The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan's Gromov-Witten Theory of orbifolds. This work will be described in detail in a subsequent paper.
The presentation here is generally based on lectures given by two of us at the Orbifold Workshop in Madison, Wisconsin. Following the spirit of the workshop, we give a nontechnical introduction to stacks, a general discussion of twisted stable maps, and a run-through of the issues that arise when generalizing Gromov-Witten theory to Deligne-Mumford stacks.
We make a special effort to make our constructions as canonical as we can, systematically using the language of algebraic stacks. Our efforts bear some concrete fruits; in particular, we are able to define the Chen-Ruan product in degree 0 (the so called stringy cohomology of the stack) with integer coefficients.
Beware - this note may be vanishing in front of your eyes: first, a somewhat less technical version intended for the workshop proceedings will shortly (we hope) be submitted. Second, a much more detailed paper is in preparation, in which all the technicalities which are swept under the rug here are discussed, as well as some cool stuff about taking roots.
In this paper, we show Langton's type theorem on separatedness and properness of moduli functor of torsion free semistable sheaves on algebraic orbifolds over an algebraically closed field k
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.
We introduce K-theoretic Gromov-Witten invariants of algebraic orbifold target spaces. Using the methods developed by Givental-Tonita we characterize Giventals Lagrangian cone of quantum K theory of orbifolds in terms of the cohomological cone.
Let X be a compact almost complex manifold with an action of a finite group G. We compute the algebra of G^n coinvariants of the stringy cohomology (math.AG/0104207) of X^n with an action of a wreath product of G. We show that it is isomorphic to the algebra A{S_n} defined by Lehn and Sorger (math.AG/0012166) where we set A to be the orbifold cohomology of [X/G]. As a consequence, we verify a special case of Ruan's cohomological hyper-kaehler conjecture (math.AG/0201123).
We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle TX. In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by D and multiplicities are bounded by the ramification indices of the components of D. We estimate precisely the curvature tensor of the co…
These are lecture notes of a C.I.M.E. course I gave at Cetraro, June 6-11 2005. The theory described is the version of Chen-Ruan's Gromov-Witten theory of orbifolds developed by Graber, Vistoli and me in the algebraic setting, but with introduction beginning in Kontsevich's formula on rational plane curves and through Gromov-Witten theory of algebraic manifolds. As this is not a joint paper, I finally get to give direct credit to Vistoli's beautiful ideas! In the appendix, I include, with permissions, two relevant letters Kontsevich wrote to Borisov in July 1996. People who study motivic integ…
Due to the work of many authors in the last decades, given an algebraic orbifold (smooth proper Deligne-Mumford stack with trivial generic stabilizer), one can construct its orbifold Chow ring and orbifold Grothendieck ring, and relate them by the orbifold Chern character map, generalizing the fundamental work of Chen-Ruan on orbifold cohomology. In this paper, we extend this theory naturally to higher Chow groups and higher algebraic K-theory, mainly following the work of Jarvis-Kaufmann-Kimura and Edidin-Jarvis-Kimura.
We study the multiplicative structure of orbifold Hochschild cohomology in an attempt to generalize the results of Kontsevich and Calaque-Van den Bergh relating the Hochschild and polyvector field cohomology rings of a smooth variety. We introduce the concept of linearized derived scheme, and we argue that when $X$ is a smooth algebraic variety and $G$ is a finite abelian group acting on $X$, the derived fixed locus $\widetilde{X^G}$ admits an HKR linearization. This allows us to define a product on the cohomology of polyvector fields of the orbifold $[X/G]$. We analyze the obstructions to ass…
We prove that the wreath product orbifolds studied earlier by the first author provide a large class of higher dimensional examples of orbifolds whose orbifold Hodge numbers coincide with the ordinary ones of suitable resolutions of singularities. We also make explicit conjectures on elliptic genera for the wreath product orbifolds.
We work through, in detail, the orbifold quantum cohomology, with gravitational descendants, of the stack BG, the point modulo trivial action of a finite group G. We provide a simple description of algebraic structures on the state space of this theory. As a consequence, we find that multiple copies of commuting Virasoro algebras appear in this theory which completely determine the correlators of the theory.
The existence of interesting multiplicative cohomology theories for orbifolds was first suggested by string theorists. Orbifold products have been intensely studied by mathematicians for the last fifteen years. In this paper we focus on the virtual orbifold product that was first introduced in Lupercio et al. (2007). We construct a categorification of the virtual orbifold product that leverages the geometry of derived loop stacks. By work of Ben-Zvi, Francis and Nadler, this reveals connections between virtual orbifold products and Drinfeld centers of monoidal categories, thus answering a ques…
We give a purely equivariant construction of orbifold products for quotient Deligne-Mumford stacks [X/G] where G is an arbitrary linear algebraic group (not necessarily finite). The key to our construction is the definition of the "logarithmic trace" of an equivariant vector bundle. We also prove that there is an orbifold Chern character homomorphism which induces an isomorphism of a canonical summand in the orbifold Grothendieck ring with the orbifold Chow ring. As an application we obtain an associative orbifold product on the Grothendieck ring of [X/G] (as opposed to its inerita stack) take…
Chen and Ruan's orbifold cohomology of the symmetric product of a complex manifold is calculated. An isomorphism of rings (up to a change of signs) $H_{orb}^*(X^n/S_n;\complex) \cong H^*(X^{[n]};\complex)$ between the orbifold cohomology of the symmetric product of a smooth projective surface with trivial canonical class $X$ and the cohomology of its Hilbert scheme $X^{[n]}$ is obtained, yielding a positive answer to a conjecture of Ruan.
We define the notion of fundamental group of an algebraic stack, prove a comparison theorem between the fundamental group of a stack over the complex numbers and that of the associated analytic orbifold, show that this notion coincides with that defined with simplicial schemes by Oda and finally prove some short exact sequences involving these groups.
For a possibly singular complex variety $X$, generating functions of total "orbifold Chern homology classes" of symmetric products $S^nX$ are given. Those are very natural "Chern class versions" (in the sense of Schwartz-MacPherson) of known generating function formulae of (generalized) orbifold Euler characteristics of $S^nX$. In fact more generally we introduce the "class version" of the Dey-Wohlfahrt formula in classical group theory.
We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. We choose to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie, but our approach applies not only to different settings of algebraic geometry such as classical algebraic geometry, derived al…
In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, and we explicitely calculate the stringy product for a weighted projective orbifold. In the second part we conside…
In this short note we use the notion of power structure over the Grothendieck ring of complex algebraic varieties to study generating series of classes of Hilbert schemes of points on complex orbifolds.
This paper surveys the authors recent work on two variable elliptic genus of singular varieties. The last section calculates a generating function for the elliptic genera of symmetric products. This generalizes the classical results of Macdonald and Zagier.
This is a survey of the author's paper arXiv:1001.0023 on "Algebraic Geometry over C-infinity rings". If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions c : X --> R is a "C-infinity ring". That is, for each smooth function f : R^n --> R there is an n-fold operation Φ_f : C^\infty(X)^n --> C^\infty(X) acting by Φ_f: (c_1,...,c_n) |--> f(c_1,...,c_n), and these operations Φ_f satisfy many natural identities. Thus, C^\infty(X) actually has a far richer structure than the obvious R-algebra structure. We explain a version of algebraic geometry in which rings or algebras ar…
We develop a graphical representation of polynomial invariants of unitary gauge groups, and use it to find the algebraic curve corresponding to a hyperkahler quotient of a linear space. We apply this method to four dimensional ALE spaces, and for the A_k, D_k, and E_6 cases, derive the explicit relation between the deformations of the curves away from the orbifold limit and the Fayet-Iliopoulos parameters in the corresponding quotient construction. We work out the orbifold limit of E_7, E_8, and some higher dimensional examples.
Withdrawn by the authors because the results of this paper are subsumed within and improved by the two papers 1. A plethora of inertial products and 2. Chern Classes and Compatible Power Operations in Inertial K-theory
Let A_r be the minimal resolution of the cyclic quotient singularity C^2/Z_{r+1}. We study the equivariant quantum cohomology ring of the n-fold symmetric product stack [Sym^n(A_r)] of A_r. We calculate the operators of quantum multiplication by divisor classes. Under the assumption of the nonderogatory conjecture, these operators completely determine the ring structure, which provides an affirmative answer to the Crepant Resolution Conjecture on [Sym^n(A_r)] and Hilb^n(A_r). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Sym^n…
This study examines the finite $F$-representation type (abbr. FFRT) property of a two-dimensional normal graded ring $R$ in characteristic $p>0$, using notions from the theory of algebraic stacks. Given a graded ring $R$, we consider an orbifold curve $\mathfrak C$, which is a root stack over the smooth curve $C=\text{Proj} R$, such that $R$ is the section ring associated with a line bundle $L$ on $\mathfrak C$. The FFRT property of $R$ is then rephrased with respect to the Frobenius push-forwards $F^e_*(L^i)$ on the orbifold curve $\mathfrak C$. As a result, we see that if the singularity of …
We show that the generating function of electrically charged 1/2-BPS states in N=4 supersymmetric Z_N-CHL orbifolds of the heterotic string on T^6 are given by multiplicative eta-products. The eta-products are determined by the cycle shape of the corresponding symplectic involution in the dual type II picture. This enables us to complete the construction of the genus-two Siegel modular forms due to David, Jatkar and Sen [arXiv:hep-th/0609109] for Z_N orbifolds when N is non-prime. We study the Z_4 CHL orbifold in detail and show that the associated Siegel modular forms, Φ_3(Z) and \widetildeΦ_…
We construct in an abstract fashion the orbifold quantum cohomology (quantum orbifold cohomology) of weighted projective space, starting from the orbifold quantum differential operator. We obtain the product, grading, and intersection form by making use of the associated self-adjoint D-module and the Birkhoff factorization procedure. The method extends to the more difficult case of Fano hypersurfaces in weighted projective space. However, in contrast to the case of weighted projective space itself or a Fano hypersurface in projective space, a "small Birkhoff cell" can appear in the constructio…
We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T^{poly}_X on X, and is generated as a T^{poly}_X-algebra by the sum of the determinants det(N_{X^g}) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild…
