Algebraic Varieties Research Articles

The Riemann–Roch theorem for the Adams operations

We prove the classical Riemann–Roch theorems for the Adams operations ψj on K-theory: a statement with coefficients on Z[j−1], that holds for arbitrary projective morphisms, as well as another statement with integral coefficients, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann–Roch formula for one Adams operation and the analogous formula for the Chern character. To do so, we complete the elementary exposition of the work of Panin–Smirnov that was initiated by the first author in a previous paper. Their notion of oriented cohomology theory on algebraic varieties allows to use classical arguments to prove general and neat statements, which imply all the aforementioned results as particular cases.

Dynamics of Chebyshev endomorphisms on some affine algebraic varieties

The Chebyshev polynomials Td in one variable are typical chaotic maps on C. Chebyshev endomorphisms PAnd: Cn→Cn are also chaotic. We consider the action of the dihedral group Dn+1 on Cn. The endomorphism PAnd maps any Dn+1-orbit of z∈Cn to a Dn+1-orbit of PAnd(z). The endomorphism PAnd induces a mapping on Cn∕Dn+1. Using invariant theory, we embed Cn∕Dn+1 as an affine subvariety X in Cm. Then we have morphisms gd on X. We study the cases n=2 and 3. In these cases, the morphisms gd are defined over Z. We find a class of affine subvarieties V of X which are invariant under gd. These varieties are concerned with branch loci or critical loci. The class contains C2, a cuspidal cubic, a parabola, a quadric hypersurface in C4, an affine algebraic surface in C4 which is birationally equivalent to an affine quadric cone in C3, and others. For each affine variety V in the class, there exists a polynomial parameterization PV satisfying gd|V(PV(y1,…,yk))=PV(Td(y1),…,Td(yk)), where Td(z) is a Chebyshev polynomial in one variable. Then we determine the set of bounded orbits of gd|V in each invariant set V and give relations between them.

Nash conditional independence curve

We study the Spohn conditional independence (CI) variety CX of an n-player game X for undirected graphical models on n binary random variables consisting of one edge. For a generic game, we show that CX is a smooth irreducible complete intersection curve (Nash conditional independence curve) in the Segre variety (P1)n−2×P3 and we give an explicit formula for its degree and genus. We prove two universality theorems for CX: The product of any affine real algebraic variety with the real line or any affine real algebraic variety in Rm defined by at most m−1 polynomials is isomorphic to an affine open subset of CX for some game X.

Noninvariance of Weak Approximation Properties Under Extension of the Ground Field

For rational points on algebraic varieties defined over a number field K, we study the behavior of the property of weak approximation with Brauer–Manin obstruction under extension of the ground field. We construct K-varieties accompanied with a quadratic extension L|K such that the property holds over K (conditionally on a conjecture) whereas fails over L. The result is unconditional when K equals Q or certain quadratic number fields. We give an explicit example when K=Q.

Representations via differential algebras and equationally Noetherian algebras

We show that free algebras of the variety of algebras generated by the Witt algebra Wn, the left-symmetric Witt algebra Ln, and the symplectic Poisson algebra Pn can be described as subalgebras of differential polynomial algebras with respect to appropriately defined products. Using these representations, we prove that Wn, Ln, Pn, and the free algebras of the varieties of algebras generated by these algebras are equationally Noetherian.

Definable structures on flat bundles

AbstractA flat vector bundle on an algebraic variety supports two natural definable structures given by the flat and algebraic coordinates. In this note, we show these two structures are compatible, subject to a condition on the local monodromy at infinity that is satisfied for all flat bundles underlying variations of Hodge structures.

On the geometry of the Hermitian Veronese curve and its quasi-Hermitian surfaces

The complete classification of the orbits on subspaces under the action of the projective stabilizer of (classical) algebraic varieties is a challenging task, and few classifications are complete. We focus on a particular action of PGL(2,q2) (and PSL(2,q2)) arising from the Hermitian Veronese curve in PG(3,q2), a maximal rational curve embedded on a smooth Hermitian surface with some fascinating properties. The study of its orbits leads to a new construction of quasi-Hermitian surfaces: sets of points with the same combinatorial and geometric properties as a non-degenerate Hermitian surface.

The orbifold Hochschild product for Fermat hypersurface

Let G be an abelian group acting on a smooth algebraic variety X. We investigate the product structure and the bigrading on the cohomology of polyvector fields on the orbifold [X/G], as introduced by Căldăraru and Huang. In this paper, we provide many new examples given by quotients of Fermat hypersurfaces, where the product is shown to be associative. This is expected due to the conjectural isomorphism at the level of algebras between the cohomology of polyvector fields and the Hochschild cohomology of orbifolds. We prove this conjecture for the Calabi-Yau Fermat hypersurface orbifold. We also show that for Calabi-Yau orbifolds, the multiplicative bigrading on the cohomology of polyvector fields agrees with what is expected in homological mirror symmetry.

Categorical action filtrations via localization and the growth as a symplectic invariant

Abstract We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A “quantitative” refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories.

On character varieties of singular manifolds

In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of G-representation varieties over manifolds with conic singularities, which we call nodefolds. This construction is valid for any algebraic group G, in any dimension and also in the parabolic setting. In particular, this TQFT allows us to compute the virtual classes of representation varieties over complex singular planar curves. In addition, in the case G = textrm{SL}_{2}(k), the virtual class of the associated character variety over a nodal closed orientable surface is computed both in the non-parabolic and parabolic scenarios.

CONDITIONAL FLATNESS, FIBERWISE LOCALIZATIONS, AND ADMISSIBLE REFLECTIONS

Abstract We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localization, analogous results to those obtained in the category of groups hold, and we provide existence theorems for certain localization functors in specific semi-abelian categories. We prove that a Birkhoff subcategory of an ideal determined category yields a conditionally flat localization, and explain how conditional flatness corresponds to the property of admissibility of an adjunction from the point of view of categorical Galois theory. Under the assumption of fiberwise localization, we give a simple criterion to determine when a (normal epi)-reflection is a torsion-free reflection. This is shown to apply, in particular, to nullification functors in any semi-abelian variety of universal algebras. We also relate semi-left-exactness for a localization functor L with what is called right properness for the L-local model structure.

Algebraic and o-Minimal Flows Beyond the Cocompact Case

Abstract Let $X \subset \mathbb{C}^{n}$ be an algebraic variety, and let $\Lambda \subset \mathbb{C}^{n}$ be a discrete subgroup whose real and complex spans agree. We describe the topological closure of the image of $X$ in $\mathbb{C}^{n} / \Lambda $, thereby extending a result of Peterzil–Starchenko in the case when $\Lambda $ is cocompact. We also obtain a similar extension when $X\subset \mathbb{R}^{n}$ is definable in an o-minimal structure with no restrictions on $\Lambda $, and as an application prove the following conjecture of Gallinaro: for a closed semi-algebraic $X\subset \mathbb{C}^{n}$ (such as a complex algebraic variety) and $\exp :\mathbb{C}^{n}\to (\mathbb{C}^{*})^{n}$ the coordinate-wise exponential map, we have $\overline{\exp (X)}=\exp (X)\cup \bigcup _{i=1}^{m} \exp (C_{i})\cdot \mathbb{T}_{i}$ where $\mathbb{T}_{i}\subset (\mathbb{C}^{*})^{n}$ are positive-dimensional compact real tori and $C_{i}\subset \mathbb{C}^{n}$ are semi-algebraic.

On Some Systems of Equations in Abelian Varieties

Abstract We solve a case of the Abelian Exponential-Algebraic Closedness Conjecture, a conjecture due to Bays and Kirby, building on work of Zilber, which predicts sufficient conditions for systems of equations involving algebraic operations and the exponential map of an abelian variety to be solvable in the complex numbers. More precisely, we show that the conjecture holds for subvarieties of the tangent bundle of an abelian variety $A$, which split as the product of a linear subspace of the Lie algebra of $A$ and an algebraic variety. This is motivated by work of Zilber and of Bays–Kirby, which establishes that a positive answer to the conjecture would imply quasiminimality of certain structures on the complex numbers. Our proofs use various techniques from homology (duality between cup product and intersection), differential topology (transversality), and o-minimality (definability of Hausdorff limits), hence we have tried to give a self-contained exposition.

Dominating real algebraic morphisms

Let X and Y be nonsingular real algebraic varieties, dim⁡X≥dim⁡Y. Assume that the variety Y is malleable, compact and connected. Our main result implies that each regular map from X to Y is homotopic to a surjective regular map. The class of malleable varieties includes all homogeneous spaces for linear real algebraic groups.

Measures of association between algebraic varieties

Measures of association between algebraic varieties

Around Exponential-Algebraic Closedness

AbstractWe present some results related to Zilber’s Exponential-Algebraic Closedness Conjecture, showing that various systems of equations involving algebraic operations and certain analytic functions admit solutions in the complex numbers. These results are inspired by Zilber’s theorems on raising to powers.We show that algebraic varieties which split as a product of a linear subspace of an additive group and an algebraic subvariety of a multiplicative group intersect the graph of the exponential function, provided that they satisfy Zilber’s freeness and rotundity conditions, using techniques from tropical geometry.We then move on to prove a similar theorem, establishing that varieties which split as a product of a linear subspace and a subvariety of an abelian variety A intersect the graph of the exponential map of A (again under the analogues of the freeness and rotundity conditions). The proof uses homology and cohomology of manifolds.Finally, we show that the graph of the modular j-function intersects varieties which satisfy freeness and broadness and split as a product of a Möbius subvariety of a power of the upper-half plane and a complex algebraic variety, using Ratner’s orbit closure theorem to study the images under j of Möbius varieties.Abstract prepared by Francesco Paolo GallinaroE-mail: francesco.gallinaro@mathematik.uni-freiburg.deURL: https://etheses.whiterose.ac.uk/31077/

MSSM: A Macaulay2 Package for the Vacuum Moduli Space

The Standard Model of particle physics has been amazingly successful at explaining interactions between three of the four fundamental forces of physics: the strong force, the weak force, and electro-magnetism. But it is an incomplete theory, failing to incorporate gravity. The Minimal Supersymmetric Standard Model (MSSM) is an approach to including gravity and supersymmetry in the standard model, by adding new particle states and interactions to the standard model. For example, supersymmetry pairs fermions (such as electrons) with bosons (such as photons). A starting point in the study of the MSSM is the Vacuum Moduli Space, which is a highly complicated algebraic variety introduced by Witten: it is the image of an affine variety X ⊆ C 49 under a symplectic quotient map to C 973 . We describe the Macaulay2 software package MSSM, which facilitates computational investigation of the Vacuum Moduli Space.

Arithmetic of Châtelet surface bundles revisited

Abstract We study the arithmetic of algebraic varieties defined over number fields by applying Lagrange interpolation to fibrations. Assuming the finiteness of the Tate–Shafarevich group of a certain elliptic curve, we show, for Châtelet surface bundles over curves, that the violation of Hasse principle being accounted for by the Brauer–Manin obstruction is not invariant under an arbitrary finite extension of the ground field.

On the exceptionality of rational APN functions

We investigate APN functions which can be represented as rational functions and we provide non-existence results exploiting the connection between these functions and specific algebraic varieties over finite fields. This approach allows to classify families of functions when previous approaches cannot be applied.

An Effective Criterion for Nielsen–Schreier Varieties

Abstract To the memory of V. A. Artamonov (1946–2021) All algebras of a certain type are said to form a Nielsen–Schreier variety if every subalgebra of a free algebra is free. This property has been perceived as extremely rare; in particular, only six Nielsen–Schreier varieties of algebras with one binary operation have been discovered in prior work on this topic. We propose an effective combinatorial criterion for the Nielsen–Schreier property in the case of algebras over a field of zero characteristic; in our approach, operads play a crucial role. Using this criterion, we show that the well-known varieties of all pre-Lie algebras and of all Lie-admissible algebras are Nielsen–Schreier, and, quite surprisingly, that there are already infinitely many non-equivalent Nielsen–Schreier varieties of algebras with one binary operation and identities of degree four.