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Given Y a non-compact manifold or orbifold, we define a natural subspace of the cohomology of Y called the narrow cohomology. We show that despite Y being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The narrow cohomology proves useful for the study of genus zero Gromov-Witten theory. When Y is a smooth complex variety or Deligne-Mumford stack, one can define a quantum D-module on the narrow cohomology of Y. This yields a new formulation of quantum Serre duality.

Consider the strong subordination of a multivariate Lévy process with a multivariate subordinator. If the subordinate is a stack of independent Lévy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Lévy process; otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Lévy process even when strong subordination does not. Here we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a Lévy process then it is necessarily equal in law to weak subordination in two cases: firstly when the subordinator is deterministic, and secondly when it is pure-jump with finite activity.

We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite-dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for any given complex elliptic curve $\mathcal {C}$, the fiber of our construction over $\mathcal {C}$ is isomorphic to Grojnowski's equivariant elliptic cohomology theory associated to $\mathcal {C}$.

We study the Hitchin map for Gℝ-Higgs bundles on a smooth curve, where Gℝ is a quasi-split real form of a complex reductive algebraic group G. By looking at the moduli stack of regular Gℝ-Higgs bundles, we prove that it induces a banded gerbe structure on a slightly larger stack, whose band is given by sheaves of tori. This characterization yields a cocyclic description of the fibres of the corresponding Hitchin map by means of cameral data. According to this, fibres of the Hitchin map are categories of principal torus bundles on the cameral cover. The corresponding points inside the stack of G-Higgs bundles are contained in the substack of points fixed by an involution induced by the Cartan involution of Gℝ. We determine this substack of fixed points and prove that stable points are in correspondence with stable Gℝ-Higgs bundles.

We give criteria for certain morphisms from an algebraic stack to a (not necessarily algebraic) stack to admit an (appropriately defined) scheme-theoretic image. We apply our criteria to show that certain natural moduli stacks of local Galois representations are algebraic (or Ind-algebraic) stacks., 137 pages; final version although not incorporating minor edits made in proof, appeared as Algebraic Geometry 8 (1) (2021) 1--132

Given a smooth projective variety $X$ and a smooth divisor $D\subset X$. We study relative Gromov-Witten invariants of $(X,D)$ and the corresponding orbifold Gromov-Witten invariants of the $r$-th root stack $X_{D,r}$. For sufficiently large $r$, we prove that orbifold Gromov-Witten invariants of $X_{D,r}$ are polynomials in $r$. Moreover, higher genus relative Gromov-Witten invariants of $(X,D)$ are exactly the constant terms of the corresponding higher genus orbifold Gromov-Witten invariants of $X_{D,r}$. We also provide a new proof for the equality between genus zero relative and orbifold Gromov-Witten invariants, originally proved by Abramovich-Cadman-Wise \cite{ACW}. When $r$ is sufficiently large and $X=C$ is a curve, we prove that stationary relative invariants of $C$ are equal to the stationary orbifold invariants in all genera., 29 pages. Revised according to referees' suggestions. Including results from arXiv:1804.09905 on target curves. To appear in Geometry & Topology

We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localising tensor subcategory generated by the motive of the curve, using Bialynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes., 32 pages. We have moved the section on Bialynicki-Birula decompositions (Section 3 in v1) to another joint paper entitled 'On the Voevodsky motive of the moduli space of Higgs bundles on a curve'

We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed $\mathrm{PGL}_2(\mathbb{C})$ local systems on the associated marked bordered surface. We prove that the image of this map is contained in the union of the domains of the cluster charts. We discuss a number of open questions concerning this monodromy map., Comment: 54 pages. Version 2: Incorporated suggestions from referee

We give an algebraic analog of the functional equation of Riemann's theta function. More precisely, we define a `theta multiplier' line bundle over the moduli stack of principally polarized abelian schemes with theta characteristic and prove that its dual is isomorphic to the determinant bundle over the moduli stack. We do so by explicitly computing with Picard groups over the moduli stack. This is all done over the ring R=Z[1/2,i]: passing to the complex numbers, we recover the classical functional equation.

The purpose of the present paper is to develop the theory of (pre-)Tango structures and (dormant generic) Miura $\mathfrak{g}$-opers (for a semisimple Lie algebra $\mathfrak{g}$) defined on pointed stable curves in positive characteristic. A (pre-)Tango structure is a certain line bundle of an algebraic curve in positive characteristic, which gives some pathological (relative to zero characteristic) phenomena. In the present paper, we construct the moduli spaces of (pre-)Tango structures and (dormant generic) Miura $\mathfrak{g}$-opers respectively, and prove certain properties of them. One of the main results of the present paper states that there exists a bijective correspondence between the (pre-)Tango structures (of prescribed monodromy) and the dormant generic Miura $\mathfrak{s} \mathfrak{l}_2$-opers (of prescribed exponent). By means of this correspondence, we achieve a detailed understanding of the moduli stack of (pre-)Tango structures. As an application, we construct a family of algebraic surfaces in positive characteristic parametrized by a higher dimensional base space whose fibers are pairwise non-isomorphic and violate the Kodaira vanishing theorem., 74 pages, revised version

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 with canonical singularities 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. This paper is mathematically identical to the previous version. The reason that the paper is replaced is in order to point out that this paper is a generalization to the case of singular surfaces with canonical singularities of the paper "Vector fields and moduli of canonically polarized surfaces in positive characteristic" with reference arXiv:1710.03076 which treated only the case of smooth surfaces. The results of this paper supercede the results of the aforementioned paper making it obsolete., This paper is a generalization to the case of singular surfaces with canonical singularities of the paper "Vector fields and moduli of canonically polarized surfaces in positive characteristic" with reference arXiv:1710.03076 which treated only the case of smooth surfaces. The results of this paper supercede the results of the aforementioned paper making it obsolete. 46 pages

For a gerbe $\mathcal{Y}$ over a smooth proper Deligne–Mumford stack $\mathcal{B}$ banded by a finite group $G$, we prove a structure result on the Gromov–Witten theory of $\mathcal{Y}$, expressing Gromov–Witten invariants of $\mathcal{Y}$ in terms of Gromov–Witten invariants of $\mathcal{B}$ twisted by various flat $U(1)$‑gerbes on $\mathcal{B}$. This can be viewed as a Leray–Hirsch type of result for Gromov–Witten theory of gerbes.

We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties. Specifically, we prove that if the stability threshold of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a $$\Theta $$ -stratification on the moduli stack of Fano varieties.

We study in this article the dual of a (strictly) commutative group stack $G$ and give some applications. Using the Picard functor and the Picard stack of $G$, we first give some sufficient conditions for $G$ to be dualizable. Then, for an algebraic stack $X$ with suitable assumptions, we define an Albanese morphism $a_X : X\to A^1(X)$ where $A^1(X)$ is a torsor under the dual commutative group stack $A^0(X)$ of $Pic_{X/S}$. We prove that $a_X$ satisfies a natural universal property. We give two applications of our Albanese morphism. On the one hand, we give a geometric description of the elementary obstruction and of universal torsors (standard tools in the study of rational varieties over number fields). On the other hand we give some examples of algebraic stacks that satisfy Grothendieck's section conjecture., 40 pages. Final version, including the comments of the referees. To appear in IMRN

We find a new presentation of the stack of hyperelliptic curves of odd genus as a quotient stack and we use it to compute its integral Chow ring by means of equivariant intersection theory., Comment: 27 pages, published on Int. Math. Res. Not

We prove that the Brauer group of the moduli stack of elliptic curves M 1 , 1 , k over an algebraically closed field k of characteristic 2 is isomorphic to Z / ( 2 ) . We also compute the Brauer group of M 1 , 1 , k where k is a finite field of characteristic 2.

Degree one twisting of Deligne cohomology, as a differential refinement of integral cohomology, was established in previous work. Here we consider higher degree twists. The Rham complex, hence de Rham cohomology, admits twists of any odd degree. However, in order to consider twists of integral cohomology we need a periodic version. Combining the periodic versions of both ingredients leads us to introduce a periodic form of Deligne cohomology. We demonstrate that this theory indeed admits a twist by a gerbe of any odd degree. We present the main properties of the new theory and illustrate its use with examples and computations, mainly via a corresponding twisted differential Atiyah-Hirzebruch spectral sequence., 25 pages, paragraph added to introduction, references added

We compute the cohomological invariants of $\mathcal{H}_g$, the moduli stack of smooth hyperelliptic curves, for every odd $g$., Comment: 42 pages, we added an introduction to Rost's theory of Chow groups with coefficients and to equivariant intersection theory. To appear in Transformation Groups

We show that the canonical bundle of the Hurwitz stack classifying covers of genus g>1 and degree k>2 of the projective line is big. We show that all coarse moduli spaces of trigonal curves of genus g>1 are of general type., 16 pages. Final version, to appear in a special issue of Mathematische Zeitschrift in honor of Olivier Debarre

We consider certain decision problems for the free model of the theory of Cartesian monoids. We introduce a model of computation based on the notion of a single stack one-way PDA due to Ginsburg, Greibach and Harrison. This model allows us to solve problems such as (1) Given a finite set B of elements and an element F, is F a product of members of B? (2) Is the submonoid generated by the finite set B infinite? for certain fragments of the free Cartesian monoid. These fragments include the submonoid of right invertible elements and so our results apply to the Thompson-Higman groups., In Proceedings ACT 2020, arXiv:2101.07888

We probe type IIA geometries with \emph{folded} semiclassical strings those encode the strong coupling dynamics associated with long operators in a class of SCFTs pertaining to spacetimes whose dimension vary from $6d$ down to $2d$. We particularly focus on folded string configurations those are extended through stack of flavor $ Dp $ ($ p=6,8 $) branes localized along the internal manifold. Considering some specific examples within the realm of $ \mathcal{N}=2 $ linear quivers, we show that the associated string spectrum exhibits a pole as the string approaches flavor D6 branes. We identify this as a \emph{fragmentation} of long operators in the dual $ \mathcal{N}=2 $ SCFTs. On the other hand, a similar analysis for $ \mathcal{N}=(1,0) $ SCFTs in $6d$ as well as $ \mathcal{N}=(0,4) $ SCFTs in $ 4d $ reveals a set of new dispersion relations at strong coupling. Based on semiclassical calculations, we set an argument that explains either of these cases., Comment: version accepted in JHEP

For a connected reductive group $G$ over a finite field, we study automorphic vector bundles on the stack of $G$-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski--Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of $G$-zips and a category of admissible modules with actions of a zero-dimensional algebraic subgroup a Levi subgroup and monodromy operators., Comment: 33 pages

In this paper we give a complete description of the integral Chow ring of the stack $\mathscr{H}_{g,1}$ of 1-pointed hyperelliptic curves, lifting relations and generators from the Chow ring of $\mathscr{H}_g$. We also give a geometric interpretation for the generators.

We previously showed that the fundamental groupoid of a topological space can be defined by the Seifert–van Kampen theorem. This allowed us to give the first axiomatization of the topological fundamental groupoid. We will prove in this paper that the analogue holds for the étale fundamental groupoid of a Noetherian scheme $X$ as well.

Suppose $(X,\Omega,g)$ is a compact Spin(7)-manifold, e.g. a Riemannian 8-manifold with holonomy Spin(7), or a Calabi-Yau 4-fold. Let $G$ be U$(m)$ or SU$(m)$, and $P\to X$ be a principal $G$-bundle. We show that the infinite-dimensional moduli space ${\mathcal B}_P$ of all connections on $P$ modulo gauge is orientable, in a certain sense. We deduce that the moduli space ${\mathcal M}_P^{Spin(7)}\subset{\mathcal B}_P$ of irreducible Spin(7)-instanton connections on $P$ modulo gauge, as a manifold or derived manifold, is orientable. This improves theorems of Cao and Leung arXiv:1502.01141 and Mu\~noz and Shahbazi arXiv:1707.02998. If $X$ is a Calabi-Yau 4-fold, the derived moduli stack $\boldsymbol{\mathscr M}$ of (complexes of) coherent sheaves on $X$ is a $-2$-shifted symplectic derived stack $(\boldsymbol{\mathcal M},\omega)$ by Pantev-To\"en-Vaqui\'e-Vezzosi arXiv:1111.3209, and so has a notion of orientation by Borisov-Joyce arXiv:1504.00690. We prove that $(\boldsymbol{\mathscr M},\omega)$ is orientable, by relating algebro-geometric orientations on $(\boldsymbol{\mathscr M},\omega)$ to differential-geometric orientations on ${\mathcal B}_P$ for U$(m)$-bundles $P\to X$, and using orientability of ${\mathcal B}_P$. This has applications to the programme of defining Donaldson-Thomas type invariants counting moduli spaces of (semi)stable coherent sheaves on a Calabi-Yau 4-fold, as in Donaldson and Thomas 1998, Cao and Leung arXiv:1407.7659, and Borisov and Joyce arXiv:1504.00690. This is the third in a series arXiv:1811.01096, arXiv:1811.02405 on orientations of gauge-theoretic moduli spaces., Comment: 57 pages. (v2) Major rewrite: new title, added author, new material on Calabi-Yau manifolds

We compute the integral Chow ring of the stack of smooth, non-hyperelliptic curves of genus three. We obtain this result by computing the integral Chow ring of the stack of smooth plane quartics, by means of equivariant intersection theory., Comment: 40 pages, comments are welcome! To appear on Transactions of the American Mathematical Society

We construct Bridgeland stability conditions on the derived category of smooth quasi-projective Deligne-Mumford surfaces whose coarse moduli spaces have ADE singularities. This unifies the construction for smooth surfaces and Bridgeland's work on Kleinian singularities. The construction hinges on an orbifold version of the Bogomolov-Gieseker inequality for slope semistable sheaves on the stack, and makes use of the To\"en-Hirzebruch-Riemann-Roch theorem., Comment: 22 pages, comments are welcome! Final version, to appear in Math. Z. Section 5 rewritten, Appendix removed and submitted independently as arXiv:2103.10767

In this paper, which is a sequel of arXiv:2002.07494, we investigate, for any reductive group $G$ over an algebraically closed field $k$, the Picard group of the universal moduli stack $\mathrm{Bun}_{G,g,n}$ of $G$-bundles over $n$-pointed smooth projective curves of genus $g$. In particular: we give new functorial presentations of the Picard group of $\mathrm{Bun}_{G,g,n}$; we study the restriction homomorphism onto the Picard group of the moduli stack of principal $G$-bundles over a fixed smooth curve; we determine the Picard group of the rigidification of $\mathrm{Bun}_{G,g,n}$ by the center of $G$ as well as the image of the obstruction homomorphism of the associated gerbe. As a consequence, we compute the divisor class group of the moduli space of semistable $G$-bundles over $n$-pointed smooth projective curves of genus $g$., Comment: 56 pages. Final version, to appear on Ann. Sc. Norm. Super. Pisa

In this paper we study the moduli stack of complexes of vector bundles (with chain isomorphisms) over a smooth projective variety X via derived algebraic geometry. We prove that if X is a Calabi–Yau variety of dimension d then this moduli stack has a ( 1 − d ) -shifted Poisson structure. In the case d = 1 , we construct a natural foliation of the moduli stack by 0-shifted symplectic substacks. We show that our construction recovers various known Poisson structures associated to complex elliptic curves, including the Poisson structure on Hilbert scheme of points on elliptic quantum projective planes studied by Nevins and Stafford, and the Poisson structures on the moduli spaces of stable triples over an elliptic curves considered by one of us. We also relate the latter Poisson structures to the semi-classical limits of the elliptic Sklyanin algebras studied by Feigin and Odesskii.

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of$G$-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type$A_{1}^{n}$,$C_{2}$, and$\mathbf{F}_{p}$-split groups of type$A_{2}$(this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular$3$-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type.

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the Hassett moduli stack of weighted stable curves, and let $\overline{M}_{g,A[n]}$ be its coarse moduli space. These are compactifications of $\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively, obtained by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \leq 1$ to the markings; they are defined over $\mathbb{Z}$, and therefore over any field. We study the first order infinitesimal deformations of $\overline{\mathcal{M}}_{g,A[n]}$ and $\overline{M}_{g,A[n]}$. In particular, we show that $\overline{M}_{0,A[n]}$ is rigid over any field, if $g\geq 1$ then $\overline{\mathcal{M}}_{g,A[n]}$ is rigid over any field of characteristic zero, and if $g+n > 4$ then the coarse moduli space $\overline{M}_{g,A[n]}$ is rigid over an algebraically closed field of characteristic zero. Finally, we take into account a degeneration of Hassett spaces parametrizing rational curves obtained by allowing the weights to have sum equal to two. In particular, we consider such a Hassett $3$-fold which is isomorphic to the Segre cubic hypersurface in $\mathbb{P}^4$, and we prove that its family of first order infinitesimal deformations is non-singular of dimension ten, and the general deformation is smooth., 14 pages

We study the stack M of cameral covers for a complex reductive group G, introduced by Donagi and Gaitsgory. We compute its rational cohomology ring. In the special case G=GL(n), M is the stack of spectral covers. We also compute the cohomology ring of the stack of abstract regular G-Higgs bundles., 30 pages

For each simply connected, simple complex group $G$ we show that the direct sum of all vector bundles of conformal blocks on the moduli stack $\bar{\mathcal{M}}_{g, n}$ of stable marked curves carries the structure of a flat sheaf of commutative algebras. The fiber of this sheaf over a smooth marked curve $(C, \vec{p})$ agrees with the Cox ring of the moduli of quasi-parabolic principal $G-$bundles on $(C, \vec{p})$. We use the factorization rules on conformal blocks to produce flat degenerations of these algebras. These degenerations are toric in the case $G = SL_2(\mathbb{C}),$ and the resulting toric varieties are shown to be isomorphic to phylogenetic algebraic varieties from mathematical biology. We conclude with a proof that the Cox ring of the moduli stack of qausi-parabolic $SL_2(\mathbb{C})$ principal bundles over a generic curve is generated by conformal blocks of levels 1 and 2 with relations generated in degrees $2, 3,$ and 4.

Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero. We follow the strategy of Mozgovoy and Schiffmann for counting Higgs bundles over finite fields. The main new ingredient is a motivic version of a theorem of Harder about Eisenstein series claiming that all vector bundles have approximately the same motivic class of Borel reductions as the degree of Borel reduction tends to $-\infty$., Comment: Minor corrections and improvements; 48 pages

For a Calabi–Yau 4-fold ( X , ω ) , where X is quasi-projective and ω is a nowhere vanishing section of its canonical bundle K X , the (derived) moduli stack of compactly supported perfect complexes M X is −2-shifted symplectic and thus has an orientation bundle O ω → M X 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 M Y × M D M Y and express these as pullbacks of Z 2 -bundles in gauge theory, constructed using positive Dirac operators on the double of X. As a result, we relate the orientation bundle O ω → M X to a gauge-theoretic orientation on the classifying space of compactly supported K-theory. Using orientability of the latter, we obtain orientability of M X . 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.

The purpose of this paper is to lay the foundations of a theory of invariants in \'etale cohomology for smooth Artin stacks. We compute the invariants in the case of the stack of elliptic curves, and we use the theory we developed to get some results regarding Brauer groups of algebraic spaces., Comment: final version

We prove a general criterion for an algebraic stack to admit a good moduli space. This result may be considered as a generalization of the Keel–Mori theorem, which guarantees the existence of a coarse moduli space for a separated Deligne–Mumford stack. We apply this result to prove that the moduli stacks $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ parameterizing $\unicode[STIX]{x1D6FC}$-stable curves introduced in [J. Alper et al., Second flip in the Hassett–Keel program: a local description, Compositio Math. 153 (2017), 1547–1583] admit good moduli spaces.

Let G be a simple simply-connected group over an algebraically closed field k, X be a smooth connected projective curve over k. In this paper we develop the theory of geometric Eisenstein series on the moduli stack Bun_G of G-torsors on X in the setting of the quantum geometric Langlands program (for \'etale l-adic sheaves) in analogy with [3]. We calculate the intersection cohomology sheaf on the version of Drinfeld compactification in our twisted setting. In the case G=SL_2 we derive some results about the Fourier coefficients of our Eisenstein series. In the case of G=SL_2 and X=P^1 we also construct the corresponding theta-sheaves and prove their Hecke property., Comment: 69 pages, v4: new results are added

We show that the failure of the usual Verdier duality on Bun(G) leads to a new duality functor on the category of D-modules, and we study its relation to the operation of Eisenstein series.

In this paper we study the homotopy limits of cosimplicial diagrams of dg-categories. We first give an explicit construction of the totalization of such a diagram and then show that the totalization agrees with the homotopy limit in the following two cases: (1) the complexes of sheaves of $\mathcal O$-modules on the \v{C}ech nerve of an open cover of a ringed space $(X, \mathcal O)$; (2) the complexes of sheaves on the simplicial nerve of a discrete group $G$ acting on a space. The explicit models we obtain in this way are twisted complexes as well as their $D$-module and $G$-equivariant versions. As an application we show that there is a stack of twisted perfect complexes., Comment: v3: 22 pages, minor changes, to appear in Homology, Homotopy and Applications. v2: a new subsection (Section 4.5) is added on the stack of twisted perfect complexes

The height filtration on the stack of formal groups $$\mathcal{M}$$ FG is well known. We explore analogous filtration on a set of morphisms of formal group laws, which extends to the stack $$\mathcal{M}$$ FG. It is correctly defined colimit object for this filtration which can be identified with the colimit $$\mathcal{M}$$ FG,∞. As a corollary we prove explicitly density of additive formal group in any group law.

We investigate the moduli of genus 10 curves that are endowed with a faithful action of the icosahedral group $\mathcal{A}_5$. We show among other things that this has the structure of a Deligne-Mumford stack whose underlying coarse moduli space essentially consists of two copies of the pencil of plane sextics that was introduced by Winger in 1924, but with the unique unstable member (a triple conic) replaced by a smooth non-planar curve. The orbifold defined by any member has genus zero and comes with 4 orbifold points. We show that by numbering the points, we get a fine moduli space whose base is naturally a finite cover of $\Mcal_{0,4}$., To be appeared on European Journal of Mathematics

Let $X$ be a compact manifold, $G$ a Lie group, $P \to X$ a principal $G$-bundle, and $\mathcal{B}_P$ the infinite-dimensional moduli space of connections on $P$ modulo gauge. For a real elliptic operator $E_\bullet$ we previously studied orientations on the real determinant line bundle over $\mathcal{B}_P$. These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson. Here we consider complex elliptic operators $F_\bullet$ and introduce the idea of spin structures, square roots of the complex determinant line bundle of $F_\bullet$. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures. Our main result identifies spin structures on $X$ with orientations on $X \times S^1$. Thus, if $P \to X$ and $Q \to X \times S^1$ are principal $G$-bundles with $Q|_{X\times\{1\}} \cong P$, we relate spin structures on $(\mathcal{B}_P,F_\bullet)$ to orientations on $(\mathcal{B}_Q,E_\bullet)$ for a certain class of operators $F_\bullet$ on $X$ and $E_\bullet$ on $X\times S^1$. Combined with arXiv:1811.02405, we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups $G=U(m), SU(m)$. In a sequel arXiv:2001.00113 we apply this to define canonical orientation data for all Calabi-Yau 3-folds $X$ over the complex numbers, as in Kontsevich-Soibelman arXiv:0811.2435, solving a long-standing problem in Donaldson-Thomas theory., Comment: 53 pages. (v3) final version, to appear in Advances in Mathematics

For a split reductive group G over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated G-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic t-structures on triangulated categories of motives. This is in accordance with general expectations on the independence of l in the Langlands correspondence for function fields., Added a comparison with equivariant higher Chow groups, further minor edits. Final version, to appear in Forum of Mathematics Sigma. The former section 6 on Motives on Witt vector affine flag varieties has been removed and will appear elsewhere

In the setting of the geometric Langlands conjecture, we argue that the phenomenon of divergence at infinity on Bun G {\operatorname{Bun}_{G}} (that is, the difference between ! {!} -extensions and * {*} -extensions) is controlled, Langlands-dually, by the locus of semisimple G ˇ {{\check{G}}} -local systems. To see this, we first rephrase the question in terms of Deligne–Lusztig duality and then study the Deligne–Lusztig functor 𝖣𝖫 G spec {\mathsf{DL}_{G}^{\mathrm{spec}}} acting on the spectral Langlands DG category IndCoh 𝒩 ⁢ ( LS G ) {{\mathrm{IndCoh}}_{\mathcal{N}}({\mathrm{LS}}_{G})} . We prove that 𝖣𝖫 G spec {\mathsf{DL}_{G}^{\mathrm{spec}}} is the projection IndCoh 𝒩 ⁢ ( LS G ) ↠ QCoh ⁢ ( LS G ) {{\mathrm{IndCoh}}_{\mathcal{N}}({\mathrm{LS}}_{G})\twoheadrightarrow{\mathrm{% QCoh}}({\mathrm{LS}}_{G})} , followed by the action of a coherent D-module St G ∈ 𝔇 ⁢ ( LS G ) {{\mathrm{St}}_{G}\in\mathfrak{D}({\mathrm{LS}}_{G})} , which we call the Steinberg D-module. We argue that St G {{\mathrm{St}}_{G}} might be regarded as the dualizing sheaf of the locus of semisimple G-local systems. We also show that 𝖣𝖫 G spec {\mathsf{DL}_{G}^{\mathrm{spec}}} , while far from being conservative, is fully faithful on the subcategory of compact objects.

We introduce a variant of the birational symbols group of Kontsevich, Pestun, and the second author, and use this to define birational invariants of algebraic orbifolds., Comment: 20 pages

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $\Theta$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space., Comment: 32 pages. Final version. To appear in Inventiones Math

We compute the Picard group of the universal abelian variety over the moduli stack $\mathscr A_{g,n}$ of principally polarized abelian varieties over $\mathbb{C}$ with a symplectic principal level $n$-structure. We then prove that over $\mathbb{C}$ the statement of the Franchetta conjecture holds in a suitable form for $\mathscr A_{g,n}$., Comment: 17 pages

Let G be a connected reductive complex algebraic group, and E a complex elliptic curve. Let G E denote the connected component of the trivial bundle in the stack of semistable G-bundles on E. We introduce a complex analytic uniformization of G E by adjoint quotients of reductive subgroups of the loop group of G. This can be viewed as a nonabelian version of the classical complex analytic uniformization E ≃ C ⁎ / q Z . We similarly construct a complex analytic uniformization of G itself via the exponential map, providing a nonabelian version of the standard isomorphism C ⁎ ≃ C / Z , and a complex analytic uniformization of G E generalizing the standard presentation E ≃ C / ( Z ⊕ Z τ ) . Finally, we apply these results to the study of sheaves with nilpotent singular support. As an application to Betti geometric Langlands conjecture in genus 1, we define a functor from S h N ( G E ) (the semistable part of the automorphic category) to IndCoh N ˇ ( Locsys G ˇ ( E ) ) (the spectral category).