Zariski Open Subset Research Articles

A Bochner principle and its applications to Fujiki class 𝒞 manifolds with vanishing first Chern class

We prove a Bochner-type vanishing theorem for compact complex manifolds [Formula: see text] in Fujiki class [Formula: see text], with vanishing first Chern class, that admit a cohomology class [Formula: see text] which is numerically effective (nef) and has positive self-intersection (meaning [Formula: see text], where [Formula: see text]). Using it, we prove that all holomorphic geometric structures of affine type on such a manifold [Formula: see text] are locally homogeneous on a non-empty Zariski open subset. Consequently, if the geometric structure is rigid in the sense of Gromov, then the fundamental group of [Formula: see text] must be infinite. In the particular case where the geometric structure is a holomorphic Riemannian metric, we show that the manifold [Formula: see text] admits a finite unramified cover by a complex torus with the property that the pulled back holomorphic Riemannian metric on the torus is translation invariant.

Counting rational points on biquadratic hypersurfaces

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses the Hardy–Littlewood circle method.

Deformations of $$\mathbb {A}^1$$-cylindrical varieties

An algebraic variety is called $$\mathbb {A}^{1}$$ -cylindrical if it contains an $$\mathbb {A}^{1}$$ -cylinder, i.e. a Zariski open subset of the form $$Z\times \mathbb {A}^{1}$$ for some algebraic variety Z. We show that the generic fiber of a family $$f:X\rightarrow S$$ of normal $$\mathbb {A}^{1}$$ -cylindrical varieties becomes $$\mathbb {A}^{1}$$ -cylindrical after a finite extension of the base. This generalizes the main result of Dubouloz and Kishimoto (Nagoya Math J 223:1–20, 2016) which established this property for families of smooth $$\mathbb {A}^{1}$$ -cylindrical affine surfaces. Our second result is a criterion for existence of an $$\mathbb {A}^{1}$$ -cylinder in X which we derive from a careful inspection of a relative Minimal Model Program run from a suitable smooth relative projective model of X over S.

Analytic Continuation of Equivariant Distributions

Abstract We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein’s theory of analytic continuation of holonomic distributions. We use this to construct H-equivariant functionals on principal series representations of G, where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p-adic case using a recent result of Hong and Sun.

Perverse obstructions to flat regular compactifications

Suppose \(\pi :W\rightarrow S\) is a smooth, surjective, proper morphism to a variety S contained as a Zariski open subset in a smooth, complex variety \({\bar{S}}\). The goal of this note is to consider the question of when \(\pi \) admits a regular, flat compactification. In other words, when does there exists a flat, proper morphism \({\bar{\pi }}:\overline{W}\rightarrow {\bar{S}}\) extending \(\pi \) with \(\overline{W}\) regular? One interesting recent example of this occurs in the preprint of Laza et al. (Acta Mathematica. arXiv:1602.05534, 2016) where \(\pi \) is a family of abelian fivefolds over a Zariski open subset S of \({\bar{S}}=\mathbb {P}^5\). In that paper, the authors construct \(\overline{W}\) using the theory of compactified Prym varieties and show that it is a holomorphic symplectic manifold (deformation equivalent to O’Grady’s 10-dimensional example). In this note I observe that non-vanishing of the local intersection cohomology of \(R^1\pi _*\mathbb {Q}\) in degree at least 2 provides an obstruction to finding a \({\bar{\pi }}\). Moreover, non-vanishing in degree 1 provides an obstruction to finding a \({\bar{\pi }}\) with irreducible fibers. Then I observe that, in some cases of interest, results of Brylinski (Asterisque 140–141:3–134, 251, 1986) Beilinson (Regulators 571:19–23, 2012) and Schnell (Math Ann 354(2):727–763, 2012) can be used to compute the intersection cohomology. I also give examples involving cubic fourfolds motivated by Laza et al. (Acta Mathematica. arXiv:1602.05534, 2016) and ask a question about palindromicity of hyperplane sections.

Families of Riemann surfaces, uniformization and arithmeticity

A consequence of the results of Bers and Griffiths on the uniformization of complex algebraic varieties is that the universal cover of a family of Riemann surfaces, with base and fibers of finite hyperbolic type, is a contractible 2 − 2- dimensional domain that can be realized as the graph of a holomorphic motion of the unit disk. In this paper we determine which holomorphic motions give rise to these uniformizing domains and characterize which among them correspond to arithmetic families (i.e. families defined over number fields). Then we apply these results to characterize the arithmeticity of complex surfaces of general type in terms of the biholomorphism class of the 2 − 2- dimensional domains that arise as universal covers of their Zariski open subsets. For the important class of Kodaira fibrations this criterion implies that arithmeticity can be read from the universal cover. All this is very much in contrast with the corresponding situation in complex dimension one, where the universal cover is always the unit disk.

UNIFORMLY RATIONAL VARIETIES WITH TORUS ACTION

A smooth variety is called uniformly rational if every point admits a Zariski open neighborhood isomorphic to a Zariski open subset of the affine space. In this note we show that every smooth and rational affine variety endowed with an algebraic torus action such that the algebraic quotient has dimension 0 or 1 is uniformly rational.

Moduli spaces of framed symplectic and orthogonal bundles on P2 and the K-theoretic Nekrasov partition functions

Let K be the compact Lie group USp ( N / 2 ) or SO ( N , R ) . Let M n K be the moduli space of framed K -instantons over S 4 with the instanton number n . By Donaldson (1984), M n K is endowed with a natural scheme structure. It is a Zariski open subset of a GIT quotient of μ − 1 ( 0 ) , where μ is a holomorphic moment map such that μ − 1 ( 0 ) consists of the ADHM data. The purpose of the paper is to study the geometric properties of μ − 1 ( 0 ) and its GIT quotient, such as complete intersection, irreducibility, reducedness and normality. If K = USp ( N / 2 ) then μ is flat and μ − 1 ( 0 ) is an irreducible normal variety for any n and even N . If K = SO ( N , R ) the similar results are proven for low n and N . As an application one can obtain a mathematical interpretation of the K -theoretic Nekrasov partition function of Nekrasov and Shadchin (2004).

Erratum to: Holonomic D-modules on abelian varieties

Proof. — Since f is smooth, f ∗N = OA ⊗f −1OB f −1N is a holonomic DA-module, and so there is a surjective morphism f ∗N →M to a nontrivial simple holonomic DAmodule M. We will prove the assertion by showing that it is an isomorphism. The support X = SuppN is an irreducible subvariety of B. As N is holonomic, there is a dense Zariski-open subset U ⊆ B such that X ∩ U is nonsingular and such that the restriction N |U is the direct image of a holomorphic vector bundle with integrable connection (E ,∇) on X ∩ U [HTT08, Proposition 3.1.6]. This means that f ∗N is supported on f −1(X), and that its restriction to f −1(U) is the direct image of (f ∗E , f ∗∇). We observe that, on the fibers of f over points of X ∩ U, the latter is a trivial bundle of rank n = rkE . Since M is a quotient of f ∗N , its restriction to f −1(U) is also the direct image of a holomorphic vector bundle with integrable connection on f −1(X ∩ U); as a quotient of (f ∗E , f ∗∇), the restriction of this bundle to the fibers of f must be trivial of some rank k ≤ n. Now let r = dim A−dim B be the relative dimension of f . By adjunction [HTT08, Corollary 3.2.15], the surjective morphism

Analytic families of quantum hyperbolic invariants

We organize the quantum hyperbolic invariants (QHI) of $3$-manifolds into sequences of rational functions indexed by the odd integers $N\geq 3$ and defined on moduli spaces of geometric structures refining the character varieties. In the case of one-cusped hyperbolic $3$-manifolds $M$ we generalize the QHI and get rational functions $\mathcal{H}_N^{h_f,h_c,k_c}$ depending on a finite set of cohomological data $(h_f,h_c,k_c)$ called {\it weights}. These functions are regular on a determined Abelian covering of degree $N^2$ of a Zariski open subset, canonically associated to $M$, of the geometric component of the variety of augmented $PSL(2,\mathbb{C})$-characters of $M$. New combinatorial ingredients are a weak version of branchings which exists on every triangulation, and state sums over weakly branched triangulations, including a sign correction which eventually fixes the sign ambiguity of the QHI. We describe in detail the invariants of three cusped manifolds, and present the results of numerical computations showing that the functions $\mathcal{H}_N^{h_f,h_c,k_c}$ depend on the weights as $N\rightarrow + \infty$, and recover the volume for some specific choices of the weights.

Foliations and Webs Inducing Galois Coverings: Table 1.

We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on $\mathbb P^2$ admitting continuous symmetries, obtaining a complete classification of Galois homogeneous foliations.

Examples of cylindrical Fano fourfolds

We construct four different families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form $$Z\,{\times }\,{\mathbb {A}}^{1}$$ , where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective $${\mathbb {G}}_{{\text {a}}}$$ -actions. Similar constructions of cylindrical Fano threefolds were done previously in the papers by Kishimoto et al. (Affine Algebraic Geometry. CRM Proceedings & Lecture Notes, vol 54, pp 123–163, 2011; Osaka J Math 51(4):1093–1112, 2014).

Meromorphic quotients for some holomorphic G-actions

Using mainly tools from [B.13] and [B.15] we give a necessary and sufficient condition in order that a holomorphic action of a connected complex Lie group $G$ on a reduced complex space $X$ admits a strongly quasi-proper meromorphic quotient. We apply this characterization to obtain a result which assert that, when $G = K.B$ with $B$ a closed complex subgroup of $G$ and $K$ a real compact subgroup of $G$, the existence of a strongly quasi-proper meromorphic quotient for the $B-$action implies, assuming moreover that there exists a $G-$invariant Zariski open dense subset in $X$ which is good for the $B-$action, the existence of a strongly quasi-proper meromorphic quotient for the $G-$action on $X$.

On the symplectic eightfold associated to a Pfaffian cubic fourfold

Abstract We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z → Hilb 4 ⁡ ( X ) Z\to\operatorname{Hilb}^{4}(X) , where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points. This note is an appendix to http://dx.doi.org/10.1515/crelle-2014-0144.

Extrapolating an Euler class

Let R be a noetherian ring of dimension d and let n be an integer so that n≤d≤2n−3. Let (a1,…,an+1) be a unimodular row so that the ideal J=(a1,…,an) has height n. Jean Fasel has associated to this row an element [(J,ωJ)] in the Euler class group En(R), with ωJ:(R/J)n→J/J2 given by (a¯1,…,a¯n−1,a¯na¯n+1). If R contains an infinite field F then we show that the rule of Fasel defines a homomorphism from WMSn+1(R)=Umn+1(R)/En+1(R) to En(R). The main problem is to get a well defined map on all of Umn+1(R). Similar results have been obtained by Das and Zinna [5], with a different proof. Our proof uses that every Zariski open subset of SLn+1(F) is path connected for walks made up of elementary matrices.

Dihedral group frames which are maximally robust to erasures

Let be a natural number larger than two. Let be the Dihedral group, and an -dimensional unitary representation of acting in as follows. and for . For any representation which is unitarily equivalent to we prove that when is prime, there exists a Zariski open subset of such that for any vector any subset of cardinality of the orbit of under the action of this representation is a basis for . However, when is even, there is no vector in which satisfies this property. As a result, we derive that if is prime, for almost every (with respect to Lebesgue measure) vector in , the -orbit of is a frame which is maximally robust to erasures. We also consider the case where is equivalent to an irreducible unitary representation of the Dihedral group acting in a vector space and we provide conditions under which it is possible to find a vector such that has the Haar property.

Noncommutative affine spaces and Lie-complete rings

In this paper, we investigate the structure sheaves of an (infinite-dimensional) affine NC-space Ancx, affine Lie-space Aliehx, and their nilpotent perturbations Anc,qx and Alieh,qx, respectively. We prove that the schemes Ancx and Aliehx are identical if and only if x is a finite set of variables, that is, when we deal with finite-dimensional noncommutative affine spaces. For each (Zariski) open subset U⊆X=Spec(C[x]), we obtain the precise descriptions of the algebras Onc(U), Onc,q(U), Olieh,q(U) and Olieh,q(U) of noncommutative regular functions on U associated with the schemes Ancx, Anc,qx, Alieh,qx and Aliehx, respectively. The obtained result for Onc(U) generalizes Kapranov's formula in the finite-dimensional case. Our approach to the matter is based on a noncommutative holomorphic functional calculus in Fréchet algebras.

Integration of complex polynomial differential equations in higher dimension, a first step: Lie transverse structure

We prove that a one-dimensional holomorphic foliationswith generic singularities on a complex projective space ℂPm+1, m ≥ 2, exhibiting a Lie group transverse structure in some Zariski open subset, is logarithmic. That is, it is given by a system of m closed rational one-forms with simple poles. The foliation is given by a linear vector field in some affine space ℂm+1 ⊂ ℂPm+1 if, and only if, it exhibits only one singularity in this affine space. An application to foliations invariant under Lie group transverse actions is given.

Affine varieties with exotic models

We show that for every $n\ge 7$ there is a smooth rational affine variety with exotic model. Moreover, we show that for every $n\ge 6$ there are Zariski open subsets $U_1, U_2$ of $\mathbb {C}^n$, such that they are holomorphically but not algebraically equivalent.

Hodge theory and derived categories of cubic fourfolds

Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics—conjecturally, the ones that are rational—have specific K3 surfaces associated to them geometrically. Hassett has studied cubics with K3 surfaces associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3 surfaces associated to them at the level of derived categories. These two notions of having an associated K3 surface should coincide. We prove that they coincide generically: Hassett’s cubics form a countable union of irreducible Noether–Lefschetz divisors in moduli space, and we show that Kuznetsov’s cubics are a dense subset of these, forming a nonempty, Zariski-open subset in each divisor.