Pure Codimension Research Articles

On the p-essential normality of principal submodules of the Bergman module on strongly pseudoconvex domains

In this paper, we show that under a mild condition, a principal submodule of the Bergman module on a bounded strongly pseudoconvex domain with smooth boundary in Cn is p-essentially normal for all p>n. This improves a previous result by the first author and K. Wang, in which it was shown that any polynomial-generated principal submodule of the Bergman module on the unit ball Bn is p-essentially normal for all p>n. As a consequence, we show that the submodule of La2(Bn) consisting of functions vanishing on an analytic subset of pure codimension 1 is p-essentially normal for all p>n.

Complete complex hypersurfaces in the ball come in foliations

In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n (n \geq 2)$ is a level set of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose level sets are complete. This shows that $\mathbb{B}_n$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in $\mathbb{B}_n$ of this type can be embedded into such a foliation. We establish a more general result in which neither completeness nor smoothness of the given hypersurface is required. Furthermore, we obtain a similar result for complex submanifolds of arbitrary positive codimension and prove the existence of a nonsingular holomorphic submersion foliation of $\mathbb{B}_n$ by smooth complete closed complex submanifolds of any pure codimension $q \in \lbrace 1, \dotsc , n-1 \rbrace$.

Wild holomorphic foliations of the ball

We prove that the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ admits a nonsingular holomorphic foliation $\mathcal F$ by closed complex hypersurfaces such that both the union of the complete leaves of $\mathcal F$ and the union of the incomplete leaves of $\mathcal F$ are dense subsets of $\mathbb{B}_n$. In particular, every leaf of $\mathcal F$ is both a limit of complete leaves of $\mathcal F$ and a limit of incomplete leaves of $\mathcal F$. This gives the first example of a holomorphic foliation of $\mathbb{B}_n$ by connected closed complex hypersurfaces having a complete leaf that is a limit of incomplete ones. We obtain an analogous result for foliations by complex submanifolds of arbitrary pure codimension $q$ with $1\le q<n$.

Open Subsets in a Stein Space with Singularities

Serre proved that a domain $Y$ in $\mathbb{C}^n$ is Stein if and only if $H^i(Y,\mathcal{O}_Y) = 0$ for all $i \gt 0$. Laufer showed that if $Y$ is an open subset of a Stein manifold of dimension $n$ and $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$, then $Y$ is Stein. Vâjâitu generalized these theorems to singular Stein space of dimension $n$. In this paper, we consider singular Stein spaces $X$ with arbitrary dimension and give necessary and sufficient conditions for an open subset $Y$ in $X$ to be Stein. We show that if $Y$ is an open subset of a reduced Stein space $X$ with arbitrary dimension and singularities, then $Y$ is Stein if and only if $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$. Without cohomology condition, if $X-Y$ is a closed subspace of $X$, then we show that the geometric condition of the boundary $X-Y$ determines the Steinness of $Y$. More precisely, we show that if $X$ is normal and the boundary $X-Y$ is the support of an effective $\mathbb{Q}$-Cartier divisor, or $X-Y$ is of pure codimension $1$ and does not contain any singular points of $X$, then $Y$ is Stein.

Good Formal Structures for Flat Meromorphic Connections, III: Irregularity and Turning Loci

Given a formal at meromorphic connection over an excellent scheme over a field of characteristic zero, in a previous paper we established existence of good formal structures and a good Deligne–Malgrange lattice after suitably blowing up. In this paper, we reinterpret and refine these results by introducing some related structures. We consider the turning locus , which is the set of points at which one cannot achieve a good formal structure without blowing up. We show that when the polar divisor has normal crossings, the turning locus is of pure codimension 1 within the polar divisor, and hence of pure codimension 2 within the full space; this had been previously established by André in the case of a smooth polar divisor. We also construct an irregularity sheaf and its associated b-divisor , which measure irregularity along divisors on blowups of the original space; this generalizes another result of André on the semicontinuity of irregularity in a curve fibration. One concrete consequence of these refinements is a process for resolution of turning points which is functorial with respect to regular morphisms of excellent schemes; this allows us to transfer the result from schemes to formal schemes, complex analytic varieties, and nonarchimedean analytic varieties.

On the nonexistence of some open immersions

In this paper, we will prove a sufficient condition for that there does not exist an open immersion between two affine schemes of finite type over a field $k$ with the same dimension. The proof is based on the following fact: the complement of an open affine subset in a noetherian integral separated scheme has pure codimension 1. We will first prove it when $k$ is a finite field, the key to the proof of this part is Lang-Weil estimation. Then we'll prove a general result over an arbitrary field by reducing to the case when $k$ is finite. And the proof of the general result is much more complicated. In order to use the result over a finite field, at some point we must produce a scheme that is defined over $\mathbf{F}_{q}$ and an open immersion, also defined over $\mathbf{F}_{q}$. One important lemma is that a morphism $f:\text{Spec}(B) \longrightarrow \text{Spec}(A)$ between two integral domains with the same quotient field $K$ is an open immersion if and only if $B$ is a birational extension of $A$ in $K$ and $B$ is flat over $A$. According to the general result, the following open immersions do not exist: $SL_{n/k} \hookrightarrow \mathbf{A}_{k}^{n^{2}-1}$, $Sp_{n/k} \hookrightarrow \mathbf{A}_{k}^{2n^{2}+n}$, $SO_{n/k} \hookrightarrow \mathbf{A}_{k}^{\frac{n^{2}-n}{2}}$, $PGL_{n/k} \hookrightarrow \mathbf{A}_{k}^{n^{2}-1}$, where $k$ is an arbitrary field.

Graded basic elements

We prove the existence theorem for basic elements in the quasi-projective case, extending results of Eisenbud-Evans and Bruns from the affine case. We give several geometric applications. For example, we show that every local complete intersection of pure codimension two in a smooth projective variety of dimension d over an infinite field is the degeneracy locus of d−1 sections of a rank d bundle.

THE MODULI SPACE OF TWISTED CANONICAL DIVISORS

The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in $\overline{{\mathcal{M}}}_{g,n}$ which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the moduli spaces of canonical divisors.In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus $g$ curves are of pure codimension $g$ in $\overline{{\mathcal{M}}}_{g,n}$. In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix A, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors.As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases).

The Gieseker-Petri divisor in $${\mathcal{M}_g}$$ for g ≤ 13

The Gieseker-Petri locus GPg is defined as the locus inside \({\mathcal{M}_g}\) consisting of curves which violate the Gieseker-Petri Theorem. It is known that GPg has always some divisorial components and it has been conjectured that it is of pure codimension 1 inside \({\mathcal{M}_g}\) . We prove that this holds true for genus up to 13.

Koszulness, Krull dimension, and other properties of graph-related algebras

The algebra of basic covers of a graph G, denoted by $\bar{A}(G)$ , was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper we compute the Krull dimension of $\bar{A}(G)$ in terms of the combinatorics of G. As a consequence, we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Furthermore, we show that if the graph is bipartite, then $\bar{A}(G)$ is a homogeneous algebra with straightening laws, and thus it is Koszul. Finally, we characterize the Cohen---Macaulay property and the Castelnuovo---Mumford regularity of the edge ideal of a certain class of graphs.

Coleff-Herrera currents, duality, and noetherian operators

Let I be a coherent subsheaf of a locally free sheaf O(E-0) and suppose that I = O(E-0)/I has pure codimension. Starting with a residue current R obtained from a locally free resolution of I we construct a. vector-valued Coleff-Herrera current it with support on the variety associated to I such that phi is in I if and only if mu phi = 0. Such a current mu can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed. By a construction due to Bjork one gets Noetherian operators for I from the current mu. The current R also provides an explicit realization of the Dickenstein-Sessa decomposition and other related canonical isomorphisms.

On the Zero Locus of Normal Functions and the Etale Abel-Jacobi Map

Let X → S be a family of complex smooth projective varieties over a quasi-projective base, and let Z ↪→ X be a flat family of cycles of pure codimension i which are homologically equivalent to zero in the fibers of the family. For any point s of S, the Abel–Jacobi map associates to the cycle Zs a point in the intermediate Jacobian J(Xs) of Xs, which is a complex torus (see part 2 for details). This construction works in family, yielding a bundle of complex tori, the Jacobian fibration Ji(X/S), and a normal function νZ , which is the holomorphic section of Ji(X/S) associated to Z ↪→ X. We can attach to νZ an admissible variation H of mixed Hodge structures on S, see [22], fitting in an exact sequence

Sur la transformation d'Abel-Radon des courants localement résiduels

We give in this note a generalisation of the following theorem of Henkin and Passare (cf. (7) and (8)) : Let be Y an analytic subvariety of pure codimension p in a linearly p−concave domain U, and ω a meromorphic q−form (q > 0) on Y ; if the Abel-Radon transform R(ω ∧ (Y )), which is meromorphic on U ∗ , has a meromorphic prolongation to ˜ U ∗ containing U ∗ , then Y extends as an analytic subvariety ˜

Intersection Homology $\mathcal D$-Module and Bernstein Polynomials Associated with a Complete Intersection

Let X be a complex analytic manifold. Given a closed subspace Y\subset X of pure codimension p ≥ 1 , we consider the sheaf of local algebraic cohomology H^p_{[Y]}(\mathcal O_X) , and \mathcal L(Y,X)\subset H^p_{[Y ]}(\mathcal O_X) the intersection homology \mathcal D_X -Module of Brylinski–Kashiwara. We give here an algebraic characterization of the spaces Y such that \mathcal L(Y,X) coincides with H^p_{[Y]}(\mathcal O_X) , in terms of Bernstein–Sato functional equations.

Sur la transformation d'Abel–Radon de courants localement résiduels

The aim of this Note is to give a generalisation of the following theorem of Henkin and Passare: let Y be an analytic subvariety of pure codimension p in a linearly p-concave domain U, and ω a meromorphic q-form ( q>0) on it; if the Abel–Radon transform A(ω∧[Y]) , which is meromorphic on U ∗ , has a meromorphic prolongation to U ∗ , then Y extends to an analytic subvariety of U , and ω to a meromorphic form on it. The problem is to show the analogous statement when we replace ω∧[ Y] by a current α of a more general type, called locally residual. We give the proof if α is of bidegree ( N,1), or ( q+1,1), 0< q< N in the particular case where A(α)=0 . We conclude with some applications of the theorem. To cite this article: B. Fabre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

On the intersection of algebraic curves and hypersurfaces

Many classical theorems about the intersection of plane algebraic curves can be derived from and generalized by the algebraic theory of residues. In higher dimensional spaces one obtains results about the intersection of curves and hypersurfaces. In various degrees of generality this was pointed out f.i. in [S], [GH], [K2] and [Hu]. In the present note we wish to prove higher dimensional analogues of theorems of Waring, G. Humbert and Reiss for arbitrary curves in affine n– space and their intersection with hypersurfaces, see Theorems 4.1, 5.1 and 6.2 below. By a curve (hypersurface) we mean a closed subscheme in projective or affine n–space of pure dimension 1 (pure codimension 1) having no embedded components. Our considerations are based on the residue theorem for complete integral algebraic curves.

ℋ︁ — Cohomologies Versus Algebraic Cycles

AbstractGlobal intersection theories for smooth algebraic varieties via products in appropriate Poincaré duality theories are obtained. We assume given a (twisted) cohomology theory H* having a cup product structure and we consider the ℋ︁‐ cohomology functor X ↝ H#Zar(X, ℋ︁*) whereℋ︁* is the Zariski sheaf associated to H*. We show that the ℋ︁‐ cohomology rings generalize the classical “intersection rings” obtained via rational or algebraic equivalences. Several basic properties e. g. Gysin maps, projection formula and projective bundle decomposition, of ℋ︁‐ cohomology are obtained. We therefore obtain, for X smooth, Chern classes cP, i :Ki(X)→ Hp‐i(X,ℋ︁ p) from the Quillen K‐theory to ℋ︁‐ cohomologies according to Gillet and Grothendieck. We finally obtain the “blow‐up formula” magnified image where X′ is the blow–up of X smooth, along a closed smooth subset Z of pure codimension c. Singular cohomology of associated analityc space, étale cohomology, de Rham and Deligne–Beilinson cohomologies are examples for this setting.

Even Linkage Classes

In this paper we generalize the E \mathcal {E} and N \mathcal {N} -type resolutions used by Martin-Deschamps and Perrin for curves in P 3 \mathbb {P}^{3} to subschemes of pure codimension in projective space, and shows that these resolutions are interchanged by the mapping cone procedure under a simple linkage. Via these resolutions, Rao’s correspondence is extended to give a bijection between even linkage classes of subschemes of pure codimension two and stable equivalence classes of reflexive sheaves E \mathcal {E} satisfying H ∗ 1 ( E ) = 0 H^{1}_{*}( \mathcal {E})=0 and E x t 1 ( E ∨ , O ) = 0 \mathcal {E}xt^{1}( \mathcal {E}^{\vee }, \mathcal {O})=0 . Further, these resolutions are used to extend the work of Martin-Deschamps and Perrin for Cohen-Macaulay curves in P 3 \mathbb {P}^{3} to subschemes of pure codimension two in P n \mathbb {P}^{n} . In particular, even linkage classes of such subschemes satisfy the Lazarsfeld-Rao property and any minimal subscheme for an even linkage class links directly to a minimal subscheme for the dual class.

A homological criterion for reducibility of analytic spaces, with application to characterizing the theta divisor of a product of two general principally polarized abelian varieties

A closed subset of pure codimension one in an analytic space, consisting entirely of local normal crossings double points, is called an ordinary rank two double locus. We give a topologically computable upper bound on the number of connected components of an ordinary rank two double locus in a given space. This leads to criteria for global reducibility of spaces. The first is that a simply connected space with a non empty ordinary rank two double locus is always reducible. A finer criterion implies that a principally polarized abelian variety A is isomorphic to a product of two positive dimensional principally polarized abelian varieties, each with smooth theta divisor, if and only if the theta divisor of A contains a non empty ordinary rank two double locus. Analogous reducibility results apply to certain complete intersection varieties, and to divisors on such varieties.

On the arithmetical rank of monomial ideals

The arithmetical rank of monomial ideals has been studied by several authors. In [ 12, 131, for example, the result of the above theorem is proved in some special cases. On the other hand in [8] we stated the conjecture to the effect that every algebraic set l’c P” of pure codimension t can be set-theoretically cut out by n [n/t] + 1 hypersurfaces. Supporting evidence for this conjecture is as follows: