Reduced classes and curve counting on surfaces II: calculations
aa r X i v : . [ m a t h . AG ] M a r REDUCED CLASSES AND CURVE COUNTING ONSURFACES II: CALCULATIONS
MARTIJN KOOL AND RICHARD THOMAS
Abstract.
We calculate the stable pair theory of a projective surface S .For fixed curve class β ∈ H ( S ) the results are entirely topological, depend-ing on β , β.c ( S ), c ( S ) , c ( S ), b ( S ) and invariants of the ring structureon H ∗ ( S ) such as the Pfaffian of β considered as an element of Λ H ( S ) ∗ .Amongst other things, this proves an extension of the G¨ottsche conjectureto non-ample linear systems.We also give conditions under which this calculates the full 3-fold reducedresidue theory of K S . This is related to the reduced residue Gromov-Wittentheory of S via the MNOP conjecture. When the surface has no holomor-phic 2-forms this can be expressed as saying that certain Gromov-Witteninvariants of S are topological.Our method uses the results of [KT1] to express the reduced virtual cyclein terms of Euler classes of bundles over a natural smooth ambient space.
1. Introduction 12. The moduli space as a zero locus 42.1. Virtual normal bundle 73. Calculation with H -insertions 84. Calculation without H -insertions 105. Relation to 3-fold invariants 15References 181. Introduction
Fix a nonsingular projective surface S and a homology class β ∈ H ( S, Z )of Hodge type (1 , S for which β is no longer (1 , S and of the Calabi-Yau 3-fold X := K S vanish by deformation invariance.Similarly for the stable pair invariants of X [PT1]. By removing part of theobstruction bundle one can define “ reduced ” invariants which are only invariantunder deformations of S in the Noether-Lefschetz locus (the locus where β isof type (1 , M. KOOL AND R. P. THOMAS the introduction to [KT1] for references. In [KT1] we defined such reducedGromov-Witten and stable pair invariants under the condition that(1) H ( T S ) ∪ β −→ H ( O S ) is surjective . Here we consider β to lie in H (Ω S ), so the map is induced by the pairingΩ S ⊗ T S → O S . When h , ( S ) = 0 the reduced invariants coincide with theordinary invariants.In this paper we work with the stable pair theory. As in [KT1] one can thenwork under the weaker condition that(2) H ( L ) = 0 for all line bundles L with c ( L ) = β. The natural C ∗ -action on the fibres of X = K S lifts to an action on the modulispace of stable pairs on X . One of the connected components of the fixedpoint locus is the moduli space of stable pairs on the surface S . (There can beother components containing stable pairs supported set-theoretically but notscheme-theoretically on S .) By C ∗ -localisation we get a reduced obstructiontheory on the moduli space of stable pairs on S . In [KT1, Appendix A], writtenwith D. Panov, we are able to identify this reduced obstruction theory with theone that arises naturally in a completely different way. Namely, we take thezero locus of a section of a bundle over a natural smooth ambient space, thena section of another bundle over this zero locus cuts out the moduli space.This allows us to calculate the (reduced, residue) stable pair invariants of S in terms of integrals over the smooth ambient space against the Euler classesof the two bundles.While it is a general principle that stable pairs are easier to calculate withthan stable maps, we know of almost no other moduli problem where suchdirect calculation is possible. Usually obtaining explicit results is very com-plicated, involving various difficult degeneration and localisation tricks.In forthcoming work [PT4] this calculation provides one of the foundationsof a computation of the full stable pairs theory of the twistor family of a K3surface. Via Pandharipande and Pixton’s recent proof of the MNOP conjecturefor many 3-folds [PaPi], this then gives a proof of the famous KKV formula forthe Gromov-Witten invariants of K3 surfaces in all genera, degrees and for allmultiple covers.We split the calculation up into two cases. In the first we simplify things byusing H -insertions [BL, KT1] to cut the moduli space down to curves livingin a single linear system | L | , where c ( L ) = β . When L is sufficiently ample The genus zero Gromov-Witten theory of complete intersections in convex varieties isperhaps the only other case.
URVE COUNTING ON SURFACES II: CALCULATIONS 3 the moduli space is smooth, the reduced obstruction bundle vanishes and theexpressions reduce to the intersection numbers encountered in [KST]. In [KST]some of these intersection numbers were related to counts of nodal curves on S and used to prove the G¨ottsche conjecture. Here we work with arbitrary L satisfying Condition (2), where the invariants with H -insertions include anextension of G¨ottsche’s invariants to the non-ample case [KT1, Section 5]. Weshow that, just as in the G¨ottsche case, the invariants only depend on the fourtopological numbers β , β.c ( S ), c ( S ) , c ( S ).Let h denote the arithmetic genus of curves in class β as given by the ad-junction formula(3) 2 h − β − c ( S ) .β. Theorem 1.1.
Fix β satisfying Condition (2) . The reduced residue invariant P red − h + n,β ( S, [ γ ] . . . [ γ b ( S ) ][ pt ] m ) ∈ Z ( t ) of [KT1, Section 3.2] is the product of t m + h , ( S ) − h , ( S ) and a universal function of the variables (4) n, m, β , β.c ( S ) , c ( S ) , c ( S ) . For fixed n, m and χ ( L ) = χ ( O S ) + ( β + β.c ( S )) it is ( − χ ( L ) − − m + n timesby a polynomial in the 4 topological numbers β , β.c ( S ) , c ( S ) and c ( S ) . In the second case we work with stable pairs over the full Hilbert schemeHilb β ( S ) of curves on S with class β , with no H -insertions. This differs fromthe first case when the dimension h , ( S ) of the Picard variety Pic β ( S ) of S is positive. This time one also has to perform integrals over Pic β ( S ). Theresulting invariants are again topological, depending not only on the topologicalnumbers (4) but also on numerical invariants of the ring structure of H ∗ ( S )described as follows.Via wedging and integration over S the classes β, c ( S ) ∈ H ( S, Z ) and1 ∈ H ( S, Z ) give rise to elements(5) [ β ] , [ c ( S )] ∈ Λ H ( S, Z ) ∗ and [1] ∈ Λ H ( S, Z ) ∗ . Wedging together combinations of these we can get elements of Λ b ( S ) H ( S, Z ) ∗ :(6) Λ i [ β ] ∧ Λ j [ c ( S )] ∧ Λ k [1] where 2 i + 2 j + 4 k = b ( S ) . This is the surface part of the C ∗ -equivariant stable pair invariant of X . Up to the powerof the equivariant cohomology parameter t , and saying “virtual” at the appropriate places,it works out to be the following. Integrate the Chern class of the cotangent bundle of themoduli space of pairs over the subspace of pairs whose underlying curves live in | L | and passthrough m fixed generic points of S . The γ i form an integral oriented basis of H ( S )/torsion;their insertion cuts H β down to a single linear system | L | . M. KOOL AND R. P. THOMAS
There is a canonical isomorphism Λ b ( S ) H ( S, Z ) ∗ ∼ = Z given by evaluating onthe wedge of any integral basis of H ( S, Z ) which is compatible in H ( S, R )with the orientation provided by the complex structure. Therefore we canregard the Λ i [ β ] ∧ Λ j [ c ( S )] ∧ Λ k [1] as integers. Theorem 1.2.
Fix β satisfying (2) . The reduced residue stable pair invariant P red − h + n,β ( S, [ pt ] m ) ∈ Z ( t ) of [KT1, Section 3.2] is equal to t m − h , ( S ) times by auniversal function of m, n, b ( S ) , β , β.c ( S ) , c ( S ) , c ( S ) , (cid:8) Λ i [ β ] ∧ Λ j [ c ( S )] ∧ Λ k [1] (cid:9) i +2 j +4 k = b ( S ) . For fixed n, m, b ( S ) and χ ( L ) = χ ( O S ) + ( β + β.c ( S )) it is the product of ( − χ ( L ) − − m + n + h , ( S ) and a universal polynomial in the topological numbers β , β.c ( S ) , c ( S ) , c ( S ) and Λ i [ β ] ∧ Λ j [ c ( S )] ∧ Λ k [1] . In Section 5 we give some conditions under which the moduli space of stablepairs on S is the whole fixed point locus of the moduli space of stable pairs on X = K S . The most obvious case is when β is an irreducible class. Another iswhen K − S is nef, β is (2 δ + 1)-very ample and the number of free points of thestable pairs is ≤ δ . A third example is provided by using only moduli spacescut down by many point insertions (see also [KT1, Section 5]).So in these cases we compute the corresponding reduced stable pair invariantsof X , not just S . By the MNOP conjecture [MNOP] (proved in the toriccase [MOOP, MPT], the “G¨ottsche case” [KT1], and now for “most” compactCalabi-Yau 3-folds [PaPi]) this determines various reduced C ∗ -equivariant GW and DT invariants of X , which are therefore also topological. Note that in thetoric case h , ( S ) = 0 so these are the usual GW/DT invariants. Acknowledgements.
We would like to thank Daniel Huybrechts, VivekShende and Rahul Pandharipande for useful discussions. Both authors weresupported by EPSRC programme grant number EP/G06170X/1.2.
The moduli space as a zero locus
We fix some notation. Let S be a nonsingular projective surface with coho-mology class β ∈ H ( S, Z ). In this paper L always denotes a line bundle with c ( L ) = β and h is the arithmetic genus (3) of curves in class β . Note that H ( S, Z ) is torsion-free. This invariant should be interpreted as in footnote 2, except the curve passing through m fixed points is no longer constrained to lie in | L | . By this we mean there exists a line bundle L in Pic β ( S ) which is (2 δ + 1)-very ample.Recall [BS] that this means that H ( L ) → H ( L | Z ) is surjective for every length 2 δ + 2subscheme Z of S . These are given by reduced GW invariants of S with K S -twisted λ -classes. URVE COUNTING ON SURFACES II: CALCULATIONS 5
For stable pairs on S in class β (or on X := K S in class ι ∗ β ) with holomorphicEuler characteristic χ = 1 − h + n we refer to [KT1, PT3]. In this paper we need only their description as pairs( C, Z ) where C ⊂ S is a pure curve in class β and Z ⊂ C is a length- n subscheme. This extends to give a set-theoretic isomorphism of moduli spaces(7) P χ ( S, β ) ∼ = Hilb n ( C /H β ) . Here
C → H β is the universal curve over the Hilbert scheme H β := Hilb β ( S ) ofpure curves in class β , and Hilb n is the relative Hilbert scheme of n points onthe fibres of C . In [PT3, Appendix B] it is shown that (7) is an isomorphism ofschemes. And in [KT1, Appendix A], written with D. Panov, it is more-or-lessshown that (7) is an isomorphism of schemes with perfect obstruction theory .Here we have to take the reduced obstruction theory on the left hand side, andon the right hand side the obstruction theory arising from a natural descriptionof the relative Hilbert scheme in terms of equations. We give a brief accountof this description in 2 steps now; for full details see [KT1]. • Pick a divisor A ⊂ S , sufficiently positive that L ( A ) is very ample with nohigher cohomology for all L ∈ Pic β ( S ). Then by adding A to divisors we getan embedding of Hilb β ( S ) into Hilb γ ( S ), where γ = [ A ] + β : H β (cid:31) (cid:127) + A / / H γ . Now H γ is smooth (it is a projective bundle over Pic γ ( S )) and the image A + H β is the set of divisors D ∈ H γ which contain A , i.e. the divisors D for which(8) s D | A = 0 ∈ H ( O ( D ) | A ) , where s D ∈ H ( O ( D )) is the equation defining D . Varying (8) over H γ we geta section of a bundle(9) s D | H γ × A of F := π γ ∗ (cid:0) O ( D ) | H γ × A (cid:1) over H γ , whose zero locus is precisely H β . Here D ⊂ S × H γ is the universaldivisor, and π S , π γ are the projections from S × H γ to its two factors. The abovepushdown has no higher cohomology due to Condition (2). This description of H β in terms of equations endows it with a natural perfect obstruction theory. What is shown is that the two tangent-obstruction complexes are the same, but it is notchecked that the maps to the cotangent complex agree. This is not important for producinga virtual cycle, which only depends on the K-theory class of the obstruction complexes.
M. KOOL AND R. P. THOMAS • Secondly, we embedHilb n ( C /H β ) (cid:31) (cid:127) / / S [ n ] × H β , where S [ n ] denotes the (smooth) Hilbert scheme of n points on S . A point( Z, C ) of S [ n ] × H β is in the image if and only if Z ⊂ C , if and only if(10) s C | Z = 0 ∈ H ( O Z ( C )) . Varying (10) over S [ n ] × H β we get a section of a bundle(11) s C | Z× H β of O ( C ) [ n ] := π ∗ (cid:0) O ( C ) | Z× H β ) (cid:1) , whose zero locus is precisely Hilb n ( C /H β ). Here C ⊂ S × H β is the universaldivisor, Z ⊂ S × S [ n ] is the universal length- n subscheme of S , and π is theprojection S × S [ n ] × H β → S [ n ] × H β . This description of Hilb n ( C /H β ) interms of equations (relative to the possibly singular space H β ) endows it witha natural perfect relative obstruction theory over H β .In [KT1, Appendix A] we show how to combine these two obstruction the-ories to endow Hilb n ( C /H β ) with a perfect absolute obstruction theory, whichwe then identify with the reduced obstruction theory of stable pairs. Notingthat over S × H β the line bundle O ( D − A ) restricts to O ( C ), we see that thebundle O ( C ) [ n ] of (11) extends naturally over S [ n ] × H γ as O ( D − A ) [ n ] . (Itssection s C | Z× H β does not extend.) As a consequence we get the following. Theorem 2.1. [KT1, Theorem A.7]
Assuming Condition (2), the pushforwardof the reduced virtual cycle [ P − h + n ( S, β )] red ∈ H v ( P − h + n ( S, β )) to the smooth ambient space S [ n ] × H γ is Poincar´e dual to c r ( F ) . c n (cid:0) O ( D − A ) [ n ] (cid:1) . Here v = h − n + R β c ( S ) + h , ( S ) is the reduced virtual dimension of P − h + n ( S, β ) and r = χ ( L ( A )) − χ ( L ) is the rank of the bundle F of (9). Also,given any family L → S × B of line bundles on S , we use the notation L [ n ] forthe rank n vector bundle on S [ n ] × B defined by pulling L back to Z × B andpushing forward to S [ n ] × B . We suppress many pullbacks for readability; here A denotes π ∗ S A . URVE COUNTING ON SURFACES II: CALCULATIONS 7
Virtual normal bundle.
Integrating insertions against the reduced vir-tual class of Theorem 2.1 gives numerical invariants of S as in [KT1]. Theseare part of the full residue invariants of [KT1], but to get all of them we mustinclude the term e ( N vir ) ∈ H ∗ C ∗ ( P − h + n ( S, β )) ⊗ Z [ t ] Z ( t ) ∼ = H ∗ ( P − h + n ( S, β )) ⊗ Z Z ( t ) , which arises in the virtual localisation formula of [GP]. Here N vir is the virtualnormal bundle of the inclusion P − h + n ( S, β ) ⊂ P − h + n ( X, ι ∗ β ), i.e. the dual ofthe moving part of the reduced 3-fold stable pairs obstruction theory.Using Serre duality and the fact that X = K S has trivial canonical bundle K X ∼ = O X ⊗ t ∗ with C ∗ -action of weight −
1, it turns out [KT1, Proposition 3.4]that N vir is the ordinary (i.e. not reduced) deformation-obstruction complex E • of P − h + n ( S, β ), shifted by [ −
1] and twisted by the C ∗ -representation ofweight 1: N vir = E • [ − ⊗ t = ( Rπ P ∗ R H om ( I • S , F )) ∨ [ − ⊗ t . See [KT1] for the meaning of this notation (though we will not need it here).In [KT1, Proposition A.3] the obstruction theory E • was shown to sit in anobvious exact triangle with the usual obstruction theory ( Rπ β ∗ O C ( C )) ∨ for H β and the relative obstruction theory (cid:8)(cid:0) O ( C ) [ n ] (cid:1) ∗ ds C | Z× Hβ / / Ω S [ n ] (cid:9) of P − h + n ( S, β ) (cid:14) H β arising from the description (11). We have again sup-pressed some pullback maps, and used π β to denote the projection S × H β → H β . Combining these facts shows that at the level of K-theory,[ N vir ] = (cid:2)(cid:0) O ( C ) [ n ] (cid:1) ∗ − Ω S [ n ] − ( Rπ β ∗ O C ( C )) ∨ (cid:3) ⊗ t . This can be expressed as either(12) [ N vir ] = (cid:2)(cid:0) O ( C ) [ n ] (cid:1) ∗ − Ω S [ n ] − ( Rπ β ∗ O ( C )) ∨ + R Γ( O S ) ∨ ⊗ O (cid:3) ⊗ t , or, using the exact sequence 0 → O C ( C ) → O D ( D ) → O A ( D ) → D − A restricts to C on H β ⊂ H γ ),(13) [ N vir ] = (cid:2)(cid:0) O ( D − A ) [ n ] (cid:1) ∗ − Ω S [ n ] − ( π γ ∗ O ( D )) ∨ + R Γ( O S ) ∨ ⊗ O + F ∗ (cid:3) ⊗ t . Recall that F is the bundle (9) and π γ is the projection S × H γ → H γ . Inthe form (13) it is clear that [ N vir ] is the restriction of a class on the ambient Writing N vir as a two-term complex E → E of equivariant bundles whose weightsare all nonzero (which is possible, and ensures that the c top ( E i ) are invertible), e ( N vir ) isdefined to be c top ( E ) /c top ( E ), where c top is the top C ∗ -equivariant Chern class. M. KOOL AND R. P. THOMAS space S [ n ] × H γ , which will make calculation of the general invariants possible:see Section 4. To compute invariants of a single linear system we will find itconvenient to use the form (12) – see Section 3 – though of course we couldalso have used (13).3. Calculation with H -insertions In this Section we compute the reduced residue stable pair invariants P redχ,β ( S, [ γ ] . . . [ γ b ( S ) ][ pt ] m ) = Z [ P χ ( S,β )] red e ( N vir ) b ( S ) Y i =1 τ ( γ i ) τ ([ pt ]) m lying in Z ( t ). Recall that χ = 1 − h + n ; otherwise we use the notation of[KT1, Section 3.2]. The γ i form an integral oriented basis of H ( S )/torsion;their insertion cuts Hilb β ( S ) down to a single linear system | L | . The m pointinsertions further cut this down to a codimension- m linear subsystem. In factby [KT1, Section 4], particularly Equations (52, 54), the above equals(14) Z j ! [ P χ ( S,β )] red h m e ( N vir ) , where j ! is the refined Gysin map [Ful, Section 6.2] for the Cartesian diagram P χ ( S, | L | ) (cid:31) (cid:127) / / (cid:15) (cid:15) P χ ( S, β ) (cid:15) (cid:15) { L } (cid:31) (cid:127) j / / Pic β ( S ) , and h is the pullback of the hyperplane cohomology class from | L | to P χ ( S, | L | ) ∼ = Hilb n ( C / | L | ). Factor this through the diagram P χ ( S, | L | ) (cid:31) (cid:127) / / (cid:127) _ ι L (cid:15) (cid:15) P χ ( S, β ) (cid:127) _ ι (cid:15) (cid:15) S [ n ] × | L ( A ) | (cid:31) (cid:127) / / (cid:15) (cid:15) S [ n ] × Hilb γ ( S ) (cid:15) (cid:15) { L ( A ) } (cid:31) (cid:127) / / Pic γ ( S ) { L } (cid:31) (cid:127) j / / Pic β ( S ) . The central vertical arrows are flat, so j ! = ! . Thus by [Ful, Theorem 6.2] wehave ι L ∗ j ! [ P χ ( S, β )] red = j ! ι ∗ [ P χ ( S, β )] red = ∗ ι ∗ [ P χ ( S, β )] red . URVE COUNTING ON SURFACES II: CALCULATIONS 9
Therefore by Theorem 2.1, (14) becomes Z S [ n ] ×| L ( A ) | c r ( F ) . c n ( O ( D − A ) [ n ] ) h m e ( N vir ) , where as usual we have suppressed the pullback maps ∗ on the bundles F and O ( D − A ) [ n ] . Over S × | L ( A ) | , the line bundle O ( D ) is isomorphic to L ( A ) ⊠ O (1) as both have a section cutting out D . Hence F | S [ n ] ×| L ( A ) | ∼ = H ( L ( A ) | A ) ⊗ O (1) ∼ = O (1) ⊕ r , where r = χ ( L ( A )) − χ ( L ), and our integral becomes(15) Z S [ n ] ×| L ( A ) | h r c n ( L [ n ] (1)) h m e ( N vir ) = Z S [ n ] × P χ ( L ) − − m c n ( L [ n ] (1)) e ( N vir ) . We use the following notation. For any bundle E and variable x , set c x ( E ) :=1 + c ( E ) x + c ( E ) x + . . . . Thus if E has rank r then(16) e ( E ⊗ t ) = r X i =0 t i c r − i ( E ) = t r r X i =0 ( − /t ) r − i ( − r − i c r − i ( E ) = t r c − /t ( E ∗ ) , where t := c ( t ) is the equivariant parameter: the generator of H ∗ ( B C ∗ ). Usethis to substitute the expression (12) for [ N vir ] into (15). Since Rπ β ∗ O ( C ) = R Γ( L ) ⊗ O (1), we get t n + χ ( L ) − n − χ ( O S ) Z S [ n ] × P χ ( L ) − − m c n ( L [ n ] (1)) c − /t (cid:0) T S [ n ] (cid:1) c − /t (cid:0) O (1) ⊕ χ ( L ) (cid:1) c − /t (cid:0) L [ n ] (1) (cid:1) . Since only the degree n + χ ( L ) − − m part of the quotient contributes to theintegral we get t n + χ ( L ) − χ ( O S ) (cid:18) − t (cid:19) n + χ ( L ) − − m Z S [ n ] × P χ ( L ) − − m c n ( L [ n ] (1)) c • (cid:0) T S [ n ] (cid:1) c • (cid:0) O (1) ⊕ χ ( L ) (cid:1) c • (cid:0) L [ n ] (1) (cid:1) , where c • denotes the total Chern class. Integrating over P χ ( L ) − − m leaves( − χ ( L ) − − m + n t m +1 − χ ( O S ) Z S [ n ] (cid:20) c • ( T S [ n ] )(1 + h ) χ ( L ) P ni =0 h i c n − i ( L [ n ] ) P ni =0 (1 + h ) i c n − i ( L [ n ] ) (cid:21) h χ ( L ) − − m where the suffix means we take the coefficient of h χ ( L ) − − m in the bracketedexpression.The right hand side is a tautological integral over S [ n ] , involving only Chernclasses of L [ n ] and the tangent bundle. Applying the recursion of [EGL] n times,it becomes an integral over S n of a polynomial in c ( L ) , c ( S ) , c ( S ) (pulledback from different S factors) and ∆ ∗ , ∆ ∗ c ( S ) , ∆ ∗ c ( S ) , ∆ ∗ c ( S ) (pulledback from different S × S factors), where ∆ : S ֒ → S × S is the diagonal. The result is a degree n universal polynomial in c ( L ) , c ( L ) .c ( S ), c ( S ) and c ( S ) (see also [KST, Section 4]). This proves Theorem 1.1.4. Calculation without H -insertions Now we turn to the calculation of the reduced residue stable pair invariants(17) P redχ,β ( S, [ pt ] m ) = Z [ P χ ( S,β )] red e ( N vir ) τ ([ pt ]) m ∈ Z ( t ) . When b ( S ) > S ).Picking a Poincar´e bundle P γ over S × Pic γ ( S ) expresses Hilb γ ( S ) as aprojective bundle over Pic γ ( S ):(18) Hilb γ ( S ) = P ( p ∗ P γ ) AJ −→ Pic γ ( S ) , where p is the projection S × Pic γ ( S ) → Pic γ ( S ) and AJ is the Abel-Jacobimap. Fix a point x ∈ S . Then the locus of curves in Hilb γ ( S ) passing through x , D x := P (cid:0) p ∗ ( P γ ⊗ I { x }× Pic γ ( S ) ) (cid:1) ⊂ Hilb γ ( S ) , is a divisor since it defines a hyperplane in each projective space fibre (by thevery ampleness of the class γ ).Of course P γ is only unique up to tensoring by line bundles pulled back fromPic γ ( S ). By choosing that line bundle to be P − γ | { x }× Pic γ ( S ) if necessary, wemay assume without loss of generality that P γ is trivial at x :(19) P γ | { x }× Pic γ ( S ) ∼ = O Pic γ ( S ) . Lemma 4.1.
Under the normalisation (19) , the hyperplane line bundle O (1) of the projective bundle (18) is O ( D x ) .Proof. Using the normalisation (19), the tautological bundle O ( − ֒ → p ∗ P γ of P ( p ∗ P γ ) has a canonical map to O given by evaluation of sections at x ∈ S .Its zero locus is precisely D x . (cid:3) Corollary 4.2.
The insertion τ ([ pt ]) is the cohomology class h := c ( O (1)) pulled back to P χ ( S, β ) via P χ ( S, β ) ⊂ S [ n ] × H γ → H γ . URVE COUNTING ON SURFACES II: CALCULATIONS 11
Proof.
Recall [KT1, Section 3.2] that we use [ · ] to denote Poincar´e duals, andthat τ ([ pt ]) ∈ H ∗ ( P χ ( S, β )) is defined by the top half of the diagram SP χ ( S, β ) × S π S O O (cid:15) (cid:15) π P / / P χ ( S, β ) (cid:15) (cid:15) H β × S π β / / H β . Namely τ ([ pt ]) = π P ∗ ( π ∗ S [ pt ] · c ( F )) on P χ ( S, β ), where F is the universal sheafover S × P χ ( S, β ). But c ( F ) is the pullback of c ( O ( C )) from H β × S . So bygoing round the above Cartesian square, we find that τ ([ pt ]) is the pullbackfrom H β of π β ∗ ( c ( O ( C )) ⊠ [ pt ]).Now the terms in the above square embed (via the obvious commuting maps)in the terms in the square S [ n ] × H γ × S (cid:15) (cid:15) / / S [ n ] × H γ (cid:15) (cid:15) H γ × S π γ / / H γ . Our class π β ∗ ( c ( O ( C )) ⊠ [ pt ]) is the restriction to H β of π γ ∗ ( c ( O ( D− A )) ⊠ [ pt ]).Since the class A is pulled back from S it contributes nothing for degree reasons.And since H γ × S is smooth we can use Poincar´e duality to write the rest as thepushdown via π γ of the homology class of D intersected with that of H γ × { x } .This intersection is D x and is transverse. By the Lemma we therefore get h ∈ H ( H γ ). Pulling up to S [ n ] × H γ and restricting to P χ ( S, β ) in the previoussquare gives the result. (cid:3)
Substituting this result and the expression (13) for e ( N vir ) into (17) givesthe expression t n + χ ( L ( A )) − n − χ ( O S ) − r Z S [ n ] × H γ c r ( F ) c n (cid:0) O ( D − A ) [ n ] (cid:1) h m c − /t ( T S [ n ] ) c − /t (cid:0) π γ ∗ O ( D )) (cid:1) c − /t (cid:0) O ( D − A ) [ n ] (cid:1) c − /t ( F ) , by using the identity (16). Recall that r := rank( F ) = χ ( L ( A )) − χ ( L ). Thepiece of the quotient in the correct degree to contribute is (cid:18) − t (cid:19) n + χ ( L ) − h , ( S ) − m c • ( T S [ n ] ) c • (cid:0) π γ ∗ O ( D )) (cid:1) c • (cid:0) O ( D − A ) [ n ] (cid:1) c • ( F ) . Thus we are left with the product of ( − χ ( L ) − n − m + h , ( S ) t m − h , ( S ) and(20) Z S [ n ] × H γ c r ( F ) c n (cid:0) O ( D − A ) [ n ] (cid:1) h m c • ( T S [ n ] ) c • (cid:0) π γ ∗ O ( D )) (cid:1) c • (cid:0) O ( D − A ) [ n ] (cid:1) c • ( F ) . This takes care of the sign and power of t in Theorem 1.2. We will nowconcentrate on the integral (20), first pushing it down the projective bundle(18) to S [ n ] × Pic( S ), then to Pic( S ), then finally to a point. Integrating over the fibres of the Abel-Jacobi map.
Since the line bun-dle H om ( O ( − , P γ ) has a canonical section cutting out D , we have the iden-tity P γ (1) ∼ = O ( D ) . Substituting into (20) yields Z S [ n ] × H γ c r ( F ) c n (cid:0) P γ ( − A ) [ n ] (1) (cid:1) h m c • ( T S [ n ] ) c • (cid:0)(cid:2) Rp ∗ P γ ( − A ) (cid:3) (1) (cid:1) c • (cid:0) P γ ( − A ) [ n ] (1) (cid:1) . Expand the integrand in powers of h = c ( O (1)). Notice that everything isnow pulled back from S [ n ] × Pic γ ( S ) except c r ( F ) and the powers of h . Theseare dealt with by the following Lemma for pushing down the projective bundle(18). We use the following diagram(21) S × H γ S × AJ / / π γ (cid:15) (cid:15) S × Pic γ ( S ) p (cid:15) (cid:15) H γ AJ / / Pic γ ( S ) , Recall that the Segre classes s i ∈ H i are defined by s • = 1 /c • . Lemma 4.3.
The pushdown AJ ∗ ( c r ( F ) h j ) to Pic γ ( S ) is equal to the Segre class s j − χ ( L )+1 (cid:0) Rp ∗ P γ ( − A ) (cid:1) .Proof. Using P γ (1) ∼ = O ( D ) and diagram (21) F = π γ ∗ (cid:0) O ( D ) | H γ × A (cid:1) ∼ = AJ ∗ p ∗ (cid:0) P γ | Pic γ ( S ) × A (cid:1) (1) , and so c r ( F ) = AJ ∗ r X i =0 c r − i (cid:0) p ∗ (cid:0) P γ | Pic γ ( S ) × A (cid:1)(cid:1) h i . We can push down using the standard identity [Ful, Section 3.1](22) AJ ∗ ( h i ) = s i − χ ( L ( A ))+1 ( p ∗ P γ ) . URVE COUNTING ON SURFACES II: CALCULATIONS 13
We get AJ ∗ ( c r ( F ) h j ) = r X i =0 c r − i (cid:0) p ∗ (cid:0) P γ | Pic γ ( S ) × A (cid:1)(cid:1) s i + j − χ ( L ( A ))+1 ( p ∗ P γ )= (cid:2) c • (cid:0) p ∗ (cid:0) P γ | Pic γ ( S ) × A (cid:1)(cid:1) s • ( p ∗ P γ ) (cid:3) r + j − χ ( L ( A ))+1 = s j − χ ( L )+1 (cid:0) p ∗ P γ − p ∗ (cid:0) P γ | Pic γ ( S ) × A (cid:1)(cid:1) = s j − χ ( L )+1 (cid:0) Rp ∗ P γ ( − A ) (cid:1) . (cid:3) Remark . Note that by integrating out c r ( F ) we are passing from H γ back to(the reduced virtual cycle of ) H β . In the case where β is sufficiently ample thatno virtual technology is necessary, we could have worked directly on P ( p ∗ P β ) and pushed down h j with no c r ( F ) insertion. As in (22) this would have given s j − χ ( L )+1 ( p ∗ P β ) , the same result as in the Lemma. Thus our integral has become one over S [ n ] × Pic γ ( S ) of a polynomial Q inthe Chern classes of T S [ n ] , Rp ∗ P γ ( − A ) and P γ ( − A ) [ n ] . Since P β := P γ ( − A )is a Poincar´e bundle for Pic β ( S ), we use ⊗O ( − A ) to identify Pic γ ( S ) withPic β ( S ) to get an integral(23) Z S [ n ] × Pic β ( S ) Q (cid:16) c • ( T S [ n ] ) , c • ( Rp ∗ P β ) , c • (cid:0) P [ n ] β (cid:1)(cid:17) . By this notation we mean that Q is a polynomial in all of the components c i of c • (rather than just in the total Chern classes themselves). Notice thisexpression is now manifestly independent of A . Integrating over the Hilbert scheme of points.
We need a family versionof the recursion of [EGL].We fix an arbitrary base B , a line bundle L on S × B a cohomology class ofthe form P ( c • ( L [ n ] ) , c • ( T S [ n ] )) on S [ n ] × B, for some polynomial P . We wish to push it down to B . The recursion [EGL]is easily checked to apply (though it was actually written for the case B = pt ).The pushdown is turned first into one down S [ n − × S × B , then S [ n − × S × B ,and so on. The end result is an integral down the fibres of S n × B → B of apolynomial in • c ( L ) , c ( S ) , c ( S ) pulled back from different S × B and S factors, • ∆ ∗ , ∆ ∗ c ( S ) , ∆ ∗ c ( S ) , ∆ ∗ c ( S ) pulled back from different S × S fac-tors, where ∆ is the diagonal S ֒ → S × S . In turn this integral is easily computed as a polynomial in integrals down p : S × B → B of products of c ( S ) , c ( S ) , c ( L ) , p ∗ p ∗ (cid:0) c ( L ) i c j ( S ) k (cid:1) . Appliedto B = Pic β ( S ) and L = P β , (23) becomes a polynomial in terms Z S × Pic β ( S ) M (cid:16) c ( S ) , c ( S ) , c • ( Rp ∗ P β ) , c ( P β ) , p ∗ p ∗ (cid:0) c ( P β ) i c j ( S ) k (cid:1)(cid:17) , where M is any monomial and j, k = 0 , , Integrating over the Picard variety.
Next we apply Grothendieck-Rie-mann-Roch, ch ( Rp ∗ P β ) = p ∗ (cid:2) exp( c ( P β )) Td( S ) (cid:3) , and the decomposition c ( P β ) = ( β, id , ∈ H ( S ) ⊕ (cid:0) H ( S ) ⊗ H (Pic β ( S )) (cid:1) ⊕ H (Pic β ( S )) . Here we use the canonical identification H (Pic β ( S )) ∼ = H ( S ) ∗ (so that id ∈ End H ( S )) and the normalisation (19). The upshot is a polynomial in theintegrals Z S × Pic β ( S ) M (cid:16) c ( S ) , c ( S ) , β, id , p ∗ p ∗ (cid:0) β i . id j . c k ( S ) l (cid:1)(cid:17) , for arbitrary i, j and k, l = 0 , ,
2. For degree reasons, pushing down to S weget a polynomial in the terms β , β.c ( S ) , c ( S ) , c ( S ) and(24) Z Pic β ( S ) M (cid:0) p ∗ (id ) , p ∗ ( β. id ) , p ∗ ( c ( S ) . id ) (cid:1) . Using the identification Λ H ( S, R ) ∗ ∼ = H (Pic β ( S ) , R ) we obtain p ∗ ( β. id ) = − β ] ,p ∗ ( c ( S ) . id ) = − c ( S )] ,p ∗ (id ) = 24[1] . where [ β ], [ c ( S )], [1] are the classes defined in (5) in the Introduction. (To getthe precise coefficients it is perhaps easiest to express everything in terms of abasis for H ( S ) and its dual basis for H ( S ) ∗ ∼ = H (Pic β ( S )) and then do thecalculation.)Finally the canonical identification Λ b ( S ) H ( S, Z ) ∗ ∼ = Z given by wedgingtogether an oriented integral basis is the same as the identification given byintegrating over Pic β ( S ). Therefore the integrals in (24) are the numbersΛ i [ β ] ∧ Λ j [ c ( S )] ∧ Λ k [1] , i + 2 j + 4 k = b ( S ) , of (6). This proves Theorem 1.2. URVE COUNTING ON SURFACES II: CALCULATIONS 15
Remarks . The invariants Λ i [ β ] ∧ Λ j [ c ( S )] ∧ Λ k [1] are in general distinctfrom β , β.c ( S ), c ( S ) , c ( S ) as can be seen by the following example. Let S = Σ g × P , where Σ g is a smooth projective curve of genus g . Under theidentification H ( S, Z ) ∼ = H (Σ g ) ⊕ H ( P ) ∼ = Z ⊕ Z , we write β = ( β , β ). Then β = 2 β β , β.c ( S ) = 2 β + (2 − g ) β , c ( S ) = 8 − g, c ( S ) = 4 − g. On the other hand, using the usual basis of a - and b -cycles for H ( S ) ∗ ∼ = H (Σ g )one computes that [1] = 0 andΛ i [ β ] ∧ Λ g − i [ c ( S )] = 2 g − i g ! β i . For S an abelian surface, however, the invariants Λ i [ β ] ∧ Λ j [ c ( S )] ∧ Λ k [1]can all be expressed in terms of β , β.c ( S ), c ( S ) , c ( S ) (i.e. just β since theothers vanish). Indeed using the standard basis arising from a homeomorphismto ( S ) it is easy to see that [1] = 1, [ c ( S )] = 0 andΛ [ β ] = Z S β . Relation to 3-fold invariants
In this section we discuss some cases in which the reduced residue stable pairinvariants of S computed in Theorem 1.1 and Theorem 1.2 are equal to thereduced residue stable pair invariants of the 3-fold X = K S . Let ι : S ֒ → X denote the inclusion.First we give cases where the fixed point locus P χ ( X, ι ∗ β ) C ∗ has no compo-nents other than P χ ( S, β ). Proposition 5.1.
There is an isomorphism P χ ( X, ι ∗ β ) C ∗ ∼ = P χ ( S, β ) if either • β is irreducible, or • K − S is nef, β is (2 δ + 1) -very ample and χ ≤ − h + δ .(As usual h is defined by h − β − c ( S ) .β . The inequality on n meansthe stable pairs have ≤ δ free points.)Proof. Let (
F, s ) ∈ P χ ( X, ι ∗ β ) C ∗ be a C ∗ -fixed stable pair in class β withscheme-theoretic support C F . Then its set-theoretic support C redF lies in S .But C F has no embedded points (by the purity of F [PT1, Lemma 1.6]) so if β is irreducible then C F is in fact reduced. Thus C F and ( F, s ) are pushed for-ward from S . So ( F, s ) ∈ P χ ( S, β ) and P χ ( S, β ) ֒ → P χ ( X, ι ∗ β ) C ∗ is a bijection. That they have the same scheme structure is proved in [KT1, Proposition 3.4].Next assume instead that β is (2 δ + 1)-very ample, that K − S is nef, and that F is not supported scheme-theoretically on S . We will show that χ ( O C F ) > − h + δ , which implies the Proposition since χ ( F ) ≥ χ ( O C F ).Let the irreducible components of C F be C F,i , with underlying reduced va-rieties C i ⊂ S . Since the C F,i are C ∗ -fixed without embedded points, there isa sequence of integers n i ≥ . . . ≥ n ir i > O C F,i = r i M k =0 O n ik C i ⊗ K − kS as a graded ring. Here n ik C i ⊂ S is the obvious divisor, and we are writing O X as L ∞ k =0 K − kS by pushing down to S ; the C ∗ -action on K S then inducesthe obvious grading by k . Since K − S is nef, we obtain χ ( O C F,i ) = r i X k =0 (cid:0) χ ( O n ik C i ) − kn ik C i .K S (cid:1) ≥ r i X k =0 χ ( O n ik C i )= − r i X k =0 (cid:0) n ik C i + n ik C i .K S (cid:1) . In turn, χ ( O C F ) = X i χ ( O C F,i ) − X i By the Hodge index theorem, a ≤ ( L.a ) /L for any positive L ∈ H , ( S ) andarbitrary a ∈ H , ( S ). (Proof: a − ( L.a ) L/L is orthogonal to L so has square ≤ L = β and a = β k gives2 (cid:0) χ ( O C F ) + h − (cid:1) ≥ β − X k ( β.β k ) β = X k ( β.β k ) (cid:18) − ( β.β k ) β (cid:19) = X k ( β.β k )( β. ( β − β k )) β = X k ( β.β k )( β. ( β − β k ))( β.β k ) + ( β. ( β − β k )) ≥ X k 12 min (cid:0) β.β k , β. ( β − β k ) (cid:1) . Both β k and β − β k are effective, since C F is not supported inside S . And thesum contains at least 2 terms. So it is ≥ β.D , for some effective divisor D .So it is sufficient to prove that β.D > δ for effective classes D ; in turn it issufficient to prove this for irreducible D .Choose 2 δ + 2 smooth points on D . By the definition of (2 δ + 1)-veryampleness, there is divisor in S in the class of β which passes through the first2 δ + 1 points, but not the last one. Therefore the divisor does not contain D ,and β.D ≥ δ + 1, as required. (cid:3) The previous proposition is false for arbitrary surfaces. For instance if K S = O S ( C ), then consider β = nC and let C be the n -fold thickening of C alongthe fibres of K S . This is C ∗ -fixed with χ = 1 − h , but not scheme-theoreticallysupported on S . However one can often make it true again by restricting tosmall linear subsystems in the space of curves. This follows from the followingresult proved and used in [KST]. Proposition 5.2 ([KST, Proposition 2.1]) . If L is a δ -very ample line bundleon S then the general δ -dimensional linear system P δ ⊂ | L | contains a finitenumber of δ -nodal curves appearing with multiplicity 1. All other curves in P δ are reduced with geometric genus g > h − δ where h is the arithmetic genus ofcurves in | L | . (cid:3) (One can also assume that the curves in P δ are irreducible if L is (2 δ +1)-veryample, by [KT1, Proposition 5.1].) So when we deal with invariants of X for stable pairs with divisor class lying in P δ ⊂ | L | ⊂ Hilb β ( S ), any C ∗ -fixed pure curve has set-theoretic supporton S which is a reduced irreducible curve of class β . This must therefore be itsscheme theoretic support too, so we get a bijection between the cut down mod-uli spaces P χ ( S, P δ ) ∼ = P χ ( X, P δ ) C ∗ which is a scheme-theoretic isomorphism by[KT1, Proposition 3.4].Therefore in the situation of Proposition 5.1, or Proposition 5.2 with full H -insertions and χ ( L ) − − δ point insertions , the reduced residue stablepair invariants of X and S coincide. Thus these invariants of X are purelytopological and determined by the universal polynomials of Theorem 1.1 andTheorem 1.2. By the MNOP conjecture, this determines the correspondingreduced GW and DT invariants of X , which should therefore also be topolog-ical. Note that in the toric case h , ( S ) = 0, so these are the usual GW/DTinvariants. References [Beh] K. Behrend, Gromov-Witten invariants in algebraic geometry , Invent. Math. ,601–617 (1997). alg-geom/9601011.[BF] K. Behrend and B. Fantechi, The intrinsic normal cone , Invent. Math. , 45–88(1997). alg-geom/9601010.[BS] M. Beltrametti and A. J. Sommese, Zero cycles and k th order embeddings of smoothprojective surfaces. With an appendix by Lothar G¨ottsche , Problems in the theoryof surfaces and their classification (Cortona, 1988) Sympos. Math. , 33–48 Aca-demic Press (1991).[BL] J. Bryan and C. Leung, Generating functions for the number of curves on abeliansurfaces , Duke Math. J. , 311–328 (1999). math.AG/9802125.[EGL] G. Ellingsrud, L. G¨ottsche, M. Lehn, On the cobordism class of the Hilbert schemeof a surface , Jour. Alg. Geom. , 81–100 (2001). math.AG/9904095.[Ful] W. Fulton, Intersection theory , Springer-Verlag (1998).[GP] T. Graber and R. Pandharipande, Localization of virtual classes , In-vent. Math. , 487–518 (1999). alg-geom/9708001.[KST] M. Kool, V. Shende and R. P. Thomas, A short proof of the G¨ottsche conjecture ,Geom. Topol. , 397–406 (2011). arXiv:1010.3211. Given a stable pair ( F, s ) on X in class ι ∗ β , its pushdown q ∗ F to S has a divisor classdiv( q ∗ F ) ∈ Hilb β ( S ): see [KT1, Section 4]. This is basically the support with multiplicities.We use insertions to force this class to lie in P δ . The H -insertions cut the divisor class down to | L | and then the point insertions furthercut down to P δ . We call these insertions, when the conditions of Proposition 5.2 hold, the“G¨ottsche case”. There are no virtual cycles involved in this case since the resulting modulispace is smooth of the correct dimension. And in [KT1, Section 5] we show the resultinginvariants contain the Severi degrees counting nodal curves studied by G¨ottsche. URVE COUNTING ON SURFACES II: CALCULATIONS 19 [KT1] M. Kool, R. P. Thomas, Reduced classes and curve counting on surfaces I: theory ,arXiv:1112.3069.[LT] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraicvarieties , J. Amer. Math. 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Thomas, Higher genus curves on K3 surfaces and theKatz-Klemm-Vafa formula , preprint., preprint.