Constant Sheaf Research Articles

The characteristic cycle of a non-confluent ℓ-adic GKZ hypergeometric sheaf

An ℓ-adic GKZ hypergeometric sheaf is defined analogously to a GKZ hypergeometric D-module. We introduce an algorithm of computing the characteristic cycle of an ℓ-adic GKZ hypergeometric sheaf of certain type. Our strategy is to apply a formula of the characteristic cycle of the direct image of an ℓ-adic sheaf. We verify the requirements for the formula to hold by calculating the dimension of the direct image of a certain closed conical subset of cotangent bundle. We also define an ℓ-adic GKZ-type sheaf whose specialization tensored with a constant sheaf is isomorphic to an ℓ-adic non-confluent GKZ hypergeometric sheaf. On the other hand, the topological model of an ℓ-adic GKZ-type sheaf is isomorphic to the image by the de Rham functor of a non-confluent GKZ hypergeometric D-module whose characteristic cycle has been calculated. This gives an easier way to determine the characteristic cycle of an ℓ-adic non-confluent GKZ hypergeometric sheaf of certain type.

Brane structures in microlocal sheaf theory

AbstractLet be an exact Lagrangian submanifold of a cotangent bundle , asymptotic to a Legendrian submanifold . We study a locally constant sheaf of ‐categories on , called the sheaf of brane structures or . Its fiber is the ‐category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from to the ‐category of sheaves of spectra on with singular support in .

Coarse Sheaf Cohomology

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0, they see the number of ends of the space. In this paper, a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition, coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus, we can use topological tools on compact Hausdorff spaces in our computations. In particular, if the asymptotic dimension of a proper metric space is finite, then higher cohomology groups vanish. We compute a few examples. As it turns out, finite abelian groups are best suited as coefficients on finitely generated groups.

The geometry and DSZ quantization four-dimensional supergravity

We implement the Dirac-Schwinger-Zwanziger integrality condition on four-dimensional classical ungauged supergravity and use it to obtain its duality-covariant, gauge-theoretic, differential-geometric model on an oriented four-manifold $M$ of arbitrary topology. Classical bosonic supergravity is completely determined by a submersion $\pi$ over $M$ equipped with a complete Ehresmann connection, a vertical euclidean metric, and a vertically-polarized flat symplectic vector bundle $\Xi$. Building on these structures, we implement the Dirac-Schwinger-Zwanziger integrality condition through the choice of an element in the degree-two sheaf cohomology group with coefficients in a locally constant sheaf $\mathcal{L}\subset \Xi$ valued in the groupoid of integral symplectic spaces. We show that this data determines a Siegel principal bundle $P_{\mathfrak{t}}$ of fixed type $\mathfrak{t}\in \mathbb{Z}^{n_v}$ whose connections provide the global geometric description of the local electromagnetic gauge potentials of the theory. Furthermore, we prove that the Maxwell gauge equations of the theory reduce to the polarized self-duality condition determined by $\Xi$ on the connections of $P_{\mathfrak{t}}$. In addition, we investigate the continuous and discrete U-duality groups of the theory, characterizing them through short exact sequences and realizing the latter through the gauge group of $P_{\mathfrak{t}}$ acting on its adjoint bundle. This elucidates the geometric origin of U-duality, which we explore in several examples, illustrating its dependence on the topology of the fiber bundles $\pi$ and $P_{\mathfrak{t}}$ as well as on the isomorphism type of $\mathcal{L}$.

Perverse Sheaves and the Cohomology of Regular Hessenberg Varieties

We use the Springer correspondence to give a partial characterization of the irreducible representations which appear in the Tymoczko dot action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. In type A, we apply these techniques to prove that all irreducible summands which appear in the pushforward of the constant sheaf on the universal Hessenberg family have full support. We also observe that the recent results of Brosnan and Chow, which apply the local invariant cycle theorem to the family of regular Hessenberg varieties in type A, extend to arbitrary Lie type. We use this extension to prove that regular Hessenberg varieties, though not necessarily smooth, always have the “Kähler package.”

On Continuity of Local Epsilon Factors of ℓ-adic Sheaves

Abstract Let $S$ be a noetherian scheme and $ Y\to S$ be a smooth morphism of relative dimension $1$. For a locally constant sheaf on the complement of a divisor in $Y$ flat over $S$, Deligne and Laumon proved that the universal local acyclicity is equivalent to the local constancy of Swan conductors. In this article, assuming the universal local acyclicity, we show an analogous result of continuity of local epsilon factors. We also give a generalization of this result to a family of isolated singularities.

A symplectic look at the Fargues–Fontaine curve

Abstract We study a version of the Fukaya category of a symplectic 2-torus with coefficients in a locally constant sheaf of rings. The sheaf of rings includes a globally defined Novikov parameter that plays its usual role in organising polygon counts by area. It also includes a ring of constants whose variation around the the torus can be encoded by a pair of commuting ring automorphisms. When these constants are perfectoid of characteristic p, one of the holonomies is trivial and the other is the $p^{th}$ power map, it is possible in a limited way to specialise the Novikov parameter to 1. We prove that the Dehn twist ring defined there is isomorphic to the homogeneous coordinate ring of a scheme introduced by Fargues and Fontaine: their ‘curve of p-adic Hodge theory’ for the local field $\mathbf {F}_p(\!(z)\!)$ .

A support theorem for nested Hilbert schemes of planar curves

Consider a family of integral complex locally planar curves. We show that under some assumptions on the basis, the relative nested Hilbert scheme is smooth. In this case, the decomposition theorem of Beilinson, Bernstein and Deligne asserts that the pushforward of the constant sheaf on the relative nested Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension.

Inner cohomology of GLn

We give an explicit description of the inner cohomology of an adèlic locally symmetric space of a given level structure attached to the general linear group of prime rank n with coefficients in a locally constant sheaf of complex vector spaces. We show that for all primes n the inner cohomology vanishes in all degrees for nonconstant sheaves, otherwise the quotient module of the inner cohomology classes that are not cuspidal is trivial in all degrees for primes n=2,3, and for all primes n≥5 the description of the same at degrees where it is nontrivial is given in terms of algebraic Hecke characters.

L-infinity pairs and applications to singularities

Over a field of characteristic zero, every deformation problem with cohomology constraints is controlled by a pair consisting of a differential graded Lie algebra together with a module. Unfortunately, these pairs are usually infinite-dimensional. We show that every deformation problem with cohomology constraints is controlled by a typically finite-dimensional L∞ pair. As a first application, we show that for complex algebraic varieties with no weight-zero 1-cohomology classes, the components of the cohomology jump loci of rank one local systems containing the constant sheaf are tori. This imposes restrictions on the fundamental groups. The same holds for links and Milnor fibers.

Rational homology manifolds and hypersurface normalizations

We prove a criterion for determining whether the normalization of a complex analytic space on which the constant sheaf is perverse is a rational homology manifold, using a perverse sheaf known as the multiple-point complex. This perverse sheaf is naturally associated to any parameterized space, and has several interesting connections with the Milnor monodromy and mixed Hodge Modules.

A Microlocal Characterization of Lipschitz Continuity

We study continuous maps between differential manifolds from a microlocal point of view. In particular, we characterize the Lipschitz continuity of these maps in terms of the microsupport of the constant sheaf on their graph. Furthermore, we give lower and upper bounds on the microsupport of the graph of a continuous map and use these bounds to characterize strict differentiability in microlocal terms.

Decomposition of perverse sheaves on plane line arrangements

ABSTRACTOn the complement to a central plane line arrangement , a locally constant sheaf of complex vector spaces ℒa is associated to any multi-index a∈ℂn. Using the description of MacPherson and Vilonen of the category of perverse sheaves [7, 8], we obtain a criterion for the irreducibility and number of decomposition factors of the direct image as a perverse sheaf, where j:X→ℂ2 is the canonical inclusion.

FROM COMPLETE TO PARTIAL FLAGS IN GEOMETRIC EXTENSION ALGEBRAS

AbstractA geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g., a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous vector bundles over homogeneous spaces. In this paper, we study the relationship between partial flag and complete flag cases. Our main result is that the locally finite modules over the geometric extension algebras are related by a recollement. As examples, we investigate parabolic affine nil Hecke algebras, geometric extension algebras associated with parabolic Springer maps and an example of Reineke of a parabolic quiver-graded Hecke algebra.

Legendrian knots and constructible sheaves

We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of the Chekanov-Eliashberg differential graded algebra. We also find two connections to topological knot theory. First, drawing a positive braid closure on the annulus, the moduli space of rank-n objects maps to the space of local systems on a circle. The second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the (colored-by-n) triply-graded Khovanov-Rozansky homology. Second, drawing a positive braid closure in the plane, the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in 'a' of the HOMFLY polynomial of the braid closure.

The derived Picard group of an affine Azumaya algebra

We describe the derived Picard group of an Azumaya algebra A on an affine scheme X in terms of global sections of the constant sheaf of integers on X, the Picard group of X, and the stabilizer of the Brauer class of A under the action of $$\mathrm {Aut}(X)$$ . In particular, we find that the derived Picard group of an Azumaya algebra is generally not isomorphic to that of the underlying scheme. In the case of the trivial Azumaya algebra, our result refines Yekutieli’s description of the derived Picard group of a commutative algebra. We also get, as a corollary, an alternate proof of a result of Antieau which relates derived equivalences to Brauer equivalences for affine Azumaya algebras. The example of a Weyl algebra in finite characteristic is examined in some detail.

The Calabi complex and Killing sheaf cohomology

It has recently been noticed that the degeneracies of the Poisson bracket of linearized gravity on constant curvature Lorentzian manifold can be described in terms of the cohomologies of a certain complex of differential operators. This complex was first introduced by Calabi and its cohomology is known to be isomorphic to that of the (locally constant) sheaf of Killing vectors. We review the structure of the Calabi complex in a novel way, with explicit calculations based on representation theory of GL(n), and also some tools for studying its cohomology in terms of of locally constant sheaves. We also conjecture how these tools would adapt to linearized gravity on other backgrounds and to other gauge theories. The presentation includes explicit formulas for the differential operators in the Calabi complex, arguments for its local exactness, discussion of generalized Poincar\'e duality, methods of computing the cohomology of locally constant sheaves, and example calculations of Killing sheaf cohomologies of some black hole and cosmological Lorentzian manifolds.

Twisted cohomology of the complement of theta divisors in an abelian surface

Let [Formula: see text] be an abelian surface, and [Formula: see text] be the sum of [Formula: see text] distinct theta divisors having normal crossings. We set [Formula: see text]. We study the structure of the nonvanishing twisted cohomology group [Formula: see text], where [Formula: see text] denotes a locally constant sheaf over [Formula: see text] defined by a multiplicative meromorphic function on [Formula: see text] infinitely ramified just along the divisor [Formula: see text] (as will be seen below, we will take as such a function a product of complex powers of theta functions). The de Rham complex on [Formula: see text] with logarithmic poles along [Formula: see text], associated to the twisted cohomology groups [Formula: see text], is [Formula: see text]-valued, where [Formula: see text] denotes a topologically trivial (i.e. Chern class zero) line bundle over [Formula: see text] determined by the locally constant sheaf [Formula: see text]. Therefore the main results of this paper, which give us information on the order of poles of meromorphic 2-forms on [Formula: see text] generating the cohomology group [Formula: see text], are divided into Theorems 4.5 and 4.6, according as the de Rham complex on [Formula: see text] with logarithmic poles along [Formula: see text] takes the values in a holomorphically nontrivial line bundle [Formula: see text] or a holomorphically trivial one [Formula: see text] ([Formula: see text] denoting the holomorphically trivial line bundle [Formula: see text]). Such a phenomenon does not occur in the case of the twisted cohomology of complex projective space with hyperplane arrangement.

A Poincaré lemma for Whitney–de Rham complex

Let M be a real analytic manifold, Z a closed subanalytic subset of M . We show that the Whitney–de Rham complex over Z is quasi-isomorphic to the constant sheaf \mathbb C_Z .

$V$-filtrations in positive characteristic and test modules

Let $R$ be a ring essentially of finite type over an $F$-finite field. Given an ideal $\mathfrak {a}$ and a principal Cartier module $M$ we introduce the notion of a $V$-filtration of $M$ along $\mathfrak {a}$. If $M$ is $F$-regular, then this coincides with the test module filtration. We also show that the associated graded induces a functor $Gr^{[0,1]}$ from Cartier crystals to Cartier crystals supported on $V(\mathfrak {a})$. This functor commutes with finite pushforwards for principal ideals and with pullbacks along essentially étale morphisms. We also derive corresponding transformation rules for test modules generalizing previous results by Schwede and Tucker in the étale case. If $\mathfrak {a} = (f)$ defines a smooth hypersurface and $R$ is in addition smooth, then for a Cartier crystal corresponding to a locally constant sheaf on $\operatorname {Spec} R_{\acute {e}t}$ the functor $Gr^{[0,1]}$ corresponds, up to a shift, to $i^!$, where $i: V(\mathfrak {a}) \to \operatorname {Spec} R$ is the closed immersion.