aa r X i v : . [ m a t h . AG ] J u l Contemporary Mathematics
Quasimaps and Some Examples of Stacks for Everybody
Rachel Webb
Abstract.
This note introduces the theory of quasimaps to GIT quotientswith intuition and concrete examples, with the goal of explaining a closedformula for the quasimap I -function. Along the way, it emphasizes aspects ofthis story that illustrate general stacky concepts.
1. Introduction
In modern algebraic geometry, stacks are pervasive. To name just a few ap-pearances, they arise as quotients of varieties by group actions, as the algebraicanalog of orbifold curves, and as moduli spaces of objects with nontrivial auto-morphism groups. Unfortunately the theory is laden with technicalities, making itdifficult for newcomers. This note provides concrete examples of stacks that arisein the theory of quasimaps to GIT quotients.
Quasimaps are themselves a topic inGromov-Witten theory with a growing list of applications.Primarily a friendly introduction to quasimaps, this note is also a “secondcourse” to Fantechi’s brief introduction to stacks [ ], a tour of some examplesof stacks “in the wild.” As an introduction to quasimaps, the goal of this noteis to present a closed formula for the quasimap I -function. Along the way, we’llhighlight examples that illustrate general stacky concepts with the heading StackExample . We will always work over C .
2. Motivation for the study of quasimaps
The theory of ǫ -stable quasimaps to GIT quotients, as presented in this note,was introduced by Ciocane-Fontanine, Kim, and Maulik in a series of papers: [ ],[ ], [ ], [ ], and [ ]. For the sake of novel and concrete exposition, this noteomits much of the historical and mathematical context of the theory, which can befound in the original papers and in the survey articles [ ] and [ ].Gromov-Witten theory begins with the study of maps from a smooth genus- g curve with n marks to a projective target, for example to P n . The set ofsuch maps having a fixed degree d forms a moduli space, denoted (in this ex-ample) M g,n ( P n , d ). Unfortunately, M g,n ( P n , d ) is not compact; in order to defineGromov-Witten invariants of P n as integrals over a moduli space of maps, we must Mathematics Subject Classification.
Primary 14-02.
Key words and phrases.
Quasimaps, abelian/nonabelian correspondence, algebraic stacks.The first author was supported in part by NSF RTG grant 1045119. replace M g,n ( P n , d ) with a compactification. One such compactification is the mod-uli space of Kontsevich-stable maps M g,n ( P n , d ). These stable map moduli spacesare the central objects of Gromov-Witten theory.The Kontsevich moduli space compactifies M g,n ( P n , d ) by allowing the sourcecurve to be nodal, but there are other ways to compactify. One may see hints ofthese other ways by looking at possible limits of families of maps in M g,n ( P n , d ).Let C be equal to P with homogeneous coordinates [ x : y ] and markings at [1 : 1]and [2 : 1]. Define(2.1) φ : C ∗ × P → P φ a ( x, y ) = [ ax : xy : y ] for a ∈ C ∗ . This is a family in M , ( P ,
2) with base C ∗ . To extend it over the origin to a flatfamily over C , we need to define a map φ . The natural choice seems to be(2.2) φ ( s, t ) = [0 : xy : y ] , but this has a basepoint (is undefined) at y = 0. To recover the limit in M , ( P , φ : A × P P (given by (2.1) and (2.2)) by blowingup this basepoint, adding an extra rational curve in the fiber over 0. This producesa morphism e φ from the blowup to P , and e φ has a source curve that is two copies of P , glued at a node. The limiting map has degree 1 on each copy. This is depictedin Figure 1. a = 0 e φ Figure 1.
The morphism e φ maps from Bl a = y =0 A × P on theleft to P on the right, depicted here in the chart of P where themiddle coordinate is nonzero. The varying shades of gray show thefibers of this family of stable maps. The fiber over a = 0 is a mapfrom a nodal curve.However, what happens if we compactify M , ( P ,
2) by allowing basepoints?That is, what if we take φ ( x, y ) = [0 : xy : y ] as a rational map from P to P tobe the limit of (2.1)? Indeed, this rational map is a stable quasimap , depicted inFigure 2.We will see that with the right definitions, stable quasimaps form a moduli spacethat is just as well-behaved as the Kontsevich moduli spaces. In fact, stability forquasimaps depends on a postive rational parameter ǫ , giving us a whole collectionof moduli spaces! This collection has the following advantages:(1) When ǫ >
2, the quasimap moduli space is equal to the familiar Kontsevichmoduli space, and its invariants are Gromov-Witten invariants.(2) When ǫ is sufficiently small, certain quasimap invariants (the genus-0 in-variants) are easier to compute. A generating function for these explicitinvariants is called the quasimap I-function . UASIMAPS AND SOME EXAMPLES OF STACKS FOR EVERYBODY 3 a = 0 φ Figure 2.
The rational map φ takes A × P on the left to P onthe right. The varying shades of gray show the fibers of this familyof quasimaps. In particular, the fiber over a = 0 is a rational mapof degree 1 with a basepoint of length 1 (see Definition 5.3).(3) One can “cross the wall,” relating quasimap invariants for differing valuesof ǫ , thereby (roughly) expressing Gromov-Witten invariants in terms ofthe quasimap I -function.These statements are a heuristic only; for careful statements and their proofs, see [ ]and [ ]. For certain targets, invariants with small ǫ and g = 0 were first studiedby Givental [ ], while the strategy in (3) above for computing Gromov-Witteninvariants was first employed by Bertram [ ].
3. Quotients in algebraic geometry
When a reductive algebraic group G acts on an affine variety X , we’d liketo take the quotient, producing an algebraic object X/G . Unfortunately, theremay not be a scheme
X/G satisfying the universal property of a quotient—if itexists, this scheme is called a categorical quotient . Even if a categorical quotientscheme exists, it may not have the expected topology (e.g., be a geometric quotient ).Therefore, defining a quotient either requires us to modify the original data X and G , or to leave the category of schemes. Both strategies arise in quasimap theory.The resulting quotients are respectively called geometric invariant theory (GIT)quotients and stack quotients. We briefly summarize the key definitions of GIT for affine X when G acts withno kernel, as found in [ ]. Fix a character θ of G . Given a nonnegative integer n ,a function f ∈ Γ( X, O X ) is a relative invariant of weight θ n if for every x ∈ X wehave f ( g · x ) = θ ( g ) n f ( x ) . Definition . A point x ∈ X is θ -semistable if there exists an integer n ≥ f of weight θ n such that f ( x ) = 0. If moreover thedimension of the orbit G · x is equal to the dimension of G and the G -action on { y ∈ X | f ( y ) = 0 } is closed, then x is θ -stable .The upshot of these definitions is that we get an open locus X ssθ ( G ) ⊂ X ofsemistable points with respect to θ , and a smaller locus X sθ ( G ) ⊂ X ssθ ( G ) of stablepoints. We define the locus of unstable points to be X usθ ( G ) = X \ X ssθ ( G ). Thesets X sθ ( G ) and X ssθ ( G ) can be computed with [ , Prop 2.5]. We include thegroup G in our notation here because later we will vary it; when there is no risk ofmisunderstanding, we will omit both the character and the group from the notation.In this note, we will assume • X ssθ ( G ) = X sθ ( G ) and this set is nonempty and smooth RACHEL WEBB • G acts freely on X ssθ ( G )In this case, the GIT quotient is the smooth variety X// θ G := X sθ ( G ) /G where the right hand side is defined to be the categorical quotient (it is a theoremthat this exists as a scheme). As a topological space, X// θ G is the usual topologicalquotient X sθ ( G ) /G . Again, when θ is understood, we will omit it from the notation.In this note, we will also assume that X//G is a projective variety. In general it isprojective over the affine quotient Spec(Γ( X, O X ) G ).Whereas the GIT quotient “forgets” the unstable locus X us , this informationis retained in the stack quotient. The objects of the stack quotient [ X/G ] over ascheme S are diagrams P XS where P is a principal G -bundle (locally trivial in the ´etale topology) and P → X is a G -equivariant map. Morphisms in this category are given by fiber diagrams.One should picture the GIT quotient X//G as an open subset of the stackquotient [
X/G ]. In a moment we will define quasimaps to
X//G to be certain“stable” maps to the stack quotient [
X/G ]. Motivation for this definition comesfrom the previous section and the next example.
Stack Example . Let C ∗ act on C n +1 by(3.1) λ · ( x , . . . , x n ) = ( λx , . . . , λx n ) . Then if θ : C ∗ → C ∗ is the identity, we have ( C n +1 ) ss = C n +1 \{ } and C n +1 // θ C ∗ = P n . A map from a scheme S to P n is given by a line bundle L on S and n + 1sections of L that do not simultaneously vanish.On the other hand, the stack [ C n +1 / C ∗ ] “remembers the origin.” By definition,a map from a scheme S to [ C n +1 / C ∗ ] is a principal C ∗ -bundle P on S and anequivariant map P → C n +1 . As we will see in Example 4.2, this equivariantmap is equivalent to a section of P × C ∗ C n +1 = L ⊕ n +1 → S where L is the linebundle associated to P . Comparing this to the data of a map to C n +1 // C ∗ , wesee that the only difference is that now, the sections of L are allowed to vanishsimultaneously—i.e., we allow the map to “hit the origin.”We close this section with an important example in which a desirable quotientdoes exist as a scheme, namely certain mixing spaces . Let P → S be a principal G -bundle with a left G -action and let X be an affine G -variety. Then P × X carriesa natural G -action defined on closed points by g · ( p, x ) = ( gp, gx ) . We define the mixing space to be the categorical quotient P × G X := ( P × X ) /G. One can show that the categorical quotient exists as a scheme, and in fact it is ageometric quotient (see for example the discussion in [ , Sec 3]). It has a naturalmap to P /G = S , with closed fibers isomorphic to X . In particular, when X is avector space and G acts linearly on X , the mixing space P × G V is the total spaceof a vector bundle on S . UASIMAPS AND SOME EXAMPLES OF STACKS FOR EVERYBODY 5
In fact, there is a bijection between principal GL r -bundles and rank- r vectorbundles on S given by sending P to P × G C r , where C r is a GL r -module via leftmultiplication. We call P × G C r the vector bundle associated to P , and P the underlying principal bundle of P × G C r .
4. Quasimaps
This note will discuss the theory of quasimaps to
V //G when V is a vectorspace. (The original papers present this theory in more generality, allowing V tobe an affine variety with l.c.i. singularities contained in V us .) Let V be a vectorspace and G a reductive algebraic group that acts linearly from the left on V . We’veseen that a quasimap to P is, loosely speaking, a rational map to P , and that sucha map can be realized as a map to the stack quotient [ C / C ∗ ]. This motivates thefollowing definition, which essentially says that a prestable quasimap to V //G is amap to the stack quotient [
V /G ] that recovers a rational map.
Definition . A prestable quasimap to V //G is data (
C, x , . . . , x n , P , ˜ u )where • C is a curve with at worst nodal singularities and marked points x , . . . , x n • ̺ : P → C is a principal G -bundle with a left G -action • ˜ u : P → V is a G -equivariant map with ̺ (˜ u − ( V us )) a finite set that isdisjoint from nodes and markings.The genus of the quasimap is the genus of C , and the degree of the quasimap is thedegree of P , i.e., a group homomorphism β ∈ Hom ( χ ( G ) , Z ) defined by β ( ω ) = deg C ( P × G C ω )where ω ∈ χ ( G ) is a character of G and C ω is the corresponding 1-dimensionalrepresentation g · z = ω ( g ) z for g ∈ G and z ∈ C ω . To write down examples of quasimaps, it is helpful to replace ˜ u with an asso-ciated section of P × G V → C , which we will typically denote u . In fact this isequivalent data, as explained in the example below. Stack Example . If ̺ : P → C is a principal G -bundle, then equivariant maps P → V are in bijection with sections of theassociated vector bundle P × G V → C . I’ll sketch this bijection.From the universal property of fiber products, we have a natural bijectionbetween Hom( P , V ) and Sec( P , P × V ) where Sec denotes the space of sections.Letting G act on morphisms by conjugation, this is a G -equivariant bijection, sowe have Hom G ( P , V ) ∼ = Sec G ( P , P × V ) . where the subscript G indicates the G -invariant part (in particular, Hom G ( P , V )is the set of G -equivariant maps). On the other hand, I’ll sketch a bijectionSec( C, P × G V ) ∼ = Sec G ( P , P × V )by interpreting P × G V as the stack quotient [( P × V ) /G ]. A map from C to P × G V is by definition a principal G -bundle Q on C and an equivariant map to RACHEL WEBB P × V ; to say that it is a section means that composition with projection to C isthe identity: Q P × V P C C
Then the induced map Q → P is a morphism of principal bundles, hence anisomorphism. After identifying P with Q this way, the map Q → P × V becomesa G -equivariant section P → P × V . The reader is invited to find the inverse tothis correspondence. Example . For two GIT quotients
V //G , we’ll write down all quasimapsfrom P to V //G as vectors of homogeneous polynomials. This example is general-ized in Section 8.(1) If V = C n +1 and G = C ∗ with action (3.1), and if θ is the identitycharacter of C ∗ , then V ss is V \ { } and V // θ G is P n . A quasimap to P n of degree d ∈ Hom( Z , Z ) = Z with source curve P is • A principal C ∗ -bundle P such that deg P ( P × C ∗ C θ ) = d . Therefore P is the underlying principal bundle of O P ( d ) . • A section of P × C ∗ C n +1 . Here the action of C ∗ on C n +1 is givenby (3.1), so that this vector bundle is n + 1 copies of the associatedbundle to P , i.e., P × C ∗ C n +1 equals O ( d ) ⊕ n +1 .Therefore, a prestable quasimap to P n of degree d ∈ Z is given by a vector( p ( x, y ) , p ( x, y ) , . . . , p n +1 ( x, y ) )of homogeneous polynomials of degree d that are not all zero. In particular(2.2) is a prestable quasimap of degree 2.(2) If V = M k × n is k × n matrices over C and G = GL k acts on V byleft multiplication and θ is the determinant character, then V ss is full-rank matrices and V // θ G is the Grassmannian Gr ( k, n ). A quasimap to Gr ( k, n ) of degree d ∈ Hom( Z , Z ) = Z with source curve P is • A principal GL k -bundle P on P such that deg P ( P × GL k C θ ) = d .Let E denote the vector bundle associated to P . From Grothendieck’sclassification of principal bundles [ ], we have E = ⊕ ki =1 O ( d i ) forsome d i with P ki =1 d i = d . • A section of P × GL k M k × n , i.e., of E ⊕ n .So a prestable quasimap to Gr ( k, n ) of degree d is given by a matrix ofpolynomials(4.1) [ p ij ( x, y )] ≤ i ≤ k, ≤ j ≤ n where p ij ( x, y ) is homogeneous of degree d i . Because M usk × n is matrices oflow rank, to define a prestable quasimap the matrix (4.1) must have lowrank on a finite set.
5. Stability and moduli spaces of quasimaps
In analogy with Gromov-Witten invariants, we want to define quasimap in-variants of a target
V //G to be integrals on certain moduli spaces. This cues theentrance of our second example of a stack: the moduli space of quasimaps. We will
UASIMAPS AND SOME EXAMPLES OF STACKS FOR EVERYBODY 7 see that this is a “master” moduli space containing M g,n ( V //G, β ) and many com-pactifications of it. Then we will define some of these compactifications (substacksof the master space) to be the ǫ -stable moduli spaces of quasimaps. Stack Example . Let M g,n ( V //G, β )be the moduli stack of prestable quasimaps of genus g , degree β , and n marks. Ob-jects in this category over a base S are families of prestable quasimaps on S ; i.e.,they are triples ( C , P , ˜ u ) where C → S is a flat family of genus- g nodal curves(not necessarily stable) on S , P → C is a principal G -bundle, ˜ u : P → V is G -equivariant, and geometric fibers over S are prestable quasimaps of degree β . Anisomorphism between objects in M g,n ( V //G, β )( S ) is a commuting diagram P ′ P V C ′ C ∼ ˜ u ′ ˜ u ∼ where C ′ ∼ −→ C commutes with the maps to S and the square is fibered (such adiagram is an isomorphisms of quasimap families). The stack M g,n ( V //G, β ) isnot Deligne-Mumford, as some prestable quasimaps have non-finite automorphismgroups. We see the familiar offenders from stable map theory: for example, adegree-0 map sending P to a point in P is invariant under the entire automorphismgroup of P . However, there are new examples as well. For instance, [ x : 3 x : x ]defines a prestable quasimap of degree 2 from P to P which is invariant under[ x : y ] [ x : ty ]. Example . Let’s find isomorphisms between the quasimaps in Example 4.3.(1) Let
V, G , and θ be as in Example 4.3 part 1. From that example, aprestable quasimap from P to P n of degree d is given by a vector ofhomogeneous degree- d polynomials ( p i ( x, y )) n +1 i =1 . Then a quasimap iso-morphism is an element α ∈ Aut( O ( d )) = C ∗ , and it sends ( p i ( x, y )) to( αp i ( x, y )).(2) Let V, G , and θ be as in Example 4.3 part 2. From that example, aprestable quasimap from P to Gr ( k, n ) of degree d is n sections of avector bundle ⊕ ki =1 O ( d i ) of degree d , which may be denoted by a k × n matrix ( p ij ( x, y )) of polynomials where p ij ( x, y ) is homogeneous of de-gree d i . Then a quasimap isomorphism is an element A of Aut( E ) =Hom( ⊕ O ( d i ) , ⊕ O ( d i )) × which we may identify with a k × k matrix ( a ℓi ( x, y ))of polynomials where a ℓi ( x, y ) is homogeneous of degree d ℓ − d i . Such anisomorphism acts on a the quasimap ( p ij ( x, y )) by matrix multiplication.(Notice that if d ≥ d ≥ . . . ≥ d k , then ( a ℓi ( x, y )) will be block uppertriangular.)For many reasons, the stack M g,n ( V //G, β ) is not the right one for defininginvariants. Instead, we impose a stability condition that cuts out a proper separatedDeligne-Mumford substack of M g,n ( V //G, β ) which contains M g,n ( V //G, β ) as anopen subset.
Definition . Let (
C, x , . . . , x n , P , ˜ u ) be a prestable quasimap to V // θ G and let L = P × G C θ . This quasimap is ǫ -stable if RACHEL WEBB (1) On every component C ′ of C we have2 g C ′ − n C ′ + ǫ deg( L | C ′ ) > g C ′ is the genus of C ′ and n C ′ is the number of marked points andnodes on C ′ , and(2) For every x ∈ C we have ℓ ( x ) ≤ /ǫ where ℓ ( x ), called the length of x , is the order of contact of u ( C ) with P × G V us , and u is the section of P × G V → C determined by ˜ u . See[ , Def 7.1.1] for more on the definition of length. For x ∈ C , the length ℓ ( x ) is nonzero if and only if u ( x ) is in P × G V us , and in this case we say x is a basepoint of the quasimap.The first condition (1) is sometimes stated as “ ω C ( P x i ) ⊗ L ǫ is ample.” A fam-ily of prestable quasimaps is ǫ -stable if every geometric fiber is ǫ -stable. The modulispace of ǫ -stable quasimaps is denoted M ǫg,n ( V //G, β ). An ǫ -stable quasimap onlyhas finitely many automorphisms. Hence the moduli spaces M ǫg,n ( V //G, β ) areDeligne-Mumford, not just Artin stacks. They also have other good geometricproperties, as stated in the following theorem.
Theorem . [ , Thm 7.1.6] The moduli space M ǫg,n ( V //G, β ) is a properseparated Deligne-Mumford stack of finite type. As explained in the introduction, the benefit of these spaces, especially in genus0, is that when ǫ is greater than 2, the space M ǫ ,n ( V //G, β ) is the familiar space M ,n ( V //G, β ). On the other hand, when ǫ is sufficiently small, the moduli spaces M ǫ ,n ( V //G, β ) do not depend on ǫ ; this is called the 0+-stable quasimap mod-uli space. The space M ,n ( V //G, β ) is more “computable” (see Example 5.7).Hence the wall-crossing theorem of [ ], which translates between invariants of M ǫ ,n ( V //G, β ) for differing values of ǫ , gives a way to relate the Gromov-Witteninvariants of V //G to invariants that are more computable.
Example . In Section 2 we described two possible limits of the family (2.1)of quasimaps to P . One limit was the stable map e φ . Because e φ has no basepoints,it satisfies condition (2) in Definition 5.3 for each ǫ . However, on the componentwith no marks—call it C ′ —we have2 g C ′ − n C ′ + ǫ deg( L | C ′ ) = − ǫ, which is possible only for ǫ >
1. So this map is ǫ stable for ǫ > φ ( x, y ) = [0 : xy : y ]. This quasimapsatisfies condition (1) of Definition 5.3 for every ǫ >
0. However, we have ℓ ([1 :0]) = 1, so that this map is ǫ -stable only when ǫ ≤ M ǫg,n ( V //G, β ) has the stable map e φ for a limit when ǫ >
1, and the rational map φ for a limit when ǫ ≤ M ǫg,n ( V //G, β ) are the ǫ -stable quasimap graph spaces . Graphspaces are used in many contexts to prove wall crossing or mirror theorems. Inparticular, we will use the graph moduli space with g = n = 0 and ǫ = 0 + inSection 7 to define the I -function, so we’ll define the quasimap graph space withthese parameters now. UASIMAPS AND SOME EXAMPLES OF STACKS FOR EVERYBODY 9
Definition . Fix a “reference copy” of P and denote it P P P . The quasimapgraph space QG ( V //G, β ) is the stack whose objects over a scheme S are prestablequasimaps ( C , P , ˜ u ), with an additional datum φ : C → P P P which restricts to an isomorphism on every geometric fiber of C → S . An isomor-phism between two objects ( C , P , ˜ u, φ ) and ( C ′ , P ′ , ˜ u ′ , φ ′ ) in QG ( V //G, β )( S ) is acommuting diagram(5.1) P ′ P V P P P C ′ C ∼ ˜ u ′ ˜ uφ ′ ∼ φ such that C ′ ∼ −→ C commutes with the maps to S and the square is fibered.When S is Spec( C ), The additional datum φ realizes C as P P P . So closed pointsof QG ( V //G, β ) are in bijection with quasimaps from P P P to V //G . In addition,notice that φ prevents the existence of automorphisms, so we do not need markedpoints to achieve stability. According to [ , Thm 7.2.2], the space QG ( V //G, d ) isa proper Deligne-Mumford stack of finite type. In the next example, we identify QG ( P n , d ) as a smooth scheme. Stack Example . We have seen(Example 4.3) that a quasimap from P to P n of degree d is a nonzero element ofΓ( P , O ( d )) ⊕ n +1 . However, two such sections define isomorphic quasimaps exactlywhen they differ by a complex scalar (Example 5.2). Hence, we naively expect QG ( P n , d ) = (cid:0) Γ( P , O ( d )) ⊕ n +1 \ { } (cid:1) / C ∗ ∼ = P N , where N = dn + d + n . Indeed, this space carries a tautological family of quasimapsas follows. On P N we have the trivial family of curves P N × P , and on this familythe vector bundle V = O P N (1) ⊕ n +1 ⊗ O P ( d ). An element of P N × P may bewritten ( σ, x ) where σ ∈ Γ( P , O ( d )) ⊕ n +1 is a vector of n + 1 degree- d homogeneouspolynomials in two variables and x = ( x, y ). The tautological quasimap is given bythe section of V sending ( σ, x ) to σ ( x ).The tautological family on P N defines a map F : P N → QG ( P n , d ) which is abijection on closed points by Examples 4.3 and 5.2. It can be shown that QG ( P n , d )is a smooth algebraic space (see Examples 6.1 and 7.1). Since P N is smooth as well,it follows that F must be an isomorphism (see [ , Lem 3.6.2]).
6. Virtual cycles via perfect obstruction theories
We want to define invariants of
X//G which are integrals over the moduli spaces M ǫg,n ( X//G, β ). But these spaces may not be smooth or even equidimensional, sothe usual definition of integration doesn’t make sense. We need to choose someclass to be the (virtual) fundamental class with which to define integration. Oneway to get such a class is via a perfect obstruction theory and the intrinsic normalcone construction of Behrend-Fantechi [ ]. We will summarize their construction;for a more thorough but still elementary introduction see [ ]. Let X be a Deligne-Mumford stack with a map to an Artin stack Y . We’llassume X and Y are separated and of locally finite type, and that the map X → Y is of relative Deligne-Mumford type , a technical condition that is always satisfiedin our examples. In this case, the relative cotangent complex of X over Y is acertain object L • X/Y in the derived category D ≤ ( X ) of quasi-coherent sheaves.One property of L • X/Y is that H ( L • X/Y ) is isomorphic to the sheaf of Kahlerdifferentials Ω X/Y .A relative perfect obstruction theory on X is a perfect complex E • ∈ D [ − , ( X )and a morphism φ : E • → L • X/Y such that h ( φ ) is an isomorphism and h − ( φ ) issurjective. From this data, Behrend-Fantechi construct • A relative intrinsic normal cone C X/Y that is an Artin stack over X • A cone stack E → X that contains the intrinsic normal cone via a closedembedding C X/Y ֒ → E By the second bullet, the intrinsic normal cone is a closed substack of E ; intuitively,the virtual class induced by E • is the intersection of the intrinsic normal cone withthe 0-section of E . Symbolically,[ X ] vir := 0 ! E [ C X/Y ] ∈ A dim Y +rk E − rk E − ( X ) . The symbol 0 ! E is defined in [ ]. One incidental use of perfect obstruction theoriesis to detect when a stack is smooth, using the following example (see [ , Prop 7.3]). Example . Let X and Y be as above and let φ : E • → L • X/Y be a relativeperfect obstruction theory. If E − = 0 then we say the obstructions vanish. Ifthis is the case and moreover E is locally free, then h ( E • ) = E is locally free,but h ( φ ) is an isomorphism, so h ( L • X/Y ) = Ω X/Y is locally free and X → Y is smooth. In this case, the construction of Behrend-Fantechi returns the usualfundamental class of X ; i.e., [ X ] vir = [ X ].There is a natural relative perfect obstruction theory on M ǫg,n ( V //G, β ). Thisstack has a forgetful map to the moduli stack of principal G -bundles on genus g curves µ : M ǫg,n ( V //G, β ) → B un G given by forgetting the section from the quasimap data. Theorem . [ , Thm 7.1.6] On M ǫg,n ( V //G, β ) let π : C → M ǫg,n ( V //G, β ) be the universal curve, let P denote the universal principal bundle, and let ̺ : P × G V → C be the associated vector bundle with T ̺ the relative tangent bundle ofthis map. Then the complex (cid:0) R • π ∗ ( u ∗ T ̺ ) (cid:1) ∨ is a µ -relative perfect obstruction theory for M ǫg,n ( V //G, β ) . According to [ ], the analogous result holds for the quasimap graph space QG ( V //G, β ). Stack Example . We compute the relativeperfect obstruction theory on QG ( P n , d ) and show that it recovers the usual fun-damental class of P N . From Stack Example 5.7, the relative perfect obstructiontheory is the complex (cid:0) R • π ∗ ( u ∗ T ̺ ) (cid:1) ∨ = (cid:0) R • π ∗ ( O P ( d ) ⊗ O P N (1) ⊕ n +1 ) (cid:1) ∨ UASIMAPS AND SOME EXAMPLES OF STACKS FOR EVERYBODY 11 where π : P N × P → P N is projection to the first factor. By the projection formula,this is the dual of the complex R π ∗ ( O P ( d )) ⊗ O P N (1) ⊕ n +1 → R π ∗ ( O P ( d )) ⊗ O P N (1) ⊕ n +1 . The second term vanishes, so the µ -relative perfect obstruction theory is the vectorbundle O ⊕ d +1 P N ⊗ O P N (1) ⊕ n +1 = O P N (1) ⊕ N +1 in degree 0. By Example 6.1 thevirtual class defined by this theory is the usual fundamental class on P N .
7. Quasimap I -functions For any positive rational ǫ , we have a proper separated Deligne-Mumford stack M ǫ ,n ( V //G, β ) with a perfect obstruction theory. This is exactly what we needto define invariants of
V //G as integrals on these spaces. As discussed at theend of Section 2, when ǫ is at least 2, these invariants are equal to Gromov-Witteninvariants. The mirror theorem of [ ] relates these invariants to localization residueson the graph moduli space QG ( V //G, β ) for the ǫ = 0 + stability parameter (seeDefinition 5.6). The residues on QG ( V //G, β ) are indexed by the quasimap I -function . In contrast to other Gromov-Witten generating functions, the advantageof the I -function is that it can be written down with a closed formula. The goal ofthis section is to define the quasimap I -function as a formal power series indexingcertain localization residues on QG ( V //G, β ).We already know from [ , Thm 7.2.2] that QG ( V //G, β ) is a Deligne-Mumfordstack, but in fact more is true: it is an algebraic space . Stack Example QG ( V //G, β ) is an algebraic space) . By [ , Thm 2.2.5]it suffices to show that a quasimap in this space has a trivial automorphism group.Essentially, this is because quasimaps are required to map into V s generically,and G acts on V s with trivial stabilizers by our assumptions in Section 2. Let σ : P → P × G V be a quasimap, and suppose φ is an automorphism of P commuting with σ . In a local chart U where P is trivial, let σ U : U → V and φ U : U → G be coordinate representations of σ and φ . Then φ U ( u ) σ U ( u ) = σ U ( u ) . For every u ∈ U that is not a basepoint, we know σ U ( u ) ∈ V s , so this equationimplies φ U ( u ) = 1 in G . Hence φ U is the identity on a dense subset of U , hence onall of U .To define the coefficients of the I -function as localization residues, we need a C ∗ -action on QG ( V //G, β ). This space carries a C ∗ action as follows. Let x , x behomogeneous coordinates on P P P and let C ∗ act on P P P by λ · [ x : x ] = [ λx : x ] , λ ∈ C ∗ . This induces an action on QG ( V //G, β ) given by λ ( C , P , ˜ u, φ ) = ( C , P , ˜ u, λ ◦ φ ) . The fixed locus of C ∗ is the closed subspace of QG ( V //G, β ) whose objects over ascheme S are families fixed by the action. Stack Example . What are the base-points of a fixed graph quasimap? If a graph quasimap ( C , P , ˜ u, φ ) over Spec( C )is C ∗ -fixed, then for every λ ∈ C ∗ we have a diagram (5.1) with ( C ′ , P ′ , ˜ u ′ , φ ′ ) =( C , P , ˜ u, λ ◦ φ ) and φ an isomorphism. Then the map C ′ → C in (5.1) must be φ − ◦ λ ◦ φ . But φ − ◦ λ ◦ φ must fix basepoints of ˜ u , for every λ . This means thatbasepoints of ˜ u have to be φ − ([0 : 1]) or φ − ([1 : 0]).We can use this information to identify components of the fixed locus: thelengths of these basepoints ℓ ([0 : 1]) and ℓ ([1 : 0]) are constant in families, sospecifying these integers specifies a component of the fixed locus.As explained in the previous example, the fixed locus of C ∗ acting on QG ( V //G, β ) has components determined by the length of the basepoints at [0 : 1]and [1 : 0]. Let F β be the component where all the basepoints are at [0 : 1] ∈ P P P .This component has a natural map ev • to V //G . Stack Example . Define a map ev • : F β → V //G asfollows. Let ( C , P , ˜ u, φ ) be an object of F β lying over S . Recall from Example 4.2that ˜ u defines a section u of P × G V → C ; since [1 : 0] is not a basepoint, u ([1 : 0])is in P × G V s . So define ev • to send ( C , P , ˜ u, φ ) to the morphism S → V //G apparent in the following diagram: P × G V P × G V s V s /G = V //GS × { [1 : 0] } C u Finally we can define the I -function as follows. The µ -relative perfect obstruc-tion theory on QG ( V //G, β ) defines an absolute perfect obstruction theory which is C ∗ -equivariant. The moving part of this equivariant theory may be used to definethe euler class of the normal bundle to the fixed locus, denoted e C ∗ ( N vir F β ). The fixedpart, when restricted to F β , is a perfect obstruction theory for F β , and hence de-fines a virtual class [ F β ] vir (see [ , Sec 3]). With these definitions, we may write the I -function of V //G as a formal sum: we use formal symbols q β to index coefficientsthat are localization residues on the stacks F β . Precisely,(7.1) I V //G ( z ) = 1 + X β =0 q β I V //Gβ ( z ) where I V //Gβ ( z ) = ( ev • ) ∗ [ F β ] vir e C ∗ ( N virF β ) ! . The promise of 0 + -stable quasimaps was that their invariants would be com-putable. Indeed, the moduli spaces F β in the formula (7.1) are represented bysmooth schemes, as stated in the following theorem: Theorem . The moduli space F β is a disjoint union of flag bundles onsmooth subvarieties of V //G . In [ ] this identification is explicit. The strategy of the proof is the same asthat used in Example 5.7: show that F β is a smooth algebraic space, and then finda tautological family on the desired smooth variety (a closed subscheme of a flagbundle on V //G ) whose geometric fibers are in bijection with objects parametrizedby (a component of) F β . This theorem can be used to derive an explicit formulafor the quasimap I -function, a formula we’ll write down in the next section. Example . Let us describe F d when the target is the Grassmannian Gr ( k, n ). Recall from Example 4.3 that a degree- d graph quasimap to Gr ( k, n )is n sections of a vector bundle ⊕ P P P ( O ( d i )) with P d i = d , which may be repre-sented by a k × n matrix of homogeneous polynomials. For convenience assume UASIMAPS AND SOME EXAMPLES OF STACKS FOR EVERYBODY 13 d ≥ d ≥ . . . ≥ d k . Clearly, full-rank matrices ( c ij x i ) for complex numbers c ij define elements of F β . Referring to Example 5.2 we see that invertible matrices ofthe form ( a ij x d i − d j ) define isomorphisms of this collection of elements of F β , where a ij are complex numbers with a ij = 0 if d j > d i . From this, one may naively guessthat there is a component of F β given by F d ,...,d k = (cid:0) M k × n \ ∆ (cid:1) /U where ∆ ⊂ M k × n is matrices of rank less than k and U is the subgroup of GL k equal to block upper triangular matrices with block sizes given by the multiplicitiesof the d i . In fact, this identification is correct (see [ ]).
8. A formula for the I -function The quasimap I -function was advertised as a more tractable member of theGromov-Witten family, a generating function that recovers Gromov-Witten invari-ants after a mirror map, but which (supposedly) has a closed formula. But how, inpractice, can one write down the quasimap I -function of V //G ?When G is abelian, the answer is well-known (see [ ] or [ ]). To handlenonabelian groups, the strategy is to prove an abelian/nonabelian correspondence for I -functions. “Abelian/nonabelian correspondences” can be found throughoutrepresentation theory and geometry; the basic idea of these results, if T is a maximaltorus of G , is to relate some data determined by G to the corresponding datadetermined by T . In our situation, a character θ of G restricts to a character of T , so one may attempt to recover the various invariants (including the I -function)of V // θ G from the more accessible invariants of V // θ T . Most abelian/nonabeliancorrespondences use the Weyl group W of T in G , which equals the quotient of thenormalizer of T by the torus T itself: W = N G ( T ) /T . Example . We describe an “abelian/nonabelian correspondence” for prin-cipal bundles on P . Let T be a principal T -bundle; then if we let T act on G by left multiplication, the associated bundle T × T G is a principal G -bundle.There is a group homomorphism τ : Hom( χ ( T ) , Z ) → Hom( χ ( G ) , Z ) induced byrestriction of characters, and if T has degree ˜ β ∈ Hom( χ ( T ) , Z ), then the degreeof T × T G is τ ( ˜ β ). Moreover, the Weyl group of T in G acts on Hom( χ ( T ) , Z ).Grothendieck’s classification theorem [ ] says that every principal G -bundle maybe written T × T G for some T , and that the isomorphism class of T × T G isdetermined by the Weyl orbit of ˜ β in Hom( χ ( T ) , Z ).We may understand the cohomology of V //G as follows. There is a rational map p : V //T
V //G defined on A = V s ( G ) /T ⊂ V s ( T ) /T . Let W denote the Weylgroup of T in G ; then W acts on A and hence on H ∗ ( A, Q ). A classical argumentshows that p ∗ defines an isomorphism from H ∗ ( V //G, Q ) to H ∗ ( A, Q ) W (this maybe read as an abelian/nonabelian correspondence for cohomology ). This is usefulbecause we can write down explicit classes in H ∗ ( A, Q ) from characters of T : If ξ ∈ χ ( T ), let L ξ denote the line bundle V s ( G ) × C ∗ C ξ on A . This gives a class c ( L ξ ), and with the right definitions, the map ξ c ( L ξ ) is W -equivariant. The full relationship of H ∗ ( V //G, Q ) and H ∗ ( V //T, Q ) (which requires understanding therestriction H ∗ ( V //T, Q ) → H ∗ ( A, Q )) is worked out in [ ] by Martin; the analogous result inChow is proved in [ ] by Ellingsrud-Stromme. With this language, we can now write down a formula for the quasimap I -function. Because the formula expresses the I -function for V //G in terms of the I -function for V //T and the Weyl group, we call it an abelian/nonabelian corre-spondence for I -functions. Theorem . [ ] Let G be a reductive algebraic group and let T be a maximaltorus of G . Suppose G acts on a vector space V , and that the weights of therestriction of this action to T are ξ , . . . , ξ n ∈ χ ( T ) . Then the coefficients of the I -function of V //G satisfy (8.1) I V //Gβ ( z ) = X ˜ β → β Y α Q ˜ β · c ( L α ) k = −∞ ( c ( L α ) + kz ) Q k = −∞ ( c ( L α ) + kz ) I V //T ˜ β ( z ) where (8.2) I V //T ˜ β ( z ) = n Y i =1 Q k = −∞ ( c ( L ξ i ) + kz ) Q ˜ β ( ξ i ) k = −∞ ( c ( L ξ i ) + kz ) . The sum is over all effective ˜ β in τ − ( β ) , the variable α runs over the roots of G ,and the equality in (8.1) is equality after pulling both sides back to H ∗ ( A, Q ) . The formula (8.2) for the coefficients of a toric I -function was originally givenby Givental in [ ] (note that the quasimap I -function differs from Givental’s I -function by an exponential factor). Theorem 8.2 has natural generalizations totwisted and equivariant I -functions.One application of this theorem is an abelian/nonabelian correspondence inquantum cohomology for a large class of targets. Indeed, for these targets, the cor-respondence was reduced to a correspondence of small J -functions in [ ], and thewall-crossing result of [ ] translates this to the relationship of small I -functions inTheorem 8.2. (A more general abelian/nonabelian correspondence theorem in quan-tum cohomology was proved using symplectic geometry by Gonzalez-Woodward in[ ].) A second application, due to Kalashnikov [ ], is to identify nonisomor-phic Fano 4-folds by comparing their quantum periods, which are determined bytheir I -functions (this reference also provides an independent proof of Theorem8.2 when the target is a quiver flag variety). Thirdly, the proof of Theorem 8.2is very geometric, and so translates easily to the setting of K-theory, providingan analogous formula for the K-theoretic I -function. By providing explicit formu-las for I -functions, Theorem 8.2 should facilitate the computation of examples inGromov-Witten theory, and may lead to the discovery of new relationships.We close this paper with a sketch of the proof of Theorem 8.2. Sketch of proof of Theorem 8.2.
To compute the I -function coefficients I V //Gβ ( z ) in (7.1), the key step is to identify the fixed locus F β . In fact, connectedcomponents of F β are indexed by the isomorphism type of the principal bundle P of the quasimap, or equivalently by the Weyl orbits in τ − ( β ) (see Example 8.1).Choose representatives ˜ β i for these orbits and let F ˜ β i represent the correspondingcomponent of the fixed locus. As we did in Example 7.5 for Grassmannians, we canna¨ıvely guess what variety F ˜ β i should be.Generalizing Example 4.3, a quasimap from P to V //G may be written as avector of homogeneous polynomials. If it is in F β , then it has a unique basepointat [0 : 1] , so we hope that we can choose the polynomials to be polynomials in x UASIMAPS AND SOME EXAMPLES OF STACKS FOR EVERYBODY 15 only. Hence, if T acts on V with weights ξ , . . . , ξ n , define V ˜ β i to be the vectorspace with elements(8.3) ( u ( x ) , u ( x ) , . . . , u n ( x )) , where u j ( x ) is a constant multiple of x ˜ β i ( ξ j ) if ˜ β i ( ξ j ) ≥ , and u j ( x ) is zero other-wise. There is an evaluation map ev • : V ˜ β i → V sending (8.3) to ( u (1) , . . . , u n (1)).This map embeds V ˜ β as the subspace of V where ˜ β i ( ξ j ) ≥
0, which allows us tointersect V ˜ β i with V s ( G ) and obtain a space V s ˜ β i . Our guess is that every closedpoint of F ˜ β i may be represented by a point of V s ˜ β i .However, distinct points of V s ˜ β i may not represent distinct quasimaps. General-izing Example 5.2, an isomorphism of quasimaps in V s ˜ β i is an invertible n × n matrixof polynomials ( a ℓj ( x )) ℓj where a ℓj ( x ) is homogeneous of degree ˜ β i ( ξ ℓ ) − ˜ β i ( ξ j ) ifthis quantity is nonnegative, and a ℓj ( x ) = 0 otherwise. Let P ˜ β i denote this groupof matrices; we also have an evaluation map ev • : P ˜ β i → Aut( V ) by evaluating amatrix at 1. In fact, the image of ev • is in G , and ev • identifies P ˜ β i with a parabolicsubgroup of G .Finally we notice that V s ˜ β i /P ˜ β i carries a tautological family of quasimaps withthe following section:(8.4) U : V s ˜ β i × ( C \ { } )( u , x ) ∼ ( A u , t x ) −→ V s ˜ β i × ( C \ { } ) × V ( u , x , v ) ∼ ( A u , t x , [ t ˜ β ( ξξξ ) ] A ( x ) v )( u , x ) ( u , x , u ( x ))where ( u , x , v ) is in V s ˜ β i × ( C \ { } ) × V and ( A, t ) ∈ P ˜ β i × C ∗ , and [ t ˜ β i ( ξξξ ) ] denotesthe matrix with diagonal ( t ˜ β i ( ξ ) , . . . , t ˜ β i ( ξ n ) ) . This matrix factor guarantees thatevery geometric fiber of this family of vector bundles has isomorphism type ˜ β i .This tautological family defines a map from V s ˜ β i /P ˜ β i to F ˜ β i , and the hardest partof Theorem 8.2 is showing that this morphism is an isomorphism. The proof usesthat obstructions on F ˜ β i vanish, and moreover this stack is smooth.From here, the computation of the formula in Theorem 8.2 is relatively straight-forward. The evaluation map V s ˜ β i /P ˜ β i → V //G (which sends x to 1) factors as V s ˜ β i /P ˜ β i i −→ V s /P ˜ β i h −→ V s /G, where the first map is a closed embedding (recall V ˜ β i is a subspace of V ) and h is aflag bundle ( P ˜ β i is a parabolic subgroup of G ). Recall that the equality (8.1) holdsonly after pulling back both sides to V s ( G ) /T , so let Ψ : V s ( G ) /T → V s ( G ) /G bethe projection. We computeΨ ∗ I V //Gβ ( z ) = Ψ ∗ X ˜ β i ( ev • ) ∗ [ F ˜ β i ] vir e C ∗ ( N vir F ˜ βi ) = X ˜ β i Ψ ∗ h ∗ i ∗ e C ∗ ( N vir F ˜ βi )where we have used that [ F ˜ β i ] vir = 1 because F ˜ β i is smooth and the obstructionsvanish (see Example 6.1).To compute e C ∗ ( N vir F ˜ βi ), we use that the restriction of the absolute obstructiontheory to F ˜ β i is R • π ∗ F , where F is the sheaf on ( V s ˜ β i /P ˜ β i ) × P is defined by the exact sequence 0 → A → B → F → A = V s ˜ β i × ( C \ { } ) × g ( u , x , X ) ∼ ( A u , t x , [ t ˜ β i ( ξξξ ) ] A ( x ) · X ) and B = V s ˜ β i × ( C \ { } ) × V ( u , x , v ) ∼ ( A u , t x , [ t ˜ β i ( ξξξ ) ] A ( x ) · v ) . So e C ∗ ( N vir F ˜ βi ) is a certain ratio of the moving parts of the pushforwards of A and B (see (8.5)). We compute i ∗ with the projection formula, noting that V s ˜ β i is the zerolocus of the tautological section of the bundle V s × ( V /V ˜ β i ) on V s . We computeΨ ∗ h ∗ with a lemma of Brion [ ], which introduces an additional sum over the Weylgroup. After these steps we have(8.5)Ψ ∗ I V //Gβ ( z ) = X ˜ β i X ω ∈ W/W Li ω " e ( V s × T ( V /V ˜ β i )) e C ∗ ( R π ∗ A ) mov e C ∗ ( R π ∗ B ) mov Q α ∈ R + \ R Li c ( L α ) e C ∗ ( R π ∗ B ) mov e C ∗ ( R π ∗ A ) mov where A and B now denote the analogous bundles on V s ( G ) /T , and L i is the Levisubgroup of P ˜ β i containting T with roots R L i and Weyl group W L i , and R + arethe opposite roots of a Borel subgroup of G contained in P ˜ β i .One can check that W L i is precisely the stabilizer of ˜ β i in Hom( χ ( T ) , Z ) andthat the double sum and group action may be simplified to a single sum over all ˜ β in τ − ( β ), givingΨ ∗ I V //Gβ ( z ) = X ˜ β → β e ( V s × T ( V /V ˜ β )) e C ∗ ( R π ∗ A ) mov e C ∗ ( R π ∗ B ) mov Q α ∈ R + \ R L ˜ β c ( L α ) e C ∗ ( R π ∗ B ) mov e C ∗ ( R π ∗ A ) mov . From here it remains to compute the euler classes. One finds that the classes comingfrom A and the flag pushforward h ∗ yield the twist factor in (8.1): e C ∗ ( R π ∗ A ) mov Q α ∈ R + \ R L ˜ β c ( L α ) e C ∗ ( R π ∗ A ) mov = Y α Q ˜ β · c ( L α ) k = −∞ ( c ( L α ) + kz ) Q k = −∞ ( c ( L α ) + kz ) while the classes coming from B and the inclusion pushforward i ∗ yield the toric I -function: e ( V s × T ( V /V ˜ β )) e C ∗ ( R π ∗ B ) mov e C ∗ ( R π ∗ B ) mov = I V //T ˜ β ( z ) . (cid:3) References
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Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Current address : Department of Mathematics, University of California, Berkeley, Berkeley,California, 94720-3840
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