that the cells are the orbits of this group. Lastly, we provide an embedding into the affine flag varieties for the affine symplectic group.

Gromov-Witten Invariants in Complex and Morava-Local K-Theories

AbstractGiven a closed symplectic manifold X, we construct Gromov-Witten-type invariants valued both in (complex) K-theory and in any complex-oriented cohomology theory $\mathbb{K}$ which is Kp(n)-local for some Morava K-theory Kp(n). We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee’s work for the quantum K-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantum K-theory and quantum $\mathbb{K}$-theory as commutative deformations of the corresponding (generalised) cohomology rings of X; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input of these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus to X. On the algebraic side, in order to establish a common framework covering both ordinary K-theory and Kp(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts.

Smooth connectivity in real algebraic varieties

AbstractA standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. This manuscript provides algorithms for computing the number of connected components, the Euler characteristic, and deciding the connectivity between two points for a smooth manifold arising as the complement of a real hypersurface of a real algebraic variety. When considering the complement of the set of singular points of a real algebraic variety, this yields an approach for determining smooth connectivity in a real algebraic variety. The method is based upon gradient ascent/descent paths on the real algebraic variety inspired by a method proposed by Hong, Rohal, Safey El Din, and Schost for complements of real hypersurfaces. Several examples are included to demonstrate the approach.

Adding an implication to logics of perfect paradefinite algebras

Abstract Perfect paradefinite algebras are De Morgan algebras expanded with an operation that allows for the full behavior of classical negation to be restored. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and single-conclusion () order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix. We studied these logics extensively in Gomes et al. ((2022). Electronic Proceedings in Theoretical Computer Science357 56–76.) from both the algebraic and the proof-theoretical perspectives. In the present paper, we continue that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand the language of the latter algebra with this connective and study the (self-extensional) Set-Set and order-preserving and $\top$ -assertional logics of the variety induced by the resulting algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil’s “symmetric modal logic.” For the order-preserving Set-Set logic, in particular, we obtain a Set-Set axiomatization that is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property.

Approximation of semistable bundles on smooth algebraic varieties

We prove some strong results on approximation of strongly semistable bundles with vanishing numerical Chern classes by filtrations, whose quotients are line bundles of similar slope. This generalizes some earlier results of Parameswaran–Subramanian in the curve case and Koley–Parameswaran in the surface case and it confirms the conjecture posed by Koley and Parameswaran.

Hilbert 17th property and central cones

This paper is dedicated to the study of the converse implication in Hilbert 17th problem for a general commutative ring. We introduce the notions of central and precentral prime cones which generalize the notion of central real points of irreducible real algebraic varieties. We study these families of prime cones which both live in the real spectrum of the ring and allow to state new Positivstellensätze and to obtain an equivalence in Hilbert 17th problem.

Plücker coordinates and the Rosenfeld planes

The exceptional compact hermitian symmetric space EIII is the quotient E6/Spin(10)×Z4U(1). We introduce the Plücker coordinates which give an embedding of EIII into CP26 as a projective subvariety. The subvariety is cut out by 27 Plücker relations. We show that, using Clifford algebra, one can solve this over-determined system of relations, giving local coordinate charts to the space.Our motivation is to understand EIII as the complex projective octonion plane (C⊗O)P2, whose construction is somewhat scattered across the literature. We will see that the EIII has an atlas whose transition functions have clear octonion interpretations, apart from those covering a sub-variety X∞ of dimension 10. This subvariety is itself a hermitian symmetric space known as DIII, with no apparent octonion interpretation. We give detailed analysis of the geometry in the neighbourhood of X∞.We further decompose X=EIII into F4-orbits: X=Y0∪Y∞, where Y0∼(OP2)C is an open F4-orbit and is the complexification of OP2, whereas Y∞ has co-dimension 1, thus EIII could be more appropriately denoted as (OP2)C‾. This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer [2].

The Brasselet–Schürmann–Yokura Conjecture on L-Classes of Projective Rational Homology Manifolds

Abstract In 2010, Brasselet, Schürmann, and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky–MacPherson $L$-class $L_{*}(X)$ and the Hirzebruch homology class $T_{1,*}(X)$ for a compact complex algebraic variety $X$ that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions that we find of independent interest.

On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn

In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside étale cohomology of certain algebraic varieties. Recently, a p-adic version of this theory started to emerge: there are p-adic Deligne–Lusztig spaces, whose cohomology encodes representation theoretic information for p-adic groups – for instance, it partially realizes the local Langlands correspondence with characteristic zero coefficients. However, the parallel case of coefficients of positive characteristic ℓ≠p has not been inspected so far. The purpose of this article is to initiate such an inspection. In particular, we relate cohomology of certain p-adic Deligne–Lusztig spaces to Vignéras's modular local Langlands correspondence for GLn.

Difference Schemes for Differential Equations with a Polynomial Right-Hand Side, Defining Birational Correspondences

This paper explores the numerical intergator of ODE based on combination of Appelroth’s quadratization of dynamical systems with polynomial right-hand sides and Kahan’s discretization method. Utilizing Appelroth’s technique, we reduce any system of ordinary differential equations with a polynomial right-hand side to a quadratic form, enabling the application of Kahan’s method. In this way, we get a difference scheme defining the one-to-one correspondence between the initial and final positions of the system (Cremona map). It provides important information about the Kahan method for differential equations with a quadratic right-hand side, because we obtain dynamical systems with a quadratic right-hand side that have movable branch points. We analyze algebraic properties of solutions obtained through this approach, showing that (1) the Kahan scheme describes the branch points as poles, significantly deviating from the behavior of the exact solution of the problem near these points, and (2) it disrupts algebraic invariant variety, in particular integral relations describing the relationship between old and Appelroth’s variables. This study advances numerical methods, emphasizing the possibility of designing difference schemes whose algebraic properties differ significantly from those of the initial dynamical system.

Introduction to Algebraic Deformation Theory and The Case of k-Linear Categories

Deformation theory is a branch of mathematics which studies how mathematical objects, such as algebraic varieties, schemes, algebras, or categories, can be deformed “continuously” depending on a space of parameter while preserving certain algebraic or geometric structures. Algebraic deformation theory, which was pioneered by Murray Gerstenhaber in 1960s-1970s, established its role as a cornerstone in modern mathematics and theoretical physics. This theory provides a powerful framework for understanding the subtle variations and deformations of mathematical and physical objects depending on a parameter space. Lying at the interface of algebra, geometry and topology this theory has been being studied extensively worldwide and obtained many applications in various areas of mathematics and theoretical physics, such as the study of Calabi-Yau manifolds, mirror symmetry, quantum physics. In such context, this article aims to introduce this important mathematical theory to the community of Vietnamese mathematicians in a hope to bring more attention of Vietnamese mathematicians and math students to this vibrant research area. We also expect that this topic will be taught at universities in Vietnam in the near future.

The moment map for the variety of associative algebras

The moment map for the variety of associative algebras

Representations of complex tori and GL(2, C)

Groups and their representations have been studied for a long time. One can extend the notion of a group by asking the group axioms to hold in other categories. A group in the category of smooth manifolds is a Lie group, and a group in the category of algebraic varieties is an algebraic group. In this paper, we discuss the representation theory of algebraic groups, in particular complex tori and GL(2,C).

Symbolic computation in algebra, geometry, and differential equations

In this survey article we describe how symbolic computation in algebra and geometry leads to symbolic, i.e., formula solutions of algebraic differential equations. Symbolic solutions of algebraic differential equations can be derived from parametrizations of corresponding algebraic varieties. Such parametrizations in turn can be computed by elimination methods, i.e., methods for solving systems of polynomial equations.

On the integral part of A-motivic cohomology

The deepest arithmetic invariants attached to an algebraic variety defined over a number field $F$ are conjecturally captured by the integral part of its motivic cohomology. There are essentially two ways of defining it when $X$ is a smooth projective variety: one is via the $K$ -theory of a regular integral model, the other is through its $\ell$ -adic realization. Both approaches are conjectured to coincide. This paper initiates the study of motivic cohomology for global fields of positive characteristic, hereafter named $A$ -motivic cohomology, where classical mixed motives are replaced by mixed Anderson $A$ -motives. Our main objective is to set the definitions of the integral part and the good $\ell$ -adic part of the $A$ -motivic cohomology using Gardeyn's notion of maximal models as the analogue of regular integral models of varieties. Our main result states that the integral part is contained in the good $\ell$ -adic part. As opposed to what is expected in the number field setting, we show that the two approaches do not match in general. We conclude this work by introducing the submodule of regulated extensions of mixed Anderson $A$ -motives, for which we expect the two approaches to match, and solve some particular cases of this expectation.

Algebraic characterizations of homeomorphisms between algebraic varieties

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 a 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 the complex case and which comes from the theory of real closed fields otherwise.

On the subvarieties with nonsingular real loci of a real algebraic variety

On the subvarieties with nonsingular real loci of a real algebraic variety

Algebraic Varieties in Quantum Chemistry

AbstractWe develop algebraic geometry for coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schrödinger equation are approximated by a hierarchy of polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Plücker embedding. We explain how to derive Hamiltonians, we offer a detailed study of truncation varieties and their CC degrees, and we present the state of the art in solving the CC equations.