LOGARITHMIC QUASIMAPS

A BSTRACT . We construct a proper moduli space which is a Deligne–Mumford stack parametrising quasimaps relative to a simple normal crossings divisor in any genus using logarithmic geometry. We show this moduli space admits a virtual fundamental class of the expected dimension leading to numerical invariants which agree with the theory of Battistella–Nabijou where the latter is deﬁned.

INTRODUCTION 0.1.Results.Let X be a GIT quotient W/ /G of an affine variety by a reductive group, satisfying the assumptions of [20].Examples include toric varieties, partial flag varieties and complete intersections in these spaces.In addition, suppose X is a subvariety of V / /G, a vector space quotient, and let D = D 1 + • • • + D r be a simple normal crossings divisor pulled back from V / /G.We build the theory of logarithmic quasimaps for (X, D).Theorem 0.1.Fix non-negative integers g, n, an effective curve class β and α = (α i,j ) i,j a matrix of non-negative integers with n j=1 α i,j = D i • β.The moduli space Q log g,α (X|D, β) parametrising logarithmic quasimaps to (X, D) of genus g, degree β with contact orders α is a proper Deligne-Mumford stack.
Given a quasimap to X from a curve C, each D i induces a line bundle-section pair on C, thought of as the pullback of the pair cutting out D i .The moduli space parametrises quasimaps to X together with a logarithmic enhancement (with contact order α) of the morphism C → [A r /G r m ] induced by these line bundle-section pairs.This compactifies the space of maps from smooth curves to X with contact order α along D. As is standard, the moduli space is not smooth admitting a fundamental class, but we show it admits a virtual fundamental class.
Theorem 0.2.The moduli space Q log g,α (X|D, β) admits a perfect obstruction theory over M log g,α ([A r /G r m ]) leading to a virtual fundamental class [Q log g,α (X|D, β)] vir .
In [6], the authors build a theory of relative quasimaps for a smooth projective toric variety relative to a smooth, very ample divisor in genus zero.Theorem 0.3.For X a smooth projective toric variety, g = 0 and D smooth and very ample, this theory coincides with the theory of Battistella-Nabijou. 0.2.Why Quasimaps?Relative or logarithmic Gromov-Witten theory has had a tremendous influence on enumerative geometry in recent years.It has proved important for modern constructions in mirror symmetry [28], for determining ordinary Gromov-Witten invariants via the degeneration formula [3,4,14,33,37,49], and for providing insights about the moduli space of curves [24].Quasimap theory provides an alternative curve counting framework [17,20,40] when the target admits a certain GIT presentation, by incorporating basepoints.The resulting quasimap invariants, have also proved important in the context of mirror symmetry, as well as for studying the moduli space of curves.One example is their use in determining relations in the κ ring [47].Most crucially though, there are wall-crossing formulas relating Gromov-Witten invariants and quasimap invariants [18,19,56].We expect that a theory of logarithmic quasimaps will be able to produce new insights in logarithmic Gromov-Witten theory and its neighbouring areas.
Logarithmic Wall-Crossing.Computations in logarithmic Gromov-Witten theory are well sought after.The case of a smooth pair is well understood, yet explicit computations in the simple normal crossings case have proved to be more elusive.The few tools available use tropical geometry [38,43,48], scattering diagrams [11,25,26] and, more recently, rank reduction [42].For ordinary (non-logarithmic) Gromov-Witten theory, almost all results that allow us to compute genus-zero Gromov-Witten invariants rely at heart on the wall-crossing formula [18,19,56], the comparison between quasimap invariants and Gromov-Witten invariants.Battistella and Nabijou have proposed a similar program of computing logarithmic Gromov-Witten invariants by proving a wallcrossing formula relating logarithmic quasimaps and logarithmic Gromov-Witten invariants [6].Furthermore, they found as evidence for their proposal that a certain generating function of genus zero quasimap invariants relative to a smooth divisor coincided with the relative I-function of [21], which in turn can be obtained from a generating function of relative Gromov-Witten invariants by a change of variables.A full theory of logarithmic quasimaps now allows this avenue to be pursued.
Mirror Symmetry.Quasimaps have a fundamental connection to mirror symmetry.The wallcrossing formula is exactly the mirror map for Calabi-Yau threefolds [18], and so the quasimap invariants coincide on the nose with B-model invariants of the mirror.The (conjectural) holomorphic anomaly equation provides remarkable structure to the Gromov-Witten theory of a Calabi-Yau, coming from the B-model.The link between the quasimap and B-model invariants has been utilised to give a direct geometric proof of the holomorphic anomaly equation for local P 2 [35].In another direction, the authors of [12] prove a holomorphic anomaly equation for the logarithmic Gromov-Witten theory of P 2 relative an elliptic curve.This provides interesting directions for holomorphic anomaly equations for logarithmic quasimaps.

Logarithmic-Orbifold Comparison.
There is another approach to counting curves with tangency conditions, using orbifolds [13,52].Here, one takes a pair (X, D 1 + • • • + D r ) and replaces the divisor components with roots of the divisor, introducing non-trivial isotropy groups.In this orbifold compactification the tangency gets recorded in the group homomorphism between isotropy groups, using the technology of orbifold stable maps.If the divisor is smooth, then these two approaches give the same invariants in genus zero [1].Roughly speaking, this equivalence can be seen by taking the coarse moduli map of an orbifold stable map, and noting that there is a logarithmic lift.This is no longer true in the simple normal crossings case [8,42].The orbifold theory is more computable since it satisfies a product formula, but gives the wrong answers from the logarithmic perspective.In [7], noting that the logarithmic theory is invariant under logarithmic modifications [5] but that the orbifold theory is not, the authors show that after a certain blow-up the two theories coincide.A related approach is suggested in [55], where the indication is that structures in orbifold Gromov-Witten theory will be preserved under modification.These approaches can be summarised as fixing the moduli problem (Gromov-Witten theory) and altering the target.A complimentary strategy is to fix the target and alter the moduli problem to a situation where the orbifold and logarithmic theories coincide.In [42,Remark 5.4], the authors observe in examples, that the error terms occur in the presence of components of the moduli space consisting of stable maps with rational tails.Quasimap theory is designed to remove rational tails.If we consider quasimap theory as our moduli problem instead it is likely that there will be significantly fewer correction terms between the logarithmic and orbifold theories, the latter of which was developed in [16].
At least in genus zero, the proposed picture is [ 16,56] This paper constructs the bottom left-hand corner of the diagram.The expected wall-crossing formula relating logarithmic Gromov-Witten theory to logarithmic quasimap theory would be the left-hand vertical arrow; the bottom arrow would be the comparison between logarithmic and orbifold quasimap theory.We expect this last comparison to be simpler, and in certain situations, trivial, which we will show in an example.
A parallel direction is the local-logarithmic correspondence [54], which exhibits some of the beautiful geometry in logarithmic Gromov-Witten theory of a smooth pair (X, D).There are counterexamples to a natural generalisation when D is a simple normal crossings divisor, which are the same counterexamples used in the logarithmic-orbifold comparison [42], since the local and orbifold theories coincide [8].
Calculations in [51] indicate that these are no longer counterexamples in the quasimap setting.Therefore in an analogue comparison to the bottom arrow of 1, it is likely that the local-logarithmic correspondence will hold in greater generality in the quasimap setting.0.3.Outline.We begin in Section 1 by recalling the definition of quasimaps and presenting a way to incorporate divisors using maps to [A r /G r m ].In Section 2 we build the moduli space of logarithmic quasimaps and show it is a proper, Deligne-Mumford stack.The difficulty in the definition is that a priori it was not clear how the logarithmic structure should interact with the basepoints.The key is to recognise that in the Gromov-Witten setting a (stable) logarithmic map can be decomposed into the underlying stable map together with a logarithmic morphism to the Artin fan (or [A r /G r m ]).We use this decomposition as an analogy for defining a logarithmic quasimap.Moreover, this approach allows for a streamlined presentation of the theory, which relies on the existence and properties of the space of logarithmic maps to [A r /G r m ].In Section 3 we present three examples of the moduli space for P N relative to a collection of hyperplanes and compare them to the Gromov-Witten setting.For a single hyperplane in genus zero, we note that both spaces are irreducible but that the boundary compactification is significantly simpler in the quasimap case.In the case of multiple hyperplanes in genus zero, we give a specific instance where wall-crossing accounts for the entire discrepancy between orbifold Gromov-Witten and logarithmic Gromov-Witten theory.The final example exhibits the fact that with respect to the full toric boundary in any genus, the logarithmic Gromov-Witten and quasimap moduli spaces coincide.
In Section 4 we construct the virtual fundamental class, first in generality and then noting how the construction simplifies when working with a smooth projective toric variety relative to a toric divisor.
Finally in Section 5, we prove that this theory of logarithmic quasimaps coincides with the Battistella-Nabijou theory of relative quasimaps from [6], where the latter is defined.We do this by proving the result for P N relative to a hyperplane and then using the very ample embedding to pull the result back.0.4.History.Over the years there has been much interest in defining and computing relative Gromov-Witten invariants of a pair (X, D).When attempting to define a proper moduli space of curves with fixed contact orders to D one sees that in the limit whole components of the curve can fall into D, at which point it is no longer clear what contact order means.This poses a major difficulty.A first solution was proposed by Gathmann [23] building on work of Vakil [53] for (very) ample smooth divisors in genus zero by taking a closure inside the space of absolute stable maps.Jun Li [36] extended this to smooth divisors in all genus using expansions of the target.Since then, there have been various approaches to extend the theory to the simple normal crossings case.The first of these came from Abramovich, Chen, Gross and Siebert [2,15,27] using logarithmic structures which provide extra structure for defining contact order even if a component falls into D.There have also been approaches which combine expansions with logarithmic structures [32,49].Finally, there is the approach using orbifolds [13,52], which can give genuinely different invariants.
In the quasimap setting, building on Gathmann's approach, Battistella and Nabijou [6] developed a theory of relative quasimaps in genus zero for smooth projective toric varieties relative a smooth (very) ample divisor by taking a closure inside the absolute quasimap space.This paper provides a quasimap theory relative a s.n.c.divisor D which removes these assumptions.
Acknowledgements.I would like to wholeheartedly thank Dhruv Ranganathan for suggestions and guidance with this project as well as for invaluable feedback on drafts.I owe a great deal of thanks to Navid Nabijou for many helpful discussions as well as for feedback.I would also like to thank Tom Coates and Rachel Webb for helpful conversations.This work was supported by the EPSRC Centre for Doctoral Training in Geometry and Number Theory at the Interface, grant number EP/L015234/1 and the UKRI Future Leaders Fellowship through grant number MR/T01783X/1.

ABSOLUTE QUASIMAPS WITH DIVISORS
Quasimaps are a variation on stable maps, where, roughly, one allows rational maps.In this section we recall the definition of absolute GIT quasimaps from [20] and explain how we will incorporate divisors.Let W = Spec A be an affine algebraic variety with the action by a reductive algebraic group G. Let θ be a character inducing a linearisation for the action.We insist that Then quasimaps to W/ /G are defined as follows.
Definition 1.1.Fix non-negative integers g, n and β ∈ Hom(Pic G W, Z).An n-marked stable quasimap of genus g and degree β to W/ /G is which satisfy (1) (Non-degeneracy) there is a finite (possibly empty) set B ⊂ C, distinct from the nodes and markings, such that ∀c ∈ C \ B we have , where L θ comes from the linearisation.
Now let X = W/ /G ֒→ V / /G be a subvariety of a vector space quotient coming from W ֒→ V (the prequotient embedding always exists [20, 2.5.2]), and let D be a smooth divisor on V / /G, defined by a line bundle-section pair (O V / /G (D), s D ), which pulls back to give a divisor on W/ /G, which we also denote D. In order to build a moduli space of quasimaps to (X, D) we need to make sense of the tangency of a quasimap to X along D.
Since V is a vector space, every line bundle is isomorphic to the trivial line bundle.Consequently, line bundles on V / /G correspond to characters of G and sections of line bundles correspond to G-equivariant sections of the trivial bundle on V .Any such pair defines a line bundle and section on the stack Definition 1.2.Let (X, D) be as above and suppose we have a family of stable quasimaps over a scheme S from a family of curves C to X. Then we define the morphism induced by D to be the morphism defined by the above line bundle-section pair on C.

MODULI SPACE OF LOGARITHMIC QUASIMAPS
Let X = W/ /G ֒→ V / /G be as above and D a simple normal crossings divisor on W/ /G pulled back from V / /G.In this section we will define the moduli space of logarithmic quasimaps to (X, D).The key point will be that although there is no genuine map from a curve to X, we have the induced morphism (3) which contains all the data concerning tangency.It is this morphism which we will make logarithmic in order to compactify.
An alternative way to approach the problem is to equip the quotient stack containing X with the divisorial logarithmic structure with respect to the closure of D and form a moduli space of logarithmic maps to this quotient stack.This is essentially equivalent, but phrasing it as above allows us to take advantage of existing moduli spaces.Specifically, in this section we can take advantage of the moduli space of logarithmic maps to [A r /G r m ].Definition 2.1.Let g, n be non-negative integers and let α be an r × n matrix of non-negative integers α = (α i,j ) i,j .A family of logarithmic maps to [A r /G r m ] over (S, M S ), a fine and saturated logarithmic scheme, is a diagram where C S → S is an n-marked family of genus g, logarithmic curves and equipped with the divisorial logarithmic structure with respect to the coordinate hyperplanes, with contact orders α at the markings.

Theorem 2.2 ( [5]
). Minimal logarithmic maps form a logarithmic algebraic stack ) is the divisorial logarithmic structure with respect to the complement of the locus of maps to [A r /G r m ] from smooth curves which hit the boundary in finitely many points.With respect to this logarithmic structure Let g, n be non-negative integers, let β be an effective curve class on X and let D = D 1 + • • •+ D r be a s.n.c.divisor pulled back from V / /G, with the simplifying assumption that the intersection of any subsets of the components of D is connected.Let α be an r × n matrix of non-negative integers α = (α i,j ) i,j such that j α i,j = D i • β.Definition 2.4.Define Q log g,α (X|D, β) as the fibre product of stacks ) is the stack of maps from n-marked, genus g prestable curves to [A r /G r m ].The morphism π 1 is given by associating to a stable quasimap the induced map to [A r /G r m ] (3), and π 2 is given by forgetting the logarithmic structure.
Proof.Since Q log g,α (X|D, β) is a fibre product of algebraic stacks it is automatically algebraic.The fact that it is a Deligne-Mumford stack will follow from the fact that the morphism π 2 is representable, which is true by [27,Proposition 1.25].Given any cartesian diagram of algebraic stacks with X Deligne-Mumford and π 2 representable, then we can use the criterion [13, Lemma 3.3.2]to show that any object (x, y, λ), where x ∈ Ob(X ), y ∈ Ob(Y), λ : π 1 (x) ≃ π 2 (y) must have finitely many automorphisms (ϕ x , ϕ y ) ∈ Aut X (x) × Aut Y (y).There are finitely many automorphisms, ϕ x , as X is Deligne-Mumford, but that fixes On the other hand, π 2 is representable so it is an injection on automorphism groups, which determines ϕ y .
) is locally of finite type, so it suffices to show that ).But this follows from the fact that once we have fixed numerical data g, n, β, the number of components of the curve occurring in Q g,n (X, β) is bounded.
The papers [2,15] construct the moduli space of stable logarithmic maps by first constructing the moduli space in the case where D is a smooth divisor, and then using this to build the moduli space when D is s.n.c.The same approach would have worked in the quasimap setting.Lemma 2.9.Let α i denote the i th row of the r × n matrix α.Then the moduli space Q log g,α (X|D, β) fits into a cartesian diagram (in the fine and saturated category) (4) Proof.This is very similar to the argument in [7, 2.2].Consider the diagram ), we can conclude that the top left square is (and so all squares are) cartesian, in the fine and saturated category, by the definition of Q log g,α (X|D, β) as in 2.4.This tells us that the outer square in the following diagram is cartesian (5) ∆ Since the right hand square is also cartesian, the result follows.
Remark 2.10.The entire construction could have equally been used to build a moduli space parametrising logarithmic ǫ-quasimaps for any ǫ ∈ Q >0 .This would involve replacing Q g,n (X, β) with Q ǫ g,n (X, β) in the definitions.Moreover, the construction of the virtual fundamental class 4.11 also works for any ǫ.We will not make use of this here but it will be important in any future application for wall-crossing.

EXAMPLES
Having built these moduli spaces, we now examine their geometry in a few simple examples and compare them with the corresponding moduli spaces in Gromov-Witten theory.We will examine a smooth divisor example, a simple normal crossings divisor example and a higher genus example, all for projective space targets.The first example is strictly speaking covered by [6], but we include it here anyway to show that even in the cases where the moduli space is as nice as possible, the quasimap moduli spaces are less complex.
Example 3.1.Let X = P N , let D = H a hyperplane and suppose we are in genus 0, degree d.We let n = 2 be the number of markings and let α = (d, 0) i.e. we are considering degree d (quasi)maps to P N from rational curves with maximal tangency to the hyperplane H at the first marking.We will compare Q log 0,(d,0 d) is irreducible of the expected dimension and we will see in 5.7 that Q log 0,(d,0) (P N |H, d) is irreducible of the same dimension for the same reason.More concretely, both spaces (or rather their images in M 0,2 (P N , d) (resp.Q 0,2 (P N , d)) are the closures of the locus of maps (P 1 , p 1 , p 2 ) → P 2 which are not mapped entirely into the hyperplane and hit H with maximal tangency at p 1 .Comparing these spaces with their images in the corresponding absolute spaces is a reasonable thing to do as the morphisms of moduli spaces here are finite and generically injective.On the other hand, the quasimap spaces do not allow rational tails, curves with rational components with a single special point.Consequently, there will be fewer boundary divisors in In both cases the boundary divisors are comb loci.These are loci where the source curve is of the form Moreover, C 0 contains the non-trivial tangency marking and gets mapped entirely into H, whereas C i only hits H at the connecting node for i = 0.
A comb locus in the quasimap space can have at most two components for stability reasons.There is a divisor corresponding to when C = C 0 which falls entirely into the hyperplane and d − 1 distinct comb loci with two components C = C 0 ∪ C 1 corresponding to the different possible degrees on each branch.However, in the Gromov-Witten space there can be up to d + 1 components on the source curve of a comb locus.Each external component C i , i = 0 must have positive degree and so there is a comb locus where each of the d external components have degree 1 and the interior component carrying the maximal tangency marking is contracted.Below we include the number of boundary divisors in the quasimap and Gromov-Witten case to make the point that the quasimap spaces are much simpler.Remark 3.2.In [31], a formula is determined for the number of boundary divisors of M log 0,(d,0,...,0) (P N |H, d), i.e. the logarithmic Gromov-Witten moduli space for (P N , H) with one maximal tangency marking and any number of redundant markings in any degree.
Example 3.3.We now move on to a simple normal crossings example.In the stable maps case this example comes from [42, 1.2] and the ideas come from [41,3].Let X = P 2 and D = H 1 + H 2 be the union of two hyperplanes.Let g = 0, d = 2 and let α be given by the matrix 2 0 0 2 .
In other words we are considering degree 2 (quasi)maps to P 2 with maximal tangency to H 1 at the first marking and maximal tangency to H 2 at the second marking.As in the previous example, 2) are irreducible of the same dimension but have differing numbers of boundary divisors.Instead of enumerating them we will make a different comparison.Recall that Q log 0,α (P 2 |H 1 + H 2 , 2) is defined via the fibre product where α 1 = (2, 0) and α 2 = (0, 2) (this is equivalent to (4)), in the category of fine and saturated logarithmic stacks.Similarly M log 0,α (P 2 |H 1 + H 2 , 2) can be defined as the fibre product (in the fine and saturated category) [2] Recall from the previous example that M log 0,α i (P 2 |H i , 2) → M 0,2 (P 2 , 2) is finite and generically injective.The (image of the) ordinary fibre product would be the intersection of (the images of) M log 0,α 1 (P 2 |H 1 , 2) and M log 0,α 2 (P 2 |H 2 , 2), each of which corresponded to the closure of the loci corresponding to maps from irreducible curves which don't map into H i .So the image of the ordinary fibre product in M 0,2 (P 2 , 2) is the intersection of the closure of these loci.On the other hand M log 0,α (P 2 |H 1 + H 2 , 2) is also irreducible and maps finitely and generically injectively into M 0,2 (P 2 , 2).This image is also the closure of the locus of maps from irreducible curves which don't map entirely to either hyperplane.So comparing the ordinary fibre product and M log 0,α (P 2 |H 1 + H 2 , 2) amounts to the difference between the intersection of the closures and the closure of the intersection.In this case the ordinary fibre product is strictly larger than M log 0,α (P 2 |H 1 + H 2 , 2).By a dimension count M log 0,α (P 2 |H 1 + H 2 , 2) is 3-dimensional.On the other hand, there is a locus in the ordinary fibre product corresponding to source curves of the form C 0 ∪ C 1 ∪ C 2 , where C 0 is attached to C 1 and C 2 at nodes, both markings lie on C 0 and this component gets contracted to the point of intersection in H 1 ∩ H 2 .This locus has dimension 3 and forms the only other irreducible component of the ordinary fibre product.So in this case M log 0,α (P 2 |H 1 + H 2 , 2) is not the same as the ordinary fibre product.
We could instead compare Q log 0,α (P 2 |H 1 +H 2 , 2) with the ordinary fibre product of Q log 0,α 1 (P 2 |H 1 , 2) and Q log 0,α 1 (P 2 |H 2 , 2) over Q 0,2 (P 2 , 2).The dimension counts are all identical but due to the quasimap stability condition, even in the ordinary fibre product there can be no component with source curve C 0 ∪ C 1 ∪ C 2 with both markings on C 0 , because C 1 and C 2 contain only a single special point.The consequence is that unlike in the stable maps case, Q log 0,α (P 2 |H 1 + H 2 , 2) is the same underlying space as the ordinary fibre product.
One reason for pointing out this difference is that the ordinary fibre product can be equipped with a 'virtual' class leading to invariants [42].These invariants are equal to the orbifold invariants of the multi root stack given by rooting P 2 along H 1 and H 2 .The orbifold Gromov-Witten invariants of a root stack coincide in genus-zero with the logarithmic Gromov-Witten invariants when the divisor is smooth [1], but differ when the divisor is simple normal crossings.In the case above the difference can be attributed to the excess component in the ordinary fibre product.On the other hand there is no difference in the quasimap setting, which suggests that the difference between logarithmic and orbifold quasimap invariants may be less pronounced.Stable maps without rational tails are stable quasimaps, and stable quasimaps without basepoints are stable maps, so it suffices to show that neither of these are possible here.Since any non-constant morphism from a rational curve must hit the boundary in at least two points we must have that this rational component contains at least two special points.On the other hand, any curve component of a logarithmic stable quasimap which is external to one of the H i cannot have basepoints, because if it did, that basepoint would necessarily have to be at a point of intersection with H i , but these points are either marked or nodes.Since ∩ N i=0 H i = ∅, any curve component has to be external to at least one H i , so no such quasimap can contain basepoints.

LOGARITHMIC QUASIMAP INVARIANTS
Let Q log g,α (X|D, β) be the moduli space from Definition 2.4.We will produce a virtual fundamental class on this moduli space using the construction of Behrend and Fantechi [9].Proof.We have a morphism Moreover, this morphism is strict, so the map from the underlying curve of C to [W/G] together with a logarithmic morphism

Therefore we have a diagram
Here C Q and C M are the universal curves over the moduli spaces Q log g,α (X|D, β) and M log g,α ([A r /G r m ]) and u is the universal map defined by 4.1.Furthermore, the square is cartesian in both the fine, saturated category and the ordinary category.We impose in this section that the morphism m ] be flat, used in the proof of 4.5, but we expect this to not be a restrictive assumption.
In [46] the author introduces the stack of logarithmic structures.Namely, if (S, M S ) is a fine logarithmic scheme, then there is a fibered category Log (S,M S ) , where the objects over a morphism of schemes T → S is an enhancement of this morphism to fine logarithmic schemes (T, M T ) → (S, M S ).The main result of [46] is that Log (S,M S ) is an algebraic stack, locally of finite presentation.Using this, Olsson defines the logarithmic cotangent complex as follows.Definition 4.2.[44] Let X → Y be a morphism of fine logarithmic schemes.Let X → Log Y be the induced morphism.Define L log X/Y , the logarithmic cotangent complex, to be the ordinary cotangent complex of this morphism.
We will need an extension of this theory to certain algebraic stacks.In [27], the authors did this for Deligne-Mumford stacks by reducing to an étale cover.Our situation is slightly more general and we will just define the logarithmic cotangent complex in the same way.

Let L log
[W/G] denote the logarithmic cotangent complex of [W/G].This is defined as the ordinary cotangent complex of the morphism where the latter denotes the stack of logarithmic structures for Spec C with the trivial logarithmic structure.

Lemma 4.4.
There is a morphism in the derived category of Q log g,α (X|D, β) Proof.By the functoriality properties of the cotangent complex [34, 17.3 (2)], we get a morphism in the derived category On the other hand, base change properties of the cotangent complex [34, 17.3 (4)] imply that Tensoring this morphism with the relative dualising sheaf gives so, by applying Rπ * and using adjunction, we get a morphism in the derived category Now we need to show two things: (1) φ defines an obstruction theory. ( Proposition 4.5.The morphism φ defines an obstruction theory. Proof.This argument follows exactly as in [27, Proposition 5.1].Let T ֒→ T be a square zero extension with ideal J and let g : T → Q be a morphism.Endow T and T with logarithmic structures pulled back from M and pullback the universal curve to T and T , which we will denote C T and C T .We have a commutative diagram (6) Recall from [29, III 2.2.4] and [45, 2.21] that g extends to a morphism T → Q if and only if ω(g) ∈ Ext 1 (Lg * L Q/M , J) is 0 where ω(g) is defined by To show that φ defines an obstruction theory we use [9,Proposition 4.53].We show that an extension exists if and only if φ * ω(g) = 0 and moreover if an extension exists, the set of isomorphism classes of extensions form a torsor under Hom(φ * E • , J).As in [27, Proposition 5.1], a lift exists if and only if u T extends logarithmically to C T .But using [44,Theorem 5.9] there is a class o ∈ Ext 1 (Lu * T L log [W/G] , p * J) which vanishes if and only if there is a lift.Moreover, the lifts form a torsor under Hom(Lu * T L log [W/G] , p * J).However, just as in [27] we have Remark 4.6.We have used results from [44], in particular [46, Axiom 1.1 (ii), (iv), Theorem 5.9], which are proved in the context of logarithmic schemes.We require these results to extend to certain algebraic stacks.The same proofs go through without change, often because the results we require rely on properties of the algebraic stacks Log Γ , or because they rely on properties of the ordinary cotangent complex, which hold quite generally.First assume that W = V a vector space and so the divisor D comes from a morphism There is a distinguished triangle in the derived category of .
By [10] the tangent complex of [V /G] is given in degrees [−1, 0] by the differential of the action.Then the tangent complex is given by The same reasoning tells us that the tangent complex of e i → (0, . . ., 0, x i , 0, . . ., 0).
Pulling this back to [V /G] gives the complex Proof.Taking the dual distinguished triangle tells us that we can build T log [V /G] as the shift of the mapping cone of . Since the morphism ( 7) is injective it follows that the mapping cone actually has cohomology supported in [−1, 0] and so after shifting, Since we are only considering divisors pulled back from [V /G] we have morphisms The associated (dual) distinguished triangle is But by the cartesian diagram which by the result above must be have cohomology supported in [0, 1].

We can also conclude that Lu
Proof of 4.7.We need to show that Rπ * (Lu in the derived category.We show that R 2 Γ(F • ) = 0. We have that H 1 (F • ) is a torsion sheaf.This follows from the fact that away from the set B of finitely many basepoints on C the morphism u factors through X.Moreover, the restriction F • | C\B is quasi-isomorphic to T log X .We have the spectral sequence The second page looks like but by the above we know that Using the criterion of [30, 3.6.4],the complex Rπ * (Lu * T log [W/G] ), which is cohomologically supported in [0, 1], is of perfect amplitude contained in [0, 1] if and only if for every point p we have But this is also clear by the above.
becomes very explicit, given by the following diagram ( 12) is a subset of the toric boundary then s 1 , . . .s r are just x j 1 , . . ., x jr corresponding to the homogeneous coordinates on X.But since the right-hand vertical map is given by differentiating the sections this map becomes e i → (0, . . ., 1, . . ., 0) where 1 is in the j th i place.Consequently, by forming the mapping cone (and shifting) we have that T log [A M /G s m ] is given by the three term complex But the right-hand map is now surjective so this complex is quasi-isomorphic to a two term complex Where in the second term the first M − r copies have action according to the weight matrix and the latter r copies have the trivial action.Since the first map is injective the second part follows.
For the first part we just pullback this complex under the inclusion X ֒→ [A M /G s m ].

COMPARISON WITH BATTISTELLA-NABIJOU THEORY
In [6] the theory of relative quasimaps is developed in the case where X is a smooth projective toric variety, D ⊂ X is a smooth, very ample divisor (not necessarily toric) and in genus-zero.This is achieved by mimicking the Gromov-Witten construction in [23] and so the relative quasimap moduli space is defined as a closed substack of the space of absolute quasimaps.
Let X be a smooth projective toric variety associated to a complete fan Σ and D a smooth (very ample) divisor, cut out by a section s D ∈ H 0 (X, O X (D)).Recall that given an ordinary quasimap from a (marked) curve C to X there is an induced line bundle section pair (L D , u D ) and in the case where there are no basepoints these are just the pullbacks of O X (D) and s D respectively along the map C → X. Definition 5.1 ( [6]).Let n ≥ 2 be the number of marked points, let β be an effective curve class and α = (α 1 , . . ., α n ) such that i α i ≤ D • β.Define the moduli space of relative quasimaps Q rel 0,α (X|D, β) to be the locus of absolute quasimaps (C, p 1 , . . ., p n ), {L ρ , u ρ } ρ∈Σ (1) , {c m } m∈M in Q 0,n (X, β) such that for every Z, a connected component of u −1 D (0) we have (1) If Z is a point, then either it is unmarked or a marked point p i such that the multiplicity of but intersecting Z, and let m (i) be the multiplicity of u D at the node Then they use diagonal pull-back to define a virtual fundamental class on Q rel 0,α (X|D, β) and define relative quasimap invariants.Remark 5.4.From now on we restrict to the case where i α i = D•β in which case the inequalities above become equalities.Theorem 5.5.Let g = 0 and D ⊂ X be a smooth, very ample divisor inside a smooth projective toric variety.There is a morphism g : The strategy for proving Theorem 5.5 is as follows.We will first prove the existence of the morphism g.Then we prove Theorem 5.5 for the case of X = P N and D = H a hyperplane.We will then use the ampleness condition to pullback this result to the general case.
Proposition 5.6.The natural morphism Q log 0,α (X|D, β) → Q 0,n (X, β) given by forgetting the logarithmic map to Proof.Certainly on the locus where C ∼ = P 1 is irreducible and the section u 0 vanishes only at the marked points, this is true.Suppose now we have logarithmic quasimap which contains a connected component Z ⊂ C on which u 0 ≡ 0. Then we need to show that We have a logarithmic morphism C → [A 1 /G m ].This induces a morphism of the tropicalisations.As in toric geometry, a piecewise linear function on the tropicalisation induces a Cartier divisor.Alternatively, a logarithmic structure can be characterised by an association of a Cartier divisor to each element of the ghost sheaf.The identity function on R ≥0 induces the divisor BG m .The fact that we have a logarithmic morphism necessarily implies that the pull back of this line bundle via the morphism is the same as the line bundle associated to the pull-back piecewise linear function.
On the one hand the line bundle pulls back to L D , when restricting to Z we get L D | Z .On the other hand, the pull back piecewise linear function is defined by the slopes of the tropicalisation map on each ray.The associated line bundle on the component Z is shown to be O( i∈Z α i p i − i m (i) q i ), where q i are the nodes, in [50, 2.4 Taking the long exact sequence in cohomology it follows that H 1 (C, F) = 0 and so this moduli space is unobstructed over M log 0,α ([A 1 /G m ]), which is logarithmically smooth.
To distinguish from a general (X, D), we denote the morphism Q log 0,α (P N |H, d) → Q rel 0,α (P N |H, d) by f . .So the proposition reduces to a statement about fundamental classes.By Lemma 5.7 and [6] we know that the locus where the source curve is irreducible and the u 0 only vanishes at the marked points is dense in both spaces.Furthermore, on this locus the map is an isomorphism so the result follows.
We now use Proposition 5.8 to prove Theorem 5.5.Proof.This follows from the fact that there are cartesian diagrams Proposition 5.10.There is a perfect obstruction theory on Q log 0,α (X|D, β) relative to Q log 0,α (P N |H, d) such that the corresponding virtual class coincides with [Q log 0,α (X|D, β)] vir .
Proof.Recall O X (D) defined a morphism j : X ֒→ P N such that j −1 (H) = D.If we write X = A M / /G s m as a GIT quotient, as in 5.2, then this morphism induces a morphism of quotient stacks, which we also denote j, j : such that j −1 ( H) = D.The divisors D, H define logarithmic structures via morphisms to [A 1 /G m ] such that there is a commutative diagram Next, consider the diagram ( 14) Q log 0,α (X|D, β) Q log 0,α (P N |H, d).Li * L Q log (P)/M log L Q log (X)/M log L i . [1] [1] The first two (and last) vertical arrows come from the perfect obstruction theory from 4.11.By [39,Construction 3.13] and [39,Remark 3.15] it follows that (Rπ * Lu * T i ) ∨ defines a perfect obstruction theory.Moreover in the language of [39, Definition 4.5] the three perfect obstruction theories form a compatible triple and so by [39,Theorem 4.8]   In 5.10 we showed that [Q log 0,α (X|D, β)] vir is defined via a perfect obstruction theory for i.In actual fact this obstruction theory is pulled back from j ′ (or i ′ ).To see this, repeat the argument of 5.10, starting instead with the distinguished triangle which shows that the perfect obstruction theory for j ′ also pulls back to (Rπ * Lu * T j ) ∨ .This tells us that j ′ ![Q log 0,α (P N |H, d)] = [Q log 0,α (X|D, β)] vir and since diagonal pullback coincides with virtual pullback [6, Lemma A.0.1], we have that j ′ ![Q rel 0,α (P N |H, d)] = [Q rel 0,α (X|D, β)] vir .Therefore, the theorem follows from [39,Proposition 5.29].

Remark 1 . 3 .
If instead of a smooth divisor we have a simple normal crossings divisor D = D 1 + • • • + D r , then we get r induced maps to [A 1 /G m ], one for each component of D. Together these give a morphism (3) C → [A r /G r m ] which we once again call the morphism induced by D.

Example 3 . 4 .
Let X = P N and D = H 0 + • • • + H N be the full toric boundary, where H i denotes the i th coordinate hyperplane.Then for any valid discrete data g, n, d, α we have that M log g,α (P N |D, d) = Q log g,α (P N |D, d).

Lemma 4 . 1 .
The divisor D induces a logarithmic structure on the quotient stack [W/G] pulled back from the toric logarithmic structure on [A r /G r m ].Moreover, a logarithmic quasimap to (X, D) from a curve C induces a logarithmic morphism C → [W/G].

Example 4 . 10 .a
If V = A M and G = G s m with action described by the weight matrix 11 a 12 . . .a 1M a 21 a 22 . . .a 2M

i
and the hypotheses on W we know that T [W/G]/[V /G] has cohomology supported in degree 1.The distinguished triangle tells us we can build T log[W/G] as the mapping cone of the morphism Li

Corollary 4 . 13 .Corollary 4 . 14 .
If D is a subset of the toric boundary then the complex(Rπ * Lu * T log [A M /G s m ] ) ∨ is of perfect amplitude contained in [−1, 0].There is a relative perfect obstruction theory on Q log g,α (X|D, β) over M log g,α ([A r /G r m ]) leading to a virtual fundamental class [Q log g,α (X|D, β)] vir .

Remark 5 . 2 .Remark 5 . 3 .
The definition of absolute quasimap here is taken from[17].If we write the toric variety X as a GIT quotient A M / /G s m as prescribed by the fan Σ, then this definition is equivalent to the Definition 1.1, involving a morphism C → [A M /G s m ]. let X = P N and let D = H ∼ = P N −1 be the hyperplane given in coordinates by {x 0 = 0}.Then Q rel 0,α (P N |H, d) is irreducible of dimension dim Q 0,n (P N , d) − i α i and so has an actual fundamental class with which one can define relative quasimap invariants.For the general case of (X, D), note that O X (D) defines a map j : X ֒→ P N .Battistella and Nabijou show that the following diagram is cartesian (with d = j * β)

H
Taking the induced distinguished triangle (on tangent complexes) givesT j → T log [A M /G s m ] → Lj * T log [A N+1 /Gm] → T j [1].
Toric Divisors.On the other hand 4.7 becomes significantly simpler in the case where X is a toric variety andD = D 1 + • • • + D ris a subset of the toric boundary.Let X be a smooth projective toric variety associated to some fan Σ inducing a quotient description as in Example 4.10.If D = D 1 + • • • + D ris a subset of the toric boundary corresponding to rays ρ j 1 , . . ., ρ jr ∈ Σ(1) (set J = {j 1 , . . ., j r }), then there is a logarithmic Euler sequence .1].Since these line bundles are necessarily isomorphic, taking degrees gives us the desired equality.It follows from 4.12 that Lu * T log [A N+1 /Gm] is quasi-isomorphic to a sheaf F which fits into an exact sequence on C N |H, d) is defined by the complex Rπ * Lu * T log [A N+1 /Gm] ∨ Because P N |H admits a logarithmic Euler sequence (11) 0 → O P N → O P N ⊕ N i=1 O P N (1) → T log P N → 0.