Moduli Stack Research Articles

Picard sheaves, local Brauer groups, and topological modular forms

AbstractWe develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real ‐theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2‐torsion group.

Counting elliptic curves with a rational N-isogeny for small N

We count the number of rational elliptic curves of bounded naive height that have a rational N-isogeny, for N∈{2,3,4,5,6,8,9,12,16,18}. For some N, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack.

Components of moduli stacks of two-dimensional Galois representations

Abstract In the article [CEGS20b], we introduced various moduli stacks of two-dimensional tamely potentially Barsotti–Tate representations of the absolute Galois group of a p-adic local field, as well as related moduli stacks of Breuil–Kisin modules with descent data. We study the irreducible components of these stacks, establishing, in particular, that the components of the former are naturally indexed by certain Serre weights.

Brauer group of moduli stack of stable parabolic PGL(r)-bundles over a curve

Let [Formula: see text] be an algebraically closed field of characteristic zero. We prove that the Brauer group of the moduli stack of stable parabolic [Formula: see text]-bundles on a smooth projective curve, with full flag quasi-parabolic structures at an arbitrary parabolic divisor, coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of parabolic [Formula: see text]-bundles. We also compute the Brauer group of the smooth locus of this coarse moduli for more general quasi-parabolic types and weights satisfying certain conditions.

Generic Torelli and local Schottky theorems for Jacobian elliptic surfaces

Suppose that $f:X\to C$ is a general Jacobian elliptic surface over ${\mathbb {C}}$ of irregularity $q$ and positive geometric genus $h$. Assume that $10 h>12(q-1)$, that $h>0$ and let $\overline {\mathcal {E}\ell \ell }$ denote the stack of generalized elliptic curves. (1) The moduli stack $\mathcal {JE}$ of such surfaces is smooth at the point $X$ and its tangent space $T$ there is naturally a direct sum of lines $(v_a)_{a\in Z}$, where $Z\subset C$ is the ramification locus of the classifying morphism $\phi :C\to \overline {\mathcal {E}\ell \ell }$ that corresponds to $X\to C$. (2) For each $a\in Z$ the map $\overline {\nabla }_{v_a}:H^{2,0}(X)\to H^{1,1}_{\rm prim}(X)$ defined by the derivative $per_*$ of the period map $per$ is of rank one. Its image is a line ${\mathbb {C}}[\eta _a]$ and its kernel is $H^0(X,\Omega ^2_X(-E_a))$, where $E_a=f^{-1}(a)$. (3) The classes $[\eta _a]$ form an orthogonal basis of $H^{1,1}_{\rm prim}(X)$ and $[\eta _a]$ is represented by a meromorphic $2$-form $\eta _a$ in $H^0(X,\Omega ^2_X(2E_a))$ of the second kind. (4) We prove a local Schottky theorem; that is, we give a description of $per_*$ in terms of a certain additional structure on the vector bundles that are involved. Assume further that $8h>10(q-1)$ and that $h\ge q+3$. (5) Given the period point $per(X)$ of $X$ that classifies the Hodge structure on the primitive cohomology $H^2_{\rm prim}(X)$ and the image of $T$ under $per_*$ we recover $Z$ as a subset of ${\mathbb {P}}^{h-1}$ and then, by quadratic interpolation, the curve $C$. (6) We prove a generic Torelli theorem for these surfaces. Everything relies on the construction, via certain kinds of Schiffer variations of curves, of certain variations of $X$ for which $per_*$ can be calculated. (In an earlier version of this paper we used variations constructed by Fay. However, Schiffer variations are slightly more powerful.)

The Hilbert scheme of infinite affine space and algebraic K-theory

We study the Hilbert scheme \mathrm{Hilb}_{d}(\mathbf{A}^{\infty}) from an \mathbf{A}^{1} -homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme \mathrm{Hilb}_{d}(\mathbf{A}^{\infty}) is \mathbf{A}^{1} -equivalent to the Grassmannian of (d-1) -planes in \mathbf{A}^{\infty} . We then describe the \mathbf{A}^{1} -homotopy type of \mathrm{Hilb}_{d}(\mathbf{A}^{n}) in a certain range, for n large compared to d . For example, we compute the integral cohomology of \mathrm{Hilb}_{d}(\mathbf{A}^{n})(\mathbf{C}) in a range. We also deduce that the forgetful map \mathcal{FF}\mathrm{lat}\to\mathcal{V}\mathrm{ect} from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an \mathbf{A}^{1} -equivalence after group completion. This implies that the moduli stack \mathcal{FF}\mathrm{lat} , viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum \mathrm{kgl} representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the \mathrm{kgl} -homology of smooth proper schemes over a perfect field.

Virtual Riemann–Roch theorems for almost perfect obstruction theories

This is the third in a series of works devoted to constructing virtual structure sheaves and K-theoretic invariants in moduli theory. The central objects of study are almost perfect obstruction theories, introduced by Y.-H. Kiem and the author as the appropriate notion in order to define invariants in K-theory for many moduli stacks of interest, including generalized K-theoretic Donaldson–Thomas invariants. In this paper, we prove virtual Riemann–Roch theorems in the setting of almost perfect obstruction theory in both the non-equivariant and equivariant cases, including cosection localized versions. These generalize and remove technical assumptions from the virtual Riemann–Roch theorems of Fantechi–Göttsche and Ravi–Sreedhar. The main technical ingredients are a treatment of the equivariant K-theory and equivariant Gysin map of sheaf stacks and a formula for the virtual Todd class.

The Jordan–Chevalley decomposition for 𝐺-bundles on elliptic curves

We study the moduli stack of degree 0 0 semistable G G -bundles on an irreducible curve E E of arithmetic genus 1 1 , where G G is a connected reductive group in arbitrary characteristic. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups H H of G G (the E E -pseudo-Levi subgroups), where each stratum is computed in terms of H H -bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where E E has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra g \mathfrak {g} (respectively, algebraic group G G ). We also provide a Tannakian description of these moduli stacks and use it to show that if E E is not a supersingular elliptic curve, the moduli of framed unipotent bundles on E E are equivariantly isomorphic to the unipotent cone in G G . Finally, we classify the E E -pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples.

Two-dimensional categorified Hall algebras

In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category $\mathsf{Coh}^{\mathsf{b}}(\mathbb{R}\mathcal{M})$ of complexes of sheaves with bounded coherent cohomology on a derived moduli stack $\mathbb{R}\mathcal{M}$. In the surface case, $\mathbb{R}\mathcal{M}$ is a suitable derived enhancement of the moduli stack $\mathcal{M}$ of coherent sheaves on the surface. This construction categorifies the K-theoretical and cohomological Hall algebras of coherent sheaves on a surface of Zhao and Kapranov–Vasserot. In the curve case, we define three categorified Hall algebras associated with suitable derived enhancements of the moduli stack of Higgs sheaves on a curve $X$, the moduli stack of vector bundles with flat connections on $X$, and the moduli stack of finite-dimensional local systems on $X$, respectively. In the Higgs sheaves case we obtain a categorification of the K-theoretical and cohomological Hall algebras of Higgs sheaves on a curve of Minets and Sala–Schiffmann, while in the other two cases our construction yields, by passing to $\mathsf{K}0$, new K-theoretical Hall algebras, and by passing to $\mathsf{H}\ast^{\mathsf{BM}}$, new cohomological Hall algebras. Finally, we show that the Riemann–Hilbert and the non-abelian Hodge correspondences can be lifted to the level of our categorified Hall algebras of a curve.

The cohomological Hall algebra of a surface and factorization cohomology

In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category $\mathsf{Coh}^{\mathsf{b}}(\mathbb{R}\mathsf{M})$ of complexes of sheaves with bounded coherent cohomology on a derived moduli stack $\mathbb{R}\mathsf{M}$. In the surface case, $\mathbb{R}\mathsf{M}$ is a suitable derived enhancement of the moduli stack $\mathsf{M}$ of coherent sheaves on the surface. This construction categorifies the K-theoretical and cohomological Hall algebras of coherent sheaves on a surface of Zhao and Kapranov-Vasserot. In the curve case, we define three categorified Hall algebras associated with suitable derived enhancements of the moduli stack of Higgs sheaves on a curve $X$, the moduli stack of vector bundles with flat connections on $X$, and the moduli stack of finite-dimensional local systems on $X$, respectively. In the Higgs sheaves case we obtain a categorification of the K-theoretical and cohomological Hall algebras of Higgs sheaves on a curve of Minets and Sala-Schiffmann, while in the other two cases our construction yields, by passing to $\mathsf K_0$, new K-theoretical Hall algebras, and by passing to $\mathsf H_\ast^{\mathsf{BM}}$, new cohomological Hall algebras. Finally, we show that the Riemann-Hilbert and the non-abelian Hodge correspondences can be lifted to the level of our categorified Hall algebras of a curve.

The intersection theory of the moduli stack of vector bundles on

AbstractWe determine the integral Chow and cohomology rings of the moduli stack $\mathcal {B}_{r,d}$ of rank r, degree d vector bundles on $\mathbb {P}^1$ -bundles. We work over a field k of arbitrary characteristic. We first show that the rational Chow ring $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$ is a free $\mathbb {Q}$ -algebra on $2r+1$ generators. The isomorphism class of this ring happens to be independent of d. Then, we prove that the integral Chow ring $A^*(\mathcal {B}_{r,d})$ is torsion-free and provide multiplicative generators for $A^*(\mathcal {B}_{r,d})$ as a subring of $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$ . From this description, we see that $A^*(\mathcal {B}_{r,d})$ is not finitely generated as a $\mathbb {Z}$ -algebra. Finally, when $k = \mathbb {C}$ , the cohomology ring of $\mathcal {B}_{r,d}$ is isomorphic to its Chow ring.

Smoothing semi-smooth stable Godeaux surfaces

We show that all the semi-smooth stable complex Godeaux surfaces, classified in [FPR18a], are smoothable, and that the moduli stack is smooth of the expected dimension 8 at the corresponding points.

Cohomology of the Universal Abelian Surface with Applications to Arithmetic Statistics

Abstract The moduli stack $\mathcal{A}_2$ of principally polarized abelian surfaces comes equipped with the universal abelian surface $\pi: \mathcal{X}_2 \to \mathcal{A}_2$. The fiber of π over a point corresponding to an abelian surface A in $\mathcal{A}_2$ is A itself. We determine the $\ell$-adic cohomology of $\mathcal{X}_2$ as a Galois representation. Similarly, we consider the bundles $\mathcal{X}_2^n \to \mathcal{A}_2$ and $\mathcal{X}_2^{\textrm{Sym}(n)} \to \mathcal{A}_2$ for all $n \geq 1$, where the fiber over a point corresponding to an abelian surface A is An and $\textrm{Sym}^n A$, respectively. We describe how to compute the $\ell$-adic cohomology of $\mathcal{X}_2^n$ and $\mathcal{X}_2^{\textrm{Sym}(n)}$ and explicitly calculate it in low degrees for all n and in all degrees for n = 2. These results yield new information regarding the arithmetic statistics on abelian surfaces, including an exact calculation of the expected value and variance as well as asymptotics for higher moments of the number of Fq-points.

Chow rings of stacks of prestable curves I

Abstract We study the Chow ring of the moduli stack $\mathfrak {M}_{g,n}$ of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space $\overline {\mathcal {M}}_{g,n}$ of stable curves. This paper provides foundations for the paper [BS21]. In the appendix (jointly with J. Skowera), we develop the theory of a proper, but not necessary projective, pushforward of algebraic cycles. The proper pushforward is necessary for the construction of the tautological rings of $\mathfrak {M}_{g,n}$ and is important in its own right. We also develop operational Chow groups for algebraic stacks.

Vector fields on canonically polarized surfaces

This paper investigates the geometry of canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces. In particular, an explicit integer valued function f(x) is obtained with the following properties. If X is a canonically polarized surface defined over an algebraically closed field of characteristic \(p>0\) such that \(p>f(K_X^2) \) and X has a nontrivial global vector field, then X is unirational and the algebraic fundamental group is trivial. As a consequence of this result, large classes of canonically polarized surfaces are identified whose moduli stack is Deligne-Mumford, a property that does not hold in general in positive characteristic.

Orientations for DT invariants on quasi-projective Calabi–Yau 4-folds

For a Calabi–Yau 4-fold (X,ω), where X is quasi-projective and ω is a nowhere vanishing section of its canonical bundle KX, the (derived) moduli stack of compactly supported perfect complexes MX is −2-shifted symplectic and thus has an orientation bundle Oω→MX in the sense of Borisov–Joyce [11] necessary for defining Donaldson–Thomas type invariants of X. We extend first the orientability result of Cao–Gross–Joyce [23] to projective spin 4-folds. Then for any smooth projective compactification Y, such that D=Y﹨X is strictly normal crossing, we define orientation bundles on the stack MY×MDMY and express these as pullbacks of Z2-bundles in gauge theory, constructed using positive Dirac operators on the double of X. As a result, we relate the orientation bundle Oω→MX to a gauge-theoretic orientation on the classifying space of compactly supported K-theory. Using orientability of the latter, we obtain orientability of MX. We also prove orientability of moduli spaces of stable pairs and Hilbert schemes of proper subschemes. Finally, we consider the compatibility of orientations under direct sums.

Moduli spaces of abstract and embedded Kummer varieties

In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack [Formula: see text] of abstract Kummer varieties and the second one is the stack [Formula: see text] of embedded Kummer varieties. We will prove that [Formula: see text] is a Deligne-Mumford stack and its coarse moduli space is isomorphic to [Formula: see text], the coarse moduli space of principally polarized abelian varieties of dimension [Formula: see text]. On the other hand, we give a modular family [Formula: see text] of embedded Kummer varieties embedded in [Formula: see text], meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber. As a consequence, we construct the coarse moduli space [Formula: see text] of embedded Kummer surfaces and prove that it is obtained from [Formula: see text] by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: [Formula: see text] could be obtained from [Formula: see text] via a contraction for all [Formula: see text].

K3 curves with index k>1

Let mathcal {KC}_g ^k be the moduli stack of pairs (S, C) with S a K3 surface and Csubseteq S a genus g curve with divisibility k in mathrm {Pic}(S). In this article we study the forgetful map c_g^k:(S,C) mapsto C from mathcal {KC}_g ^k to {mathcal {M}}_g for k>1. First we compute by geometric means the dimension of its general fibre. This turns out to be interesting only when S is a complete intersection or a section of a Mukai variety. In the former case we find the existence of interesting Fano varieties extending C in its canonical embedding. In the latter case this is related to delicate modular properties of the Mukai varieties. Next we investigate whether c_g^k dominates the locus in {mathcal {M}}_g of k-spin curves with the appropriate number of independent sections. We are able to do this only when S is a complete intersection, and obtain in these cases some classification results for spin curves.

The logarithmic gauged linear sigma model

We introduce the notion of log R-maps, and develop a proper moduli stack of stable log R-maps in the case of a hybrid gauged linear sigma model. Two virtual cycles (canonical and reduced) are constructed for these moduli stacks. The main results are two comparison theorems relating the reduced virtual cycle to the cosection localized virtual cycle, as well as the reduced virtual cycle to the canonical virtual cycle. This sets the foundation of a new technique for computing higher genus Gromov-Witten invariants of complete intersections.

Smooth covers of moduli stacks of Riemann surfaces with symmetry

We construct explicitly a finite cover of the moduli stack of compact Riemann surfaces with a given group of symmetries by a smooth quasi-projective variety.