Linear categories naturally have several identification relations : isomorphisms, categorical equivalences and Morita equivalences. In this thesis, we construct the classifying stacks for these three relations ($\ukcatiso$, $\ukcateq$, $\ukcatmor$) together with the classifying stack of abelian categories ($\ukab$), the originality of the subject being that, apart from the first one, these are higher stacks.
The principal result is that, under some finiteness assumptions, these stacks are geometric in the sense of C. Simpson. In particular, one recover the Hochschild cohomology of a category $C$ as the tangent complex, i.e. the object classifying first order deformations of $C$, of these stacks at the point defined by $C$.
Moreover, there exists a natural sequence of surjective morphisms of stacks : $$\ukcatiso \tto \ukcateq \tto \ukcatmor \tto \ukab$$ for which we prove that the middle one is etale, and the right one is an equivalence.
We construct the fine moduli space of log abelian varieties with PEL structure, which gives a toroidal compactification of the moduli space of abelian varieties with PEL structure.
Periods of moduli spaces of stable sheaves on K3 surfaces were computed by Mukai, O'Grady and the author. In this paper, we shall treat moduli spaces of stable sheaves on abelian surfaces.
The moduli space of abelian surfaces with polarisation of type (1,6) and a bilevel structure has positive Kodaira dimension. By contrast, Mukai has shown that the moduli space of bilevel-t abelian sufaces is rational for t=2,3,4,5.
In this note it is shown that associating to a principally polarized abelian variety its de Rham cohomology defines a faithfully flat morphism of the moduli space of principally polarized abelian varieties in positive characteristics to the moduli space of symplectic $F$-Zips as defined in \cite{MW}.
The moduli space $\cM_g$ of nonsingular projective curves of genus $g$ is compactified into the moduli $\bcM_g$ of Deligne-Mumford stable curves of genus $g$. We compactify in a similar way the moduli space of abelian varieties by adding some mildly degenerating limits of abelian varieties. A typical case is the moduli space of Hesse cubics. Any Hesse cubic is GIT-stable in the sense that its $\SL(3)$-orbit is closed in the semistable locus, and conversely any GIT-stable planar cubic is one of Hesse cubics. Similarly in arbitrary dimension, the moduli space of abelian varieties is compactified…
We determine the Artin-Mazur étale homotopy types of moduli stacks of polarised abelian schemes using transcendental methods and derive some arithmetic properties of the étale fundamental groups of these moduli stacks. Finally we analyse the Torelli morphism between the moduli stacks of algebraic curves and principally polarised abelian schemes from an étale homotopy point of view.
We study the moduli space of abelian threefolds with Iwahori level structure in positive characteristic. We explicitly determine the fibers of the canonical projection to the moduli space of principally polarized abelian varieties and draw conclusions about the relationship between the Ekedahl-Oort, the Kottwitz-Rapoport and the Newton stratification on these spaces.
We give an example of a regular dessin d'enfant whose field of moduli is not an abelian extension of the rational numbers, namely it is the field generated by a cubic root of 2. This answers a previous question. We also prove that the underlying curve has non-abelian field of moduli itself, giving an explicit example of a quasiplatonic curve with non-abelian field of moduli. In the last section, we note that two examples in previous literature can be used to find other examples of regular dessins d'enfants with non-abelian field of moduli.
Compactifications of moduli spaces of (1,p)-polarized abelian surfaces with level structures of canonical type have been described in great detail by Hulek, Kahn and Weintraub. The aim of this paper is to determine some invariants of smooth models of these moduli spaces. In particular, a geometric description of their canonical divisors is given and their Chern numbers are computed.
We shall study the chamber structure of positive cone of the albanese fiber of the moduli spaces of stable objects on an abelian surfaces via the chamber structure of stability conditions.
In a previous paper we showed that for every polarization on an abelian variety there is a dual polarization on the dual abelian variety. In this note we extend this notion of duality to families of polarized abelian varieties. As a main consequence we obtain an involution on the set of moduli spaces of polarized abelian varieties of dimension $g$. In particular, the moduli spaces $\mathcal A_{(d_1,...,d_g)}$ and $\mathcal A_{(d_1,\frac{d_1d_g}{d_{g-1}},...,\frac{d_1d_g}{d_2},d_g)}$ are canonically isomorphic.
In this paper we give a lower bound for the codimension of the Andreotti-Mayer loci in the moduli space of principally polarized complex abelian varieties. We also present a conjecture on this codimension.
This is the sequel to arXiv:math/0001089. In this paper, we complete the promised description of moduli of abelian surfaces of low degree, covering the cases of degree (1,12), (1,14), (1,16), (1,18) and (1,20). In each case, we describe birational models for the moduli space of abelian surfaces with polarization of the given degree and either a level structure, or what we call a partial level structure. In every case, the moduli space is rational or unirational. In additional, several new Calabi-Yau threefolds fibred in abelian surfaces are described.
We compute moduli spaces of Bridgeland stable objects on an irreducible principally polarized complex abelian surface corresponding to twisted ideal sheaves. We use Fourier-Mukai techniques to extend the ideas of Arcara and Bertram to express wall-crossings as Mukai flops and show that the moduli spaces are projective.
We describe the moduli spaces of morphisms between polarized complex abelian varieties. The discrete invariants, derived from a Poincare' decomposition of morphisms, are the types of polarizations and of lattice homomorphisms occurring in the decomposition. For a given type of morphisms the moduli variety is irreducible, and is obtained from a product of Siegel spaces modulo the action of a discrete group.
The moduli space of (1,p)-polarized abelian surfaces is a quasi-projective variety. In the case when p is a prime, we study its Kodaira dimension. We show that it is of general type for p > 71 and some smaller values of p. This improves an earlier result of G. K. Sankaran.
We study the geometry of some varieties of sums of powers related to the Klein quartic. This allows us to describe the birational geometry of certain moduli spaces of abelian surfaces. In particular we show that the moduli space $\mathcal{A}_2(1,7)^{-}_{sym}$ of $(1,7)$-polarized abelian surfaces with a symmetric theta structure and an odd theta characteristic is unirational by showing that it admits a dominant morphism from a unirational conic bundle.
We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties.
We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties. We also give a generalization to higher genera of the famous formula $12 λ_1=δ$ for genus 1.
Lecture notes at a conference on Arithmetic Geometry, Goettingen, July/August 2006: Density of ordinary Hecke orbits and a conjecture by Grothendieck on deformations of p-divisible groups.
Let $\mathcal{A}_g$ denote the moduli space of principally polarized abelian varieties of dimension $g \ge 3.$ In this paper we prove the connectedness of the singular sublocus of $\mathcal{A}_g$ consisting of those abelian varieties which possess an involution different from $-id$.
We study the moduli space $\mathbf{M}_X(Λ, n)$ of semistable $Λ$-modules of vanishing Chern classes over an abelian variety $X$, where $Λ$ belongs to a certain subclass of $D$-algebras. In particular, for $Λ= \mathcal{D}_X$ (resp. $Λ= \mathrm{Sym}^\bullet \mathcal{T}X$) we obtain a description of the moduli spaces of flat connections (resp. Higgs bundles). We give a description of $\mathbf{M}_X(Λ, n)$ in terms of a symmetric product of a certain fibre bundle over the dual abelian variety $\hat{X}$. We also give a moduli interpretation to the associated Hilbert scheme as the classifying space o…
We investigate the moduli spaces of one- and two-dimensional sheaves on projective K3 and abelian surfaces that are semistable with respect to a nongeneral ample divisor with regard to the symplectic resolvability. We can exclude the existence of new examples of projective irreducible symplectic manifolds lying birationally over components of the moduli spaces of one-dimensional semistable sheaves on K3 surfaces, and over components of many of the moduli spaces of two-dimensional sheaves on K3 surfaces, in particular, of those for rank two sheaves.
Perfect quadratic forms give a toroidal compactification of the moduli space of principally polarized abelian g-folds that is Q-factorial and whose ample classes are characterized, over any base. In characteristic zero it has canonical singularities if g is at least 5, and is the canonical model (in the sense of Mori and Reid) if g is at least 12.
In this paper we consider the stratification on the moduli space of principally polarized abelian surfaces in characteristic $p>0$ defined by the height of the formal group associated to $H^2(X,O_X)$. We compute the cycle classes of the strata and compare these with those of the $p$-rank stratification.
For fixed $k<g$ and a family of polarized abelian varieties of dimension $g$ over $\mathbb{R}$, we give a criterion for the density in the parameter space of those abelian varieties over $\mathbb{R}$ containing a $k$-dimensional abelian subvariety over $\mathbb{R}$. As application, we prove density of such a set in the moduli space of polarized real abelian varieties of dimension $g$, and density of real algebraic curves mapping non-trivially to real $k$-dimensional abelian varieties in the moduli space of real algebraic curves as well as in the space of real plane curves. This extends to the …
We construct a canonical compactification $SQ^{toric}_{g,K}$ of the moduli of abelian varieties over $Z[ζ_N, 1/N]$ where $ζ_N$ is a primitive $N$-th root of unity. This is very similar to, but slightly diferent from the compactification constructed by the author in Inventiones vol. 136 (1999). Any degenerate abelian scheme on the boundary of $SQ^{toric}_{g,K}$ is reduced and singular and it is one of the stable quasi-abelian varieties introduced by Alexeev and the author (Tohoku J. 1999).
We show that the moduli spaces of sheaves on a projective K3 surface are irreducible symplectic varieties, and that the same holds for the fibers of the Albanese map of moduli spaces of sheaves on an Abelian surface.
For an abelian or a projective K3 surface $X$ over an algebraically closed field $k$, consider the moduli space $\splcpx_{X/k}\uet$ of the objects $E$ in $D^b(\mathrm{Coh}(X))$ satisfying $\Ext^{-1}_X(E,E)=0$ and $\Hom(E,E)\cong k$. Then we can prove that $\splcpx_{X/k}\uet$ is smooth and has a symplectic structure.
This paper is an expository survey of results about the effective divisors on moduli spaces, with a focus on what is known about the effective cones of moduli spaces of stable curves and of principally polarized abelian varieties. This version incorporates clarifications and suggestions of Ana-Maria Castravet and the referee, both of whom we thank. To appear in the proceedings of the conference, "Joe at 60:A Celebration of Algebraic Geometry" edited by Brendan Hassett, James McKernan, Jason Starr and Ravi Vakil.
For a finite abelian subgroup $G\subset SL_n(\mathbb{C})$, we study whether a given crepant resolution $X$ of the quotient variety $\mathbb{C}^n/G$ is obtained as a moduli space of $G$-constellations. In particular we show that, if $X$ admits a natural $G$-constellation family in the sense of Logvinenko over it with all fibers being indecomposable as $\mathbb{C}[\mathbb{C}^n]$-modules, then $X$ is isomorphic to the normalization of a fine moduli space of $G$-constellations.
In this paper, we consider moduli spaces of stable sheaves on abelian surfaces. Our main assumption is the primitivity of the associated Mukai vector. We construct many isomorphisms of muduli spaces induced by Fourier-Mukai functor. As an application, we show that deformation type of these spaces are determined by their dimension. We next show that the fiber of the albanese map is irreducible symplectic manifolds In particular, we describe the period by using Mukai lattice. We also discuss deformation type of moduli spaces of stable sheaves on K3 surfaces.
We remove the global quotient presentation input in the theory of windows in derived categories of smooth Artin stacks of finite type. As an application, we use existing results on flipping of strata for wall-crossing of Gieseker semi-stable torsion-free sheaves of rank two on rational surfaces to produce semi-orthogonal decompositions relating the different moduli stacks. The complementary pieces of these semi-orthogonal decompositions are derived categories of products of Hilbert schemes of points on the surface.
We show that the moduli spaces of Thaddeus pairs on smooth projective curves and those of dual pairs are related by d-critical flips, which are virtual birational transformations introduced by the second author. We then prove the existence of fully-faithful functors between derived categories of coherent sheaves on these moduli spaces. Our result gives an evidence of a d-critical analogue of Bondal-Orlov, Kawamata's D/K equivalence conjecture, and also a categorification of wall-crossing formula of Donaldson-Thomas type invariants on ADHM sheaves introduced by Diaconescu.
We give a generalization to higher genera of the famous formula $12 λ=δ$ for genus 1. We also compute the classes of certain strata in the Satake compactification as elements of the push down of the tautological ring.
We study the birational geometry of some moduli spaces of abelian varieties with extra structure: in particular, with a symmetric theta structure and an odd theta characteristic. For a $(d_1,d_2)$-polarized abelian surface, we show how the parities of the $d_i$ influence the relation between canonical level structures and symmetric theta structures. For certain values of $d_1$ and $d_2$, a theta characteristic is needed in order to define Theta-null maps. We use these Theta-null maps and preceding work of other authors on the representations of the Heisenberg group to study the birational geom…
This paper studies stable sheaves on abelian surfaces of Picard number one. Our main tools are semi-homogeneous sheaves and Fourier-Mukai transforms. We introduce the notion of semi-homogeneous presentation and investigate the behavior of stable sheaves under Fourier-Mukai transforms. As a consequence, an affirmative proof is given to the conjecture proposed by Mukai in the 1980s. This paper also includes an explicit description of the birational correspondence between the moduli spaces of stable sheaves and the Hilbert schemes.
We prove that the moduli space A_{11}^{lev} of (1,11) polarized abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface: a^2b+b^2c+c^2d+d^2e+e^2a=0 in P^4. Therefore, A_{11}^{lev} is unirational but not rational, and there are no Gamma_{11}-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of A_{9}^{lev}.
We study Le Potier's strange duality conjecture for moduli spaces of sheaves over generic abelian surfaces. We prove the isomorphism for abelian surfaces which are products of elliptic curves, when the moduli spaces consist of sheaves of equal ranks and fiber degree 1. The birational type of the moduli space of sheaves is also investigated. Generalizations to arbitrary product elliptic surfaces are given.
We find explicit projective models of a compact Shimura curve and of a (non-compact) surface which are the moduli spaces of principally polarised abelian fourfolds with an automorphism of order five. The surface has a 24-nodal canonical model in IP^4 which is the complete intersection of two S_5-invariant cubics. It is dominated by a Hilbert modular surface and we give a modular interpretation for this. We also determine the L-series of these varieties as well as those of several modular covers of the Shimura curve.
In this paper we study and relate the non-abelian zeta functions introduced by Weng and invariants of the moduli spaces of arbitrary rank stable pairs over curves. We prove a wall-crossing formula for the latter invariants and obtain an explicit formula for these invariants in terms of the motive of a curve. Previously, formulas for these invariants were known only for rank 2 due to Thaddeus and for rank 3 due to Muñoz. Using these results we obtain an explicit formula for the non-abelian zeta functions, we check the uniformity conjecture by Weng for the ranks 2 and 3, and we prove the countin…
Starting from certain perverse sheaves on an abelian variety, including the intersection cohomology sheaves of curves and smooth ample divisors, we construct a semisimple super-Tannakian category.
Consider all moduli points corresponding with polarized abelian varieties in characteristic p such that the associated quasi-polarized p-divisible group is geometrically isomorphic with a given one. This defines a subset C of the moduli space. We show that C is closed in the corresponding open Newton polygon stratum. In this way the stratum is the disjoint union of such "central leaves". Under local-local isogenies we consider the orbit of one point: we obtain a second type of leaves. Every stratum, up to a finite map, is the product of one of its central leaves and one of its isogeny leaves. …
Let J be an abelian surface with a generic ample line bundle O(1). For n>0, the moduli space M(2, 0, 2n) of O(1)-semistable sheaves F of rank 2 with Chern classes c_1(F) = 0, c_2(F) = 2n is a singular projective variety, endowed with a holomorphic symplectic structure on the smooth locus. In this paper, we show that there does not exist a crepant resolution of M(2; 0; 2n) for n>1. This certainly implies that there is no symplectic desingularization of M(2, 0, 2n) for n>1.
In this paper we study the second integral cohomology of moduli spaces of semistable sheaves on projective K3 surfaces. If $S$ is a projective K3 surface, $v$ a Mukai vector and $H$ a $v-$generic polarization on $S$, we show that $H^{2}(M_{v},\mathbb{Z})$ is a free $\mathbb{Z}-$module of rank 23 carrying a pure weight-two Hodge structure and a lattice structure, with respect to which $H^{2}(M_{v},\mathbb{Z})$ is Hodge isometric to the Hodge sublattice $v^{\perp}$ of the Mukai lattice of $S$. Similar results are proved for Abelian surfaces.
Given an action of a finite group $G$ on the derived category of a smooth projective variety $X$ we relate the fixed loci of the induced $G$-action on moduli spaces of stable objects in $D^b(\mathrm{Coh}(X))$ with moduli spaces of stable objects in the equivariant category $D^b(\mathrm{Coh}(X))_G$. As an application we obtain a criterion for the equivariant category of a symplectic action on the derived category of a symplectic surface to be equivalent to the derived category of a surface. This generalizes the derived McKay correspondence, and yields a general framework for describing fixed lo…
We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the theorem to (quasi-compact and separated) algebraic spaces. Using more advanced technology (and assuming flatness) we also give a proof of the folklore result that the group of autoequivalences of the category of quasi-coherent sheaves consists of automorphisms of the underlying space and twists by line bundles. We apply our strategy to prove analogous statement…
We show that the Poincaré bundle gives a fully faithful embedding from the derived category of a curve of sufficiently high genus into the derived category of its moduli space of bundles of rank $r$ with fixed determinant of degree 1. Moreover we show that a twist of the embedding, together with 2 exceptional line bundles, gives the start of a semi-orthogonal decomposition. This generalises results of Narasimhan and Fonarev-Kuznetsov, who embedded the derived category of a single copy of the curve, for rank 2.
This paper lays the foundation for determining the Kodaira dimension of the projectivized strata of Abelian differentials with prescribed zero and pole orders in large genus. We work with the moduli space of multi-scale differentials constructed in [BCGGM2] which provides an orbifold compactification of these strata. We establish the projectivity of the moduli space of multi-scale differentials, describe the locus of canonical singularities, and compute a series of effective divisor classes. Moreover, we exhibit a perturbation of the canonical class which allows the corresponding pluri-canonic…
