Zariski Open Subset Research Articles

Logarithmic Frobenius manifolds, hypergeometric systems and quantum 𝒟-modules

We describe mirror symmetry for weak Fano toric manifolds as an equivalence of filtered D \mathcal {D} -modules. We discuss in particular the logarithmic degeneration behavior at the large radius limit point and express the mirror correspondence as an isomorphism of Frobenius manifolds with logarithmic poles. The main tool is an identification of the Gauß-Manin system of the mirror Landau-Ginzburg model with a hypergeometric D \mathcal {D} -module, and a detailed study of a natural filtration defined on this differential system. We obtain a solution of the Birkhoff problem for lattices defined by this filtration and show the existence of a primitive form, which yields the construction of Frobenius structures with logarithmic poles associated to the mirror Laurent polynomial. As a final application, we show the existence of a pure polarized non-commutative Hodge structure on a Zariski open subset of the complexified Kähler moduli space of the variety.

Framed moduli spaces and tuples of operators

In this work, we address the classical problem of classifying tuples of linear operators and linear functions on a finite-dimensional vector space up to base change. Having adopted for the situation a construction of framed moduli spaces of quivers, we develop an explicit classification of tuples belonging to a Zariski open subset. For such tuples we provide a finite family of normal forms and a procedure allowing one to determine whether two tuples are equivalent.

Maximal Families of Calabi–Yau Manifolds with Minimal Length Yukawa Coupling

For each natural odd number $n\geq 3$, we exhibit a maximal family of $n$-dimensional Calabi-Yau manifolds whose Yukawa coupling length is one. As a consequence, Shafarevich's conjecture holds true for these families. Moreover, it follows from Deligne-Mostow and Mostow that, for $n=3$, it can be partially compactified to a Shimura family of ball type, and for $n=5,9$, there is a sub $\mathbb Q$-PVHS of the family uniformizing a Zariski open subset of an arithmetic ball quotient.

Nonalgebraic compactifications of quotients of the cylinder

We classify compact complex surfaces which contain a Zariski open subset whose universal covering is the cylinder DxC.

Global Hölderian Error Bound for Nondegenerate Polynomials

Let $f \colon \mathbb{R}^n \to \mathbb{R}$ be a polynomial and $S = \{x \in \mathbb{R}^n: f(x) \le 0\}$. Let $f_+(x) = \max\{0,f(x)\}$. If there exist $c > 0, \alpha >0, \beta > 0$ such that $d(x,S) \le c([f(x)]_+^\alpha + [f(x)]_+^\beta)$ for all $x \in \mathbb{R}^n$, where $d(x,S)$ denotes the distance between $x$ and $S$, then we say that $f$ has a global Hölderian error bound. We prove that if $f$ is convenient and nondegenerate with respect to its Newton boundary at infinity in the sense of Kouchnirenko [ Invent. Math., 32 (1976), pp. 1--32], then it has a global Hölderian error bound. Since the nondegenerate polynomials form a Zariski open subset in a variety of polynomials with a given convenient Newton boundary at infinity, this result shows that global Hölderian error bounds hold for a large class of polynomials, which can be relatively easy recognized.

On the volume of graded linear series and Monge–Ampère mass

We give an analytic description of the volume of a graded linear series, as the Monge–Ampere mass of a certain equilibrium metric associated to any smooth Hermitian metric on the line bundle. We also show the continuity of this equilibrium metric on some Zariski open subset, under a geometric assumption.

Lowest weights in cohomology of variations of Hodge structure

AbstractLetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

Lowest weights in cohomology of variations of Hodge structure

AbstractLetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

On the monodromy of the moduli space of Calabi–Yau threefolds coming from eight planes in $${\mathbb{P}^3}$$

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi–Yau threefolds coming from eight planes in \({\mathbb{P}^3}\) does not have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space. Besides general Hodge theory, representation theory and computational commutative algebra, one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of Hodge structures.

Rational invariants on the space of all stuctures of algebras on a two-dimensional vector space

Let V be a 2-dimensional vector space over an algebraically closed field F of characteristic different from 2 and 3. A non-empty Zariski-open subset O ⊂ ⊗V ∗ ⊗ V and four GL(V )invariant rational functions Ii,Fi:O → F for i = 1, 2 are proved to exist such that two bilinear maps t, t ∈ O are GL(V )-equivalent with respect to the tensorial representation of GL(V ) if and only if Ii(t) = Ii(t) and Fi(t) = Fi(t) for i = 1, 2. The matrix reducing t ∈ O to normal form is also studied. As the computation of the invariants Fi, i = 1, 2, is expensive, two new invariants Ij with j = 3, 4 are introduced, which are easy to be computed and have a geometric meaning. The invariants Fi, i = 1, 2, are written in terms of Ii, i = 1, . . . , 4, on a suitable Zariski-open subset O ⊂ O. Hence, they also solve the equivalence problem on O.

Big and small elements in Chevalley groups

Let $$ \tilde{G} $$ be a reductive algebraic group, which is defined and split over a field K. Here the Zariski open subset $$ \mathfrak{B} $$ of the group $$ \tilde{G} $$ that consists of elements such that their conjugacy classes intersect the Big Bruhat Cell is considered. In particular, a description is given for the set $$ \mathfrak{B}(K) $$ in the case $$ \tilde{G} = {\text{G}}{{\text{L}}_n} $$ , SL n . Bibliography: 16 titles.

Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology

In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary ∂ M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field X M on M, Witten defines an inhomogeneous coboundary operator d X M = d + ι X M on invariant forms on M. The main purpose is to adapt Belishev–Sharafutdinovʼs boundary data to invariant forms in terms of the operator d X M in order to investigate to what extent the equivariant topology of a manifold is determined by the corresponding variant of the DN map. We define an operator Λ X M on invariant forms on the boundary which we call the X M -DN map and using this we recover the X M -cohomology groups from the generalized boundary data ( ∂ M , Λ X M ) . This shows that for a Zariski-open subset of the Lie algebra, Λ X M determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the ring structure of X M -cohomology groups from Λ X M . These results explain to what extent the equivariant topology of the manifold in question is determined by Λ X M .

Cohen–Macaulay Loci of Modules

The Cohen–Macaulay locus of any finite module over a noetherian local ring A is studied, and it is shown that it is a Zariski-open subset of Spec A in certain cases. In this connection, the rings whose formal fibres over certain prime ideals are Cohen–Macaulay are studied.

Complex analytic Néron models for arbitrary families of intermediate Jacobians

Given a family of intermediate Jacobians (for a polarizable variation of integral Hodge structure of odd weight) on a Zariski-open subset of a complex manifold, we construct an analytic space that naturally extends the family. Its two main properties are: (a) the horizontal and holomorphic sections are precisely the admissible normal functions without singularities; (b) the graph of any admissible normal function has an analytic closure inside our space. As a consequence, we obtain a new proof for the zero locus conjecture of M. Green and P. Griffiths. The construction uses filtered $\mathcal {D}$ -modules and M. Saito’s theory of mixed Hodge modules; it is functorial, and does not require normal crossing or unipotent monodromy assumptions.

The orbit structure of the Gelfand–Zeitlin group onn×nmatrices

Recently, Kostant and Wallach constructed an action of a simply connected Lie group A ≅ ℂ n(n-1)/2 on gl(n) using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. They show that A-orbits of dimension n(n — 1)/2 form Lagrangian submanifolds of regular adjoint orbits in gl(n) and describe the orbits of A on a certain Zariski open subset of regular semisimple elements. In this paper, we describe all A-orbits of dimension n(n — 1)/2 and thus all polarizations of regular adjoint orbits obtained using Gelfand-Zeitlin theory.

SL2(ℂ)-character variety of a hyperbolic link and regulator

In this paper, we study the SL2(C) character variety of a hyperbolic link in S 3 . We analyze a special smooth projective variety Y h arising from some 1-dimensional irreducible slices on the character variety. We prove that a natural symbol obtained from these 1-dimensional slices is a torsion in K2(C(Y h )). By using the regulator map from K2 to the corresponding Deligne cohomology, we get some variation formulae on some Zariski open subset of Y h . From this we give some discussions on a possible parametrized volume conjecture for both hyperbolic links and knots.

Regularization of Local CR-Automorphisms of Real-Analytic CR-Manifolds

Let M be a connected generic real-analytic CR-submanifold of a finite-dimensional complex vector space E. Suppose that for every a∈M the Lie algebra \({\frak{hol}}(M,a)\) of germs of all infinitesimal real-analytic CR-automorphisms of M at a is finite-dimensional and its complexification contains all constant vector fields \(\alpha\frac {\partial}{\partial z} \), α∈E, and the Euler vector field \(z\frac{\partial}{\partial z} \). Under these assumptions we show that: (I) every \({\frak{hol}}(M,a)\) consists of polynomial vector fields, hence coincides with the Lie algebra \({\frak{hol}}(M)\) of all infinitesimal real-analytic CR-automorphisms of M, (II) every local real-analytic CR-automorphism of M extends to a birational transformation of E, and (III) the group Bir (M) generated by such birational transformations is realized as a group of projective transformations upon embedding E as a Zariski open subset into a projective algebraic variety. Under additional assumptions the group Bir (M) is shown to have the structure of a Lie group with at most countably many connected components and Lie algebra \({\frak{hol}}(M)\). All of the above results apply, for instance, to Levi non-degenerate quadrics, as well as a large number of Levi degenerate tube manifolds.

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.

Embeddings of $${\mathbb{C}^*}$$ -surfaces into weighted projective spaces

Let V be a normal affine surface which admits \({\mathbb{C}^*}\) - and \({\mathbb{C}_+}\) -actions. Such surfaces were classified e.g., in: Flenner and Zaidenberg (Osaka J Math 40:981–1009, 2003; 42:931–974), see also the references therein. In this note we show that in many cases V can be embedded as a principal Zariski open subset into a hypersurface of a weighted projective space. In particular, we recover a result of D. Daigle and P. Russell, see Theorem A in: Daigle and Russell (Can J Math 56:1145–1189, 2004)

On the Bounded Height Conjecture

We prove the Bounded Height Conjecture formulated by Bombieri, Masser, and Zannier: given an irreducible closed subvariety ⁠, and after replacing X by a natural and Zariski open subset; the set of its points contained in the union of all algebraic subgroups of codimension at least has bounded absolute Weil height. We proceed to show some finiteness results related to conjectures stated by Zilber and Pink, if the codimension of the subgroups is at least ⁠.