Coherent Sheaf Research Articles

Coherent Springer theory and the categorical Deligne-Langlands correspondence

Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$K$\\end{document}-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$F$\\end{document} with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$K$\\end{document}-theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf. As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu).In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of GLn(F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathrm{GL}_{n}(F)$\\end{document} into coherent sheaves on the stack of Langlands parameters.

Semiorthogonal decomposition via categorical action

We show that the categorical action of the shifted q=0 affine algebra can be used to construct semiorthogonal decomposition on the weight categories. In particular, this construction recovers Kapranov's exceptional collection when the weight categories are the derived categories of coherent sheaves on Grassmannians and n-step partial flag varieties. Finally, as an application, we use this result to construct a semiorthogonal decomposition on the derived categories of coherent sheaves on Grassmannians of a coherent sheaf with homological dimension ≤1 over a smooth projective variety X.

Positive Ulrich sheaves

AbstractWe provide a criterion for a coherent sheaf to be an Ulrich sheaf in terms of a certain bilinear form on its global sections. When working over the real numbers, we call it a positive Ulrich sheaf if this bilinear form is symmetric or Hermitian and positive-definite. In that case, our result provides a common theoretical framework for several results in real algebraic geometry concerning the existence of algebraic certificates for certain geometric properties. For instance, it implies Hilbert’s theorem on nonnegative ternary quartics, via the geometry of del Pezzo surfaces, and the solution of the Lax conjecture on plane hyperbolic curves due to Helton and Vinnikov.

Instanton sheaves on projective schemes

A h–instanton sheaf on a closed subscheme X of some projective space endowed with an ample and globally generated line bundle OX(h) is a coherent sheaf whose cohomology table has a certain prescribed shape. In this paper we deal with h–instanton sheaves relating them to Ulrich sheaves. Moreover, we study h–instanton sheaves on smooth curves and surfaces, cyclic n–folds, Fano 3–folds and scrolls over arbitrary smooth curves. We also deal with a family of monads associated to h–instanton bundles on varieties satisfying some mild extra technical conditions.

Geometric Invariants for a Class of Submodules of Analytic Hilbert Modules Via the Sheaf Model

Let \(\Omega \subseteq {\mathbb {C}}^m\) be a bounded connected open set and \({\mathcal {H}} \subseteq {\mathcal {O}}(\Omega )\) be an analytic Hilbert module, i.e., the Hilbert space \({\mathcal {H}}\) possesses a reproducing kernel K, the polynomial ring \(\mathbb C[{\varvec{z}}]\subseteq {\mathcal {H}}\) is dense and the point-wise multiplication induced by \(p\in {\mathbb {C}}[{\varvec{z}}]\) is bounded on \({\mathcal {H}}\). We fix an ideal \({\mathcal {I}} \subseteq {\mathbb {C}}[{\varvec{z}}]\) generated by \(p_1,\ldots ,p_t\) and let \([{\mathcal {I}}]\) denote the completion of \({\mathcal {I}}\) in \(\mathcal H\). The sheaf \({\mathcal {S}}^{\mathcal {H}}\) associated to analytic Hilbert module \({\mathcal {H}}\) is the sheaf \({\mathcal {O}}(\Omega )\) of holomorphic functions on \(\Omega \) and hence is free. However, the subsheaf \({\mathcal {S}}^{\mathcal [{\mathcal {I}}]}\) associated to \([{\mathcal {I}}]\) is coherent and not necessarily locally free. Building on the earlier work of Biswas, Misra and Putinar (Journal fr die reine und angewandte Mathematik (Crelles Journal) 662:165–204, 2012), we prescribe a hermitian structure for a coherent sheaf and use it to find tractable invariants. Moreover, we prove that if the zero set \(V_{[{\mathcal {I}}]}\) is a submanifold of codimension t, then there is a unique local decomposition for the kernel \(K_{[{\mathcal {I}}]}\) along the zero set that serves as a holomorphic frame for a vector bundle on \(V_{[{\mathcal {I}}]}\). The complex geometric invariants of this vector bundle are also unitary invariants for the submodule \([{\mathcal {I}}] \subseteq {\mathcal {H}}\).

On Yuzvinsky’s lattice sheaf cohomology for hyperplane arrangements

We establish the relationship between the cohomology of a certain sheaf on the intersection lattice of a hyperplane arrangement introduced by Yuzvinsky and the cohomology of the coherent sheaf on punctured affine space, respectively projective space associated to the module of logarithmic vector fields along the arrangement. Our main result gives a K\"unneth formula connecting the cohomology theories, answering a question by Yoshinaga. This, in turn, provides a characterization of the projective dimension of the module of logarithmic vector fields and yields a new proof of Yuzvinsky's freeness criterion. Furthermore, our approach affords a new formulation of Terao's freeness conjecture and a more general problem.

Chern currents of coherent sheaves

Given a finite locally free resolution of a coherent analytic sheaf $\mathcal F$, equipped with Hermitian metrics and connections, we construct an explicit current, obtained as the limit of certain smooth Chern forms of $\mathcal F$, that represents the Chern class of $\mathcal F$ and has support on the support of $\mathcal F$. If the connections are $(1,0)$-connections and $\mathcal F$ has pure dimension, then the first nontrivial component of this Chern current coincides with (a constant times) the fundamental cycle of $\mathcal F$. The proof of this goes through a generalized Poincar\'e-Lelong formula, previously obtained by the authors, and a result that relates the Chern current to the residue current associated with the locally free resolution.

The Role of the Gradient Term of the Bochner–Kodaira Formula in Coherent Sheaf Extension

In applying the Bochner–Kodaira formula with boundary term to solve the \({\bar{\partial }}\) equation with \(L^2\) estimates, the gradient term is usually not used. Two potentially important applications of the use of the gradient term are the strong rigidity for holomorphic vector bundles and the very ampleness part of the Fujita conjecture. In this note, we use the gradient term to construct holomorphic sections to prove the Thullen-type extension across codimension 1 for holomorphic vector bundles with Hermitian metric whose curvature is \(L^p\) for some \(p>1\). This construction of sections points out a typical way of how the gradient term can be used.

A counter-example to the Andreotti--Grauert conjecture

In 1962, Andreotti and Grauert showed that every $q$-completecomplex space $X$ is cohomologically $q$-complete, that is for every coherent analytic sheaf${\mathcal{F}}$ on $X$, the cohomology group $H^{p}(X, {\mathcal{F}})$ vanishes if $p\geq q$.Since then the question whether the reciprocal statements of these theorems are true have been subject to extensive studies, where more specific assumptions have been added. Until now it is not known if these two conditions are equivalent. Using test cohomology classes, it was shown however that if $X$ is a Stein manifold and, if $D\subset X$ is an open subset which has $C^{2}$ boundary such that $H^{p}(D, {\mathcal{O}}_{D})=0$ for all $p\geq q$, then $D$ is $q$-complete.The aim of the present article is to give a counterexample tothe conjecture posed in $1962$ by Andreotti and Grauert [1]to show that a~cohomologically $q$-complete space is not necessarily $q$-complete.More precisely, we show that there exist for each $n\geq 3$ open subsets$\Omega\subset\mathbb{C}^{n}$ such that for every ${\mathcal{F}}\in coh(\Omega)$,the cohomology groups $H^{p}(\Omega, {\mathcal{F}})$ vanish for all $p\geq n-1$but $\Omega$ is not $(n-1)$-complete.

Canonical subsheaves of torsionfree semistable sheaves

Let $F$ be a torsionfree semistable coherent sheaf on a polarized normal projective variety defined over an algebraically closed field. We prove that $F$ has a unique maximal locally free subsheaf $V$ such that $F/V$ is torsionfree and $V$ also admits a filtration of subbundles for which each successive quotient is a stable vector bundle whose slope is $\mu (F)$. We also prove that $F$ has a unique maximal reflexive subsheaf $W$ such that $F/W$ is torsionfree and $W$ admits a filtration of subsheaves for which each successive quotient is a stable reflexive sheaf whose slope is $\mu (F)$. We show that these canonical subsheaves behave well with respect to the pullback operation by étale Galois covering maps. Given a separable finite surjective map $\phi \colon Y \longrightarrow X$ between normal projective varieties, we give a criterion for the induced homomorphism of étale fundamental groups $\phi _*\colon \pi ^{\textrm {et}}_{1}(Y) \longrightarrow \pi ^{\textrm {et}}_{1}(X)$ to be surjective. The criterion in question is expressed in terms of the above mentioned unique maximal locally free subsheaf associated to the direct image $\phi _*{\mathcal O}_Y$.

Derived categories of projectivizations and flops

We prove a generalization of Orlov's projectivization formula for the derived category Dcohb(P(E)), where E does not need to be a vector bundle; Instead, E is a coherent sheaf which locally admits two-step resolutions. As a special case, this also gives Orlov's generalized universal hyperplane section formula. As applications, (i) we obtain a blowup formula for blowup along codimension two Cohen-Macaulay subschemes, (ii) we obtain new “flop-flop=twist” results for a large class of flops obtained by crepant resolutions of degeneracy loci. As another consequence, this gives a perverse schober on C. (iii) we give applications of above results to symmetric powers of curves and Θ-flops, following Toda [79].

Deformations of Polystable Sheaves on Surfaces: Quadraticity Implies Formality

We study relations between the quadraticity of the Kuranishi family of a coherent sheaf on a complex projective scheme and the formality of the DG-Lie algebra of its derived endomorphisms. In particular, we prove that for a polystable coherent sheaf on a smooth complex projective surface the DG-Lie algebra of derived endomorphisms is formal if and only if the Kuranishi family is quadratic.

Notions of Stein spaces in non-Archimedean geometry

Let k k be a non-Archimedean complete valued field and let X X be a k k -analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: (1) for every complete valued extension k ′ k’ of k k , every coherent sheaf on X × k k ′ X \times _{k} k’ is acyclic; (2) X X is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex), and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, X X is compact); (3) X X admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity. When X X has no boundary the characterization is simpler: in (2) the vanishing of higher cohomology groups of the structure sheaf is no longer needed, so that we recover the usual notion of Stein space in complex geometry; in (3) the domains considered by Liu can be replaced by affinoid domains, which leads us back to Kiehl’s definition of Stein space.

Bounding Selmer Groups for the Rankin–Selberg Convolution of Coleman Families

AbstractLet f and g be two cuspidal modular forms and let ${\mathcal {F}}$ be a Coleman family passing through f, defined over an open affinoid subdomain V of weight space $\mathcal {W}$ . Using ideas of Pottharst, under certain hypotheses on f and $g,$ we construct a coherent sheaf over $V \times \mathcal {W}$ that interpolates the Bloch–Kato Selmer group of the Rankin–Selberg convolution of two modular forms in the critical range (i.e, the range where the p-adic L-function $L_p$ interpolates critical values of the global L-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$ .

Fiber Invariants of Projective Morphisms and Regularity of Powers of Ideals

We introduce an invariant, associated to a coherent sheaf of graded modules over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of the regularity and a∗-invariant of powers of homogeneous ideals. Specifically, for an equigenerated homogeneous ideal I in a standard graded algebra over a Noetherian ring, we give bounds for the smallest values of power q starting from which a∗(Iq) and reg(Iq) become linear functions.

Local Langlands correspondence in rigid families

We show that local-global compatibility (at split primes) away from $p$ holds at all points of the $p$-adic eigenvariety of a definite $n$-variable unitary group. The novelty is we allow non-classical points, possibly non-etale over weight space. More precisely we interpolate the local Langlands correspondence for GL(n) across the eigenvariety by considering the fibers of its defining coherent sheaf. We employ techniques of Scholze from his new approach to the local Langlands conjecture.

Adic sheaves over weighted projective lines and elliptic curves

In the present paper, we study the behavior of adic sheaves over a weighted projective line of genus one or an elliptic curve, and describe the relationship between adic sheaves and generic sheaves. Moreover, we construct the generic sheaf through the coherent sheaf and adic sheaves in the sense of extension.

Kasteleyn operators from mirror symmetry

Given a consistent bipartite graph $$\Gamma $$ in $$T^2$$ with a complex-valued edge weighting $$\mathcal {E}$$ we show the following two constructions are the same. The first is to form the Kasteleyn operator of $$(\Gamma , \mathcal {E})$$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $$(\mathbb {C}^\times )^2$$ . The second is to form the conjugate Lagrangian $$L \subset T^* T^2$$ of $$\Gamma $$ , equip it with a brane structure prescribed by $$\mathcal {E}$$ , and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of $$(\mathbb {C}^\times )^2$$ determined by the Legendrian link which lifts the zig-zag paths of $$\Gamma $$ (and to which the noncompact Lagrangian L is asymptotic). We work in the setting of the coherent–constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. We also show that tensoring with line bundles on the compactification is mirror to certain Legendrian autoisotopies of the asymptotic boundary of L.

On Deformations of Pairs (Manifold, Coherent Sheaf)

AbstractWe analyse infinitesimal deformations of pairs $(X,{\mathcal{F}})$ with ${\mathcal{F}}$ a coherent sheaf on a smooth projective variety $X$ over an algebraically closed field of characteristic 0. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai–Artamkin theorem about the trace map.

2-Nilpotent co-Higgs structures

A co-Higgs sheaf on a smooth complex projective variety X is a pair of a torsion-free coherent sheaf $$\mathcal {E}$$ and a global section of $$\mathcal {E}nd(\mathcal {E})\otimes T_X$$ with $$T_X$$ the tangent bundle. We construct 2-nilpotent co-Higgs sheaves of rank two for some rational surfaces and of rank three for $$\mathbb {P}^3$$ , using the Hartshorne-Serre correspondence. Then we investigate the non-existence, especially over projective spaces.