Coherent Sheaf Research Articles

Singular support of coherent sheaves and the geometric Langlands conjecture

We define the notion of singular support of a coherent sheaf on a quasi-smooth-derived scheme or Artin stack, where “quasi-smooth” means that it is a locally complete intersection in the derived sense. This develops the idea of “cohomological” support of coherent sheaves on a locally complete intersection scheme introduced by D. Benson, S. B. Iyengar, and H. Krause. We study the behavior of singular support under the direct and inverse image functors for coherent sheaves. We use the theory of singular support of coherent sheaves to formulate the categorical geometric Langlands conjecture. We verify that it passes natural consistency tests: It is compatible with the geometric Satake equivalence and with the Eisenstein series functors. The latter compatibility is particularly important, as it fails in the original “naive” form of the conjecture.

A Krull-Schmdit type theorem for coherent sheaves

Abstract Let X be projective variety over an algebraically closed field k and G be a finite group with g.c.d.(char(k), |G|) = 1. We prove that any representations of G on a coherent sheaf, ρ : G → End(ℰ), has a natural decomposition ℰ ≃ ⊕ V ⊗ k ℱ V , where G acts trivially on ℱ V and the sum run over all irreducible representations of G over k.

-algebras associated with curves and rational functions on . I

AbstractWe consider the natural$A_{\infty }$-structure on the$\mathrm{Ext}$-algebra$\mathrm{Ext}^*(G,G)$associated with the coherent sheaf$G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$on a smooth projective curve$C$, where$p_1,\ldots,p_n\in C$are distinct points. We study the homotopy class of the product$m_3$. Assuming that$h^0(p_1+\cdots +p_n)=1$, we prove that$m_3$is homotopic to zero if and only if$C$is hyperelliptic and the points$p_i$are Weierstrass points. In the latter case we show that$m_4$is not homotopic to zero, provided the genus of$C$is greater than$1$. In the case$n=g$we prove that the$A_{\infty }$-structure is determined uniquely (up to homotopy) by the products$m_i$with$i\le 6$. Also, in this case we study the rational map$\mathcal{M}_{g,g}\to \mathbb{A}^{g^2-2g}$associated with the homotopy class of$m_3$. We prove that for$g\ge 6$it is birational onto its image, while for$g\le 5$it is dominant. We also give an interpretation of this map in terms of tangents to$C$in the canonical embedding and in the projective embedding given by the linear series$|2(p_1+\cdots +p_g)|$.

Green functions, Segre numbers, and King’s formula

Let 𝒥 be a coherent ideal sheaf on a complex manifold X with zero set Z, and let G be a plurisubharmonic function such that G=log|f|+𝒪(1) locally at Z, where f is a tuple of holomorphic functions that defines 𝒥. We give a meaning to the Monge-Ampère products (dd c G) k for k=0,1,2,..., and prove that the Lelong numbers of the currents M k 𝒥 :=1 Z (dd c G) k at x coincide with the so-called Segre numbers of J at x, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that M k 𝒥 satisfy a certain generalization of the classical King formula.

Bounded sets of sheaves on Kähler manifolds

Abstract We show that any set of quotients with fixed Chern classes of a given coherent sheaf on a compact Kähler manifold is bounded in a sense which we define. The result is proved by adapting Grothendieck's boundedness criterion expressed via the Hilbert polynomial to the Kähler set-up. As a consequence we obtain the compactness of the connected components of the Douady space of a compact Kähler manifold.

A spectral sequence and nef vector bundles of the first Chern class two on hyperquadrics

We deduce from Bondal’s theorem a spectral sequence, which yields a description, such as a resolution, of a coherent sheaf in terms of a full strong exceptional sequence. We then apply the sequence to the case of a vector bundle given with some cohomological data on a projective space; we obtain a resolution of the vector bundle in terms of exceptional line bundles, resolution which is different from that obtained by the Beilinson spectral sequence. Finally we list all known nef vector bundles of the first Chern class two on a hyperquadric of dimension greater than three.

*-Quantizations of Fourier–Mukai Transforms

We study deformations of Fourier–Mukai transforms in general complex analytic settings. Suppose X and Y are complex manifolds, and let P be a coherent sheaf on X × Y. Suppose that the Fourier–Mukai transform \({\Phi}\) given by the kernel P is an equivalence between the coherent derived categories of X and of Y. Suppose also that we are given a formal *-quantization \({\mathbb{X}}\) of X. Our main result is that \({\mathbb{X}}\) gives rise to a unique formal *-quantization \({\mathbb{Y}}\) of Y. For the statement to hold, *-quantizations must be understood in the framework of stacks of algebroids. The quantization \({\mathbb{Y}}\) is uniquely determined by the condition that \({\Phi}\) deforms to an equivalence between the derived categories of \({\mathbb{X}}\) and \({\mathbb{Y}}\) . Equivalently, the condition is that P deforms to a coherent sheaf \({\tilde{P}}\) on the formal *-quantization \({\mathbb{X} \times\mathbb{Y}^{op}}\) of X × Y; here \({\mathbb{Y}^{op}}\) is the opposite of the quantization \({\mathbb{Y}}\) .

A note on sheaves without self-extensions on the projective 𝑛-space

Let P n \textbf {P}^n be the projective n n -space over the complex numbers. In this note we show that an indecomposable rigid coherent sheaf on P n \textbf {P}^n has a trivial endomorphism algebra. This generalizes a result of Drézet for n = 2. n=2.

On the classification of the real vector subspaces of a quaternionic vector space

AbstractWe prove the classification of the real vector subspaces of a quaternionic vector space by using a covariant functor which associates, to any pair formed of a quaternionic vector space and a real subspace, a coherent sheaf over the sphere.

Line bundles with partially vanishing cohomology

Define a line bundle L on a projective variety to be q -ample, for a natural number q , if tensoring with high powers of L kills coherent sheaf cohomology above dimension q . Thus 0-ampleness is the usual notion of ampleness. We show that q -ampleness of a line bundle on a projective variety in characteristic zero is equivalent to the vanishing of an explicit finite list of cohomology groups. It follows that q -ampleness is a Zariski open condition, which is not clear from the definition.

Geometric Auslander criterion for flatness

Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let $\varphi: X \to Y$ be a morphism of complex-analytic spaces, where $Y$ is smooth. We prove that nonflatness of $\varphi$ is equivalent to a severe discontinuity of the fibres---the existence of a {\it vertical component} (a local irreducible component at a point of the source whose image is nowhere-dense in $Y$)---after passage to the $n$-fold fibred power of $\varphi$, where $n = \dim Y$. Our main theorem is a more general criterion for flatness over $Y$ of a coherent sheaf of modules $\cal{F}$ on $X$. In the case that $\varphi$ is a morphism of complex algebraic varieties, the result implies that the stalk $\cal{F}_\xi$ of $\cal{F}$ at a point $\xi \in X$ is flat over $R := \cal{O}_{Y,\varphi(\xi)}$ if and only if its $n$-fold tensor power is a torsion-free $R$-module (conjecture of Vasconcelos in the case of $\Bbb{C}$-algebras).

A local ampleness criterion of torsion free sheaves

We give an ampleness criterion of a torsion free coherent sheaf at a given point in terms of a curvature positivity of a singular Hermitian metric.

Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford

The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a cohomology with coefficients in representations and a Chern character, and we prove a Grothendieck-Riemann-Roch theorem for the Riemann-Roch transformation it defines. As a corollary we obtain an Hirzebruch-Riemann-Roch formula for the Euler characteristic of a coherent sheaf, and some formulas for the different topological Euler characteristics of complex algebraic stacks.

Bridgeland-stable moduli spaces for $K$-trivial surfaces

We give a one-parameter family of Bridgeland stability conditions on the derived category of a smooth projective complex surface S and describe "wall-crossing behavior'' for objects with the same invariants as \mathcal O_C(H) when H generates Pic (S) and C \in |H| . If, in addition, S is a K3 or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgeland-stable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover a natural generalization of Thaddeus' stable pairs for curves embedded in the moduli spaces.

Berkovich spaces and tubular descent

We consider an algebraic variety X together with the choice of a subvariety Z. We show that any coherent sheaf on X can be constructed out of a coherent sheaf on the formal neighborhood of Z, a coherent sheaf on the complement of Z, and an isomorphism between certain representative images of these two sheaves in the category of coherent sheaves on a Berkovich analytic space W which we define.

Continuous closure of sheaves

We give a purely algebraic construction of the continuous closure of any finitely generated torsion free module; a concept first studied by H.~Brenner and M.~Hochster. The construction implies that, at least in characteristic 0, taking continuous closure commutes with flat morphisms whose fibers are semi-normal. This implies that the continuous closure of a coherent ideal sheaf is again a coherent ideal sheaf (both in the Zariski and in the etale topologies) and it commutes with field extensions.

STABILITY AND HERMITIAN–EINSTEIN METRICS FOR VECTOR BUNDLES ON FRAMED MANIFOLDS

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford–Takemoto and Hermitian–Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that KX⊗ [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization KX⊗ [D] coincides with the degree with respect to the complete Kähler–Einstein metric gX\Don X\D. For stable holomorphic vector bundles, we prove the existence of a Hermitian–Einstein metric with respect to gX\Dand also the uniqueness in an adapted sense.

Detecting flatness over smooth bases

Given an essentially finite type morphism of schemes f : X → Y f\colon X\to Y and a positive integer d d , let f { d } : X { d } → Y f^{\{d\}}\colon X^{\{d\}}\to Y denote the natural map from the d d -fold fiber product X { d } = X × Y ⋯ × Y X X^{\{d\}}= X\times _{Y}\cdots \times _{Y}X and π i : X { d } → X \pi _i\colon X^{\{d\}}\to X the i i th canonical projection. When Y Y is smooth over a field and F \mathcal F is a coherent sheaf on X X , it is proved that F \mathcal F is flat over Y Y if (and only if) f { d } f^{\{d\}} maps the associated points of ⨂ i = 1 d π i ∗ F {\bigotimes _{i=1}^d}\pi _i^*{\mathcal F} to generic points of Y Y , for some d ≥ dim ⁡ Y d\ge \dim Y . The equivalent statement in commutative algebra is an analog—but not a consequence—of a classical criterion of Auslander and Lichtenbaum for the freeness of finitely generated modules over regular local rings.

Exemples de faisceaux cohérents sans résolution localement libre en dimension 3

In 1982, Schuster proved that for any compact complex surface X, every coherent sheaf F on X has global resolutions⋯→En→En−1→⋯→E1→E0→F→0 such that the Eiʼs are locally free (vector bundles). In 2002, Voisin proved that this is false for some Kähler compact complex manifolds of dimension ⩾3. In this Note, we give some new examples of non-Kähler compact complex manifolds of dimension 3 and coherent sheaves F on X having no global resolution by vector bundles. The proof that these sheaves do not admit a locally free resolution is very different from Voisinʼs argument.

Bigq-ample line bundles

AbstractA recent paper of Totaro developed a theory ofq-ample bundles in characteristic 0. Specifically, a line bundleLonXisq-ample if for every coherent sheaf ℱ onX, there exists an integerm0such thatm≥m0impliesHi(X,ℱ⊗𝒪(mL))=0 fori>q. We show that a line bundleLon a complex projective schemeXisq-ample if and only if the restriction ofLto its augmented base locus isq-ample. In particular, whenXis a variety andLis big but fails to beq-ample, then there exists a codimension-one subschemeDofXsuch that the restriction ofLtoDis notq-ample.