a reconstruction theorem for genus zero gromov witten invariants of stacks we generalize the first reconstruction theorem of kontsevich and manin in two respects first we allow the target space to be a deligne mumford stack second under some convergence assumptions we show it suffices to check the hypothesis of h 2 generation not on the cohomology ring but on an any quantum ring in the family given by small quantum cohomology as an example the latter result is used to compute genus zero gromov witten invariants of p 1 b

gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

gromov witten invariants in algebraic geometry gromov witten invariants for arbitrary non singular projective vari eties and arbitrary genus are constructed using the techniques from k behrend b fantechi the intrinsic normal cone

the boolean hierarchy and the polynomial hierarchy a closer connection we show that if the boolean hierarchy collapses to level k then the polynomial hierarchy collapses to bh 3 k where bh 3 k is the k th level of the boolean hierarchy over sigma p 2 this is an improvement over the known results which show that the polynomial hierarchy would collapse to p np np o log n this result is significant in two ways first the theorem says that a deeper collapse of the boolean hierarchy implies a deeper collapse of the polynomial hierarchy also this result points to some previously unexplored connections between the boolean and query hierarchies of delta p 2 and delta p 3 namely begin eqnarray bh k rm co bh k implies bh 3 k rm co bh 3 k jot p rm np k p rm np k 1 implies p rm np rm np k 1 p rm np rm np k 2 end eqnarray

evaluating tautological classes using only hurwitz numbers hurwitz numbers count ramified covers of a riemann surface with prescribed monodromy as such they are purely combinatorial objects tautological classes on the other hand are distinguished classes in the intersection ring of the moduli spaces of riemann surfaces of a given genus and are thus geometric localization computations in gromov witten theory provide non obvious relations between the two this paper makes one such computation and shows how it leads to a master relation theorem 0 1 that reduces the ratios of certain interesting tautological classes to the pure combinatorics of hurwitz numbers as a corollary we obtain a purely combinatorial proof of a theorem of bryan and pandharipande expressing in generating function form classical computations by faber looijenga theorem 0 2

sheaves on artin stacks we develop a theory of quasi coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of laumon and moret bailly we study basic cohomological properties of such sheaves and prove stack theoretic versions of grothendieck s fundamental theorem for proper morphisms grothendieck s existence theorem zariski s connectedness theorem as well as finiteness theorems for proper pushforwards of coherent and constructible sheaves we also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to laumon and moret bailly 1 1 in the book lm b the lisse etale topos of an algebraic stack was introduced and a theory of quasi coherent and constructible sheaves in this topology was developed unfortunately it was since observed by gabber and behrend independently that the lisse etale topos is not functorial as asserted in loc cit and hence the development of the theory of sheaves in this book is not satisfactory as is in addition since the publication of the book lm b several new results have been obtained such as finiteness of coherent and etale cohomology fa ol and various other consequences of chow s lemma ol the purpose of this paper is to explain how one can modify the arguments of lm b to obtain good theories of quasi coherent and constructible sheaves on algebraic stacks and in addition we provide an account of the theory of sheaves which also includes the more recent results mentioned above 1 2 the paper is organized as follows in section 2 we recall some aspects of the theory of cohomological descent sga4 v which will be used in what follows in section 3 we review the basic definitions of the lisse etale site cartesian sheaves over a sheaf of algebras and verify some basic properties of such sheaves in section 4 we relate the derived category of cartesian sheaves over some sheaf of rings to various derived categories of sheaves on the simplicial space obtained from a covering of the algebraic stack by an algebraic space loosely speaking the main result states that the cohomology of a complex with cartesian cohomology sheaves can be computed by restricting to the simplicial space obtained from a covering and computing cohomology on this simplicial space using the etale topology in section 5 we generalize these results to comparisons between ext groups computed in the lisse etale topos and ext groups computed using the etale topology on a hypercovering in section 6 we specialize the discussion of sections 3 5 to quasi coherent sheaves we show that if x is an algebraic stack and oxlis et denotes the structure sheaf of the lisse etale topos then the triangulated category d qcoh x of bounded below complexes of oxlis et modules with quasi coherent cohomology sheaves satisfies all the basic properties that one would expect from the theory for schemes for example we show in this section that if f x y is a quasi compact morphism of algebraic stacks and m is a quasi coherent sheaf on x date november 2 2005 1

the boolean hierarchy i structural properties in this paper we study the complexity of sets formed by boolean operations union intersection and complement on np sets these are the sets accepted by trees of hardware with np predicates as leaves and together these form the boolean hierarchy we present many results about the structure of the boolean hierarchy separation and immunity results natural complete languages and structural asymmetries between complementary classes we show that in some relativized worlds the boolean hierarchy is infinite and that for every k there is a relativized world in which the boolean hierarchy extends exactly k levels we prove natural languages variations of vertex cover complete for the various levels of the boolean hierarchy we show the following structural asymmetry though no set in the boolean hierarchy is text d text p immune there is a relativized world in which the boolean hierarchy contains text cod text p immune sets thus this paper explores the structural properties of the

topological orbifold models and quantum cohomology rings we discuss the topological sigma model on an orbifold target space we describe the moduli space of classical minima for computing correlation functions involving twisted operators and show through a detailed computation of an orbifold ofcp1 by the dihedral groupd4 how to compute the complete ring of observables through this procedure we compute all the rings of dihedralcp1 orbifolds we then considercp2 d4 and show how the techniques of topologicalanti topological fusion might be used to compute twist field correlation functions for nonabelian orbifolds

curvature estimates for minimal hypersurfaces in singular spaces we here prove pointwise curvature estimates for minimal hypersurfaces in singular spaces using the integral curvature estimate and a generalized simons inequality which were established recently a further basic ingredient is a new sobolev type inequality for stationary hypersurfaces

duality theorems for etale gerbes on orbifolds abstract let g be a finite group and y a g gerbe over an orbifold b a disconnected orbifold y and a flat u 1 gerbe c on y is canonically constructed from y motivated by a proposal in physics we study a mathematical duality between the geometry of the g gerbe y and the geometry of y twisted by c we prove several results verifying this duality in the contexts of non commutative geometry and symplectic topology in particular we prove that the category of sheaves on y is equivalent to the category of c twisted sheaves on y when y is symplectic we show by a combination of techniques from non commutative geometry and symplectic topology that the chen ruan orbifold cohomology of y is isomorphic to the c twisted orbifold cohomology of y as graded algebras

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

exterior powers of the adjoint representation and the weyl ring of e8 abstract i derive explicitly all polynomial relations in the character ring of e 8 of the form k e 8 p k 1 8 0 where k e 8 is an arbitrary exterior power of the adjoint representation and i is the ith fundamental character this has simultaneous implications for the theory of relativistic integrable systems seiberg witten theory quantum topology orbifold gromov witten theory and the arithmetic of elliptic curves the solution is obtained by reducing the problem to a large but finite dimensional linear problem which is amenable to an efficient solution via distributed computation

equivariant gw theory of stacky curves we extend okounkov and pandharipande s work on the equivariant gromov witten theory of mathbb p 1 to a class of stacky curves mathcal x our main result uses virtual localization and the orbifold elsv formula to express the tau function tau mathcal x as a vacuum expectation on a fock space

stokes matrices for the quantum cohomologies of a class of orbifold projective lines we prove the dubrovin s conjecture for the stokes matrices for the quantum cohomology of orbifold projective lines the conjecture states that the stokes matrix of the first structure connection of the frobenius manifold constructed from the gromov witten theory coincides with the euler matrix of a full exceptional collection of the bounded derived category of the coherent sheaves our proof is based on the homological mirror symmetry primitive forms of affine cusp polynomials and the picard lefschetz theory

an integral structure in quantum cohomology and mirror symmetry for toric orbifolds abstract we introduce an integral structure in orbifold quantum cohomology associated to the k group and the class in the case of compact toric orbifolds we show that this integral structure matches with the natural integral structure for the landau ginzburg model under mirror symmetry by assuming the existence of an integral structure we give a natural explanation for the specialization to a root of unity in y ruan s crepant resolution conjecture yongbin ruan the cohomology ring of crepant resolutions of orbifolds in contemp math vol 403 amer math soc providence ri 2006 pp 117 126

cycle groups for artin stacks we construct an algebraic homology functor for artin stacks of finite type over a field and we develop intersection theoretic properties

gromov witten theory of deligne mumford stacks given a smooth complex deligne mumford stack cal x with a projective coarse moduli space we introduce gromov witten invariants of cal x and prove some of their basic properties including the wdvv equation

on the dirichlet problem for minimal graphs in hyperbolic space soit r n n 2 un domaine borne on etudie le probleme de dirichlet suivant f f i f j f ij 1 df 2 n f 0 dans f 0 dans f 0 sur on considere le comportement a la frontiere de la solution

using stacks to impose tangency conditions on curves we define a deligne mumford stack xp r which depends on a scheme x an effective carrier divisor d c x and a positive integer r then we show that the abramovich vistoli mod uli stack of stable maps into xq r provides compactifications of the locally closed substacks of mgtn x 3 corresponding to relative stable maps 1 introduction this paper has two main purposes the first is to introduce the rth root construction which is a way of creating stack structure along a divisor given a scheme x an effective cartier divisor d c x and a positive integer r which is invertible on x it produces a deligne mumford stack xo r when r is not invertible the construction still makes sense and produces an artin stack which has finite diagonal over x since the output is in general an artin stack we give the definition and basic properties when x is an artin stack this simplifies the exposition since we need to apply the construction multiple times this construction was known to several mathematicians before we became aware of it but very little appeared in the literature aside from a brief mention in agv 3 5 3 secondly we compare twisted stable maps into xp r with ordinary stable maps into x the theory of twisted stable maps was developed by chen and ruan cr in the symplectic category and by abramovich and vistoli av in the algebraic category the latter showed that there is a deligne mumford stack xg n x 3 parametrizing morphisms x from twisted rc marked genus g nodal curves into a deligne mumford stack x such that 3 in the case where x is a scheme this is just the space of kontsevich stable maps mg n x 3 the theory of relative stable maps was developed by several symplectic ge ometers and the algebraic definition was worked out partially by andreas gath mann ga and in general by jun li li let x be a smooth complex projective veriety d c x a smooth divisor and 3 g b x such that d 3 0 here b x denotes the group of 1 cycles modulo algebraic equivalence choose a partition d 3 q qn where each qi is a nonnegative integer then a stable map c x from a smooth curve c having marked points x xn such that f c 3 and d y qixi ls a relative stable map the space of such mor phisms was compactified by ga and li and they used this to define relative gromov witten invariants

mori dream stacks we propose a generalisation of mori dream spaces to stacks we show that this notion is preserved under root constructions and taking abelian gerbes unlike the case of mori dream spaces such a stack is not always given as a quotient of the spectrum of its cox ring by the picard group we give a criterion when this is true in terms of mori dream spaces and root constructions finally we compare this notion with that of smooth toric deligne mumford stacks

chern classes of deligne mumford stacks and their coarse moduli spaces let x be a complex projective algebraic variety with gorenstein quotient singularities and mathcal x a smooth deligne mumford stack having x as its coarse moduli space we show that the csm class c sm x coincides with the pushforward to x of the total chern class c t i mathcal x of the inertia stack i mathcal x we also show that the stringy chern class c str x of x whenever it is defined coincides with the pushforward to x of the total chern class c t ii mathcal x of the double inertia stack ii mathcal x some consequences concerning stringy orbifold hodge numbers are deduced

erratum the polynomial time hierarchy collapses if the boolean hierarchy collapses it is shown that if the boolean hierarchy bh collapses then there exists a sparse set s such that text co np subseteq text np s and therefore the polynomial time hierarchy ph col

the quantum mckay correspondence for polyhedral singularities let g be a polyhedral group namely a finite subgroup of so 3 nakamura s g hilbert scheme provides a preferred calabi yau resolution y of the polyhedral singularity 3 g the classical mckay correspondence describes the classical geometry of y in terms of the representation theory of g in this paper we describe the quantum geometry of y in terms of r an ade root system associated to g namely we give an explicit formula for the gromov witten partition function of y as a product over the positive roots of r in terms of counts of bps states gopakumar vafa invariants our result can be stated as a correspondence each positive root of r corresponds to one half of a genus zero bps state as an application we use the crepant resolution conjecture to provide a full prediction for the orbifold gromov witten invariants of 3 g

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

a remark on gromov witten invariants of quintic threefold abstract the purpose of the article is to give a proof of a conjecture of maulik and pandharipande for genus 2 and 3 as a result it gives a way to determine gromov witten invariants of the quintic threefold for genus 2 and 3

gromov witten gauge theory we introduce a modular completion of the stack of maps from stable marked curves to the quotient stack pt c x and use this stack to construct some gauge theoretic analogues of the gromov witten invariants we also indicate the generalization of these invariants to the quotient stacks x c x where x is a smooth proper complex algebraic variety

the crepant transformation conjecture for toric complete intersections abstract let x and y be k equivalent toric deligne mumford stacks related by a single toric wall crossing we prove the crepant transformation conjecture in this case fully equivariantly and in genus zero that is we show that the equivariant quantum connections for x and y become gauge equivalent after analytic continuation in quantum parameters furthermore we identify the gauge transformation involved which can be thought of as a linear symplectomorphism between the givental spaces for x and y with a fourier mukai transformation between the k groups of x and y via an equivariant version of the gamma integral structure on quantum cohomology we prove similar results for toric complete intersections we impose only very weak geometric hypotheses on x and y they can be non compact for example and need not be weak fano or have gorenstein coarse moduli space our main tools are the mirror theorems for toric deligne mumford stacks and toric complete intersections and the mellin barnes method for analytic continuation of hypergeometric functions

the boolean hierarchy of np partitions we introduce the boolean hierarchy of k partitions over np for k 3 as a generalization of the boolean hierarchy of sets i e 2 partitions over np whereas the structure of the latter hierarchy is rather simple the structure of the boolean hierarchy of k partitions over np for k 3 turns out to be much more complicated we formulate the embedding conjecture which enables us to get a complete idea of this structure this conjecture is supported by several partial results

hurwitz hodge integrals the e6 and d4 root systems and the crepant resolution conjecture abstract let g be the group a 4 or z 2 z 2 we compute the integral of g on the hurwitz locus h g m g of curves admitting a degree 4 cover of p 1 having monodromy group g we compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the e 6 and d 4 root systems respectively as an application we prove the crepant resolution conjecture for the orbifolds c 3 a 4 and c 3 z 2 z 2

on the crepant resolution conjecture in the local case in this paper we analyze four examples of birational transformations between local calabi yau 3 folds two crepant resolutions a crepant partial resolution and a flop we study the effect of these transformations on genus zero gromov witten invariants proving the coates iritani tseng ruan form of the crepant resolution conjecture in each case our results suggest that this form of the crepant resolution conjecture may also hold for more general crepant birational transformations they also suggest that ruan s original crepant resolution conjecture should be modified by including appropriate quantum corrections and that there is no straightforward generalization of either ruan s original conjecture or the cohomological crepant resolution conjecture to the case of crepant partial resolutions our methods are based on mirror symmetry for toric orbifolds

contact cohomology of the projective plane we construct an associative ring which is a deformation of the quantum cohomology ring of the projective plane just as the quantum cohomology encodes the incidence characteris tic numbers of rational plane curves the contact cohomology encodes the tangency characteristic numbers 1 introduction in this paper we construct an associative ring which we call the contact cohomology ring of the projective plane we believe that an analogous construction should work for all homogeneous varieties but in our proof of associativity we rely on certain technical results from our earlier paper on characteristic numbers of rational plane curves ek98 as we formulate it in section 3 of the present paper the quantum product is actually a whole family of products parametrized by elements of the chow ring

on boolean lowness and boolean highness the concepts of lowness and highness originate from recursion theory and were introduced into the complexity theory by schoning sch85 informally a set is low high resp for a relativizable class k of languages if it does not add adds maximal resp power to k when used as an oracle in this paper we introduce the notions of boolean lowness and boolean highness informally a set is boolean low boolean high resp for a class k of languages if it does not add adds maximal resp power to k when combined with k by boolean operations we prove properties of boolean lowness and boolean highness which show a lot of similarities with the notions of lowness and highness using kadin s technique of hard strings see kad88 wag87 ck96 bco93 we show that the sets which are boolean low for the classes of the boolean hierarchy are low for the boolean closure of 2p furthermore we prove a result on boolean lowness which has as a corollary the best known result see bco93 in fact even a bit better on the connection of the collapses of the boolean hierarchy and the polynomial time hierarchy if bh np k then ph 2p k 1 np k

computing genus zero twisted gromov witten invariants twisted gromov witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold x which depend in addition on a vector bundle over x and an invertible multiplicative characteristic class special cases are closely related to local gromov witten invariants of the bundle and to genus zero one point invariants of complete intersections in x we develop tools for computing genus zero twisted gromov witten invariants of orbifolds and apply them to several examples we prove a quantum lefschetz theorem that expresses genus zero one point gromov witten invariants of a complete intersection in terms of those of the ambient orbifold x we determine the genus zero gromov witten potential of the type a surface singularity c2 zn we also compute some genus zero invariants of c3 z3 verifying predictions of aganagic bouchard and klemm 3 in a self contained appendix we determine the relationship between the quantum cohomology of the an surface singularity and that of its crepant resolution thereby proving the crepant resolution conjectures of ruan and of bryan and graber 12 in this case

hom stacks and restriction of scalars fix an algebraic space s and let x and y be separated artin stacks of finite presentation over s with finite diagonals over s we define a stack homs x y classifying morphisms between x and y assume that x is proper and flat over s and assume fppf locally on s that there exists a finite finitely presented flat cover z x with z an algebraic space then we show that homs x y is an artin stack with quasi compact and separated diagonal