aa r X i v : . [ m a t h . AG ] N ov NON-COMMUTATIVE WIDTH ANDGOPAKUMAR-VAFA INVARIANTS
YUKINOBU TODA
Abstract.
We show that the non-commutative widths for flop-ping curves on smooth 3-folds introduced by Donovan-Wemyss aredescribed by Katz’s genus zero Gopakumar-Vafa invariants. Introduction
Result.
Let X be a smooth quasi-projective complex 3-fold and f : X → Y a birational flopping contraction which contracts a single rational curve P ∼ = C ⊂ X to a point p ∈ Y . In the paper [DW], Donovan-Wemyssintroduced a new invariant associated to f , the contraction algebra A con , given by the universal non-commutative deformation algebra ofthe curve C in X . The algebra A con is finite dimensional, and it is com-mutative if and only if C is not a (1 , − A con iscommutative, the dimension of A con coincides with Reid’s width [Rei] of C . Based on this observation, Donovan-Wemyss defined the followinggeneralizations of Reid’s widthwid( C ) := dim C A con , cwid( C ) := dim C A abcon which they called non-commutative width and commutative width re-spectively.On the other hand, Katz [Kat08] defined genus zero Gopakumar-Vafa(GV) invariants as virtual numbers of one dimensional stable sheaveson X . For j ≥
1, the genus zero GV invariant n j ∈ Z ≥ of curve class j [ C ] on X is shown in [Kat08] to coincide with the multiplicity of theHilbert scheme of X at some subscheme C ( j ) ⊂ X with curve class j [ C ]. The purpose of this short note is to describe Donovan-Wemyss’swidths in terms of Katz’s genus zero GV invariants. The main resultis as follows: Theorem 1.1.
We have the following formulas wid( C ) = l X j =1 j · n j , cwid( C ) = n . (1) Here l is the scheme theoretic length of f − ( p ) at C . Here we remark that the identity of cwid( C ) is almost obvious fromthe definitions, and the identity of wid( C ) is more interesting. Theresult of Theorem 1.1 indicates that one can study non-commutativewidths without using non-commutative algebras . Conversely, one maycompute genus zero GV invariants by computing contraction algebras.The proof of Theorem 1.1 is an easy application of the main resultof [DW], combined with some deformation argument. By [DW], thealgebra A con defines the non-commutative twist functor, describingBridgeland-Chen’s flop-flop autoequivalence of D b Coh( X ). On theother hand, after taking the completion at p , the morphism f de-forms to flopping contractions of disjoint ( − , − − , − j [ C ] coincides with n j .Now the flop-flop autoequivalence deforms along the deformation of f ,hence the non-commutative twist functor also deforms: the resultingdeformation is a composition of Seidel-Thomas’s spherical twists along( − , − C ).1.2. Examples and a Remark.
Here we describe some examples ofTheorem 1.1.
Example 1.2.
In Theorem 1.1, we have l = 1 if and only if C is eithera ( − , − or a (0 , − -curve. In this case, we have wid( C ) = cwid( C ) ,and it coincides with Reid’s width (cf. [DW, Example 3.12] ). On theother hand, the genus zero GV invariant n also coincides with Reid’swidth as indicated in [BKL01, Section 1] . Example 1.3.
Suppose that Y = Spec R k , where R k is defined by R k = C [ u, v, x, y ] / ( u + v y = x ( x + y k +1 )) . There is a flopping contraction f : X → Y with l = 2 . The contractionalgebra A con is computed in [DW, Example 3.14] A con ∼ = C h x, y i / ( xy = − yx, x = y k +1 ) A abcon ∼ = C [ x, y ] / ( xy = 0 , x = y k +1 ) . It follows that wid( C ) = 3(2 k + 1) , cwid( C ) = 2 k + 3 . The result of Theorem 1.1 indicates that n = 2 k + 3 and n = k . We also have the following remark: Wemyss pointed out to the author that the non-commutative widths are com-mutative things, as they are computed using some Ext-groups on commutativealgebras. See [DW, Remark 5.2].
ON-COMMUTATIVE WIDTH AND GOPAKUMAR-VAFA INVARIANTS 3
Remark 1.4.
We have n j ≥ for ≤ j ≤ l . So Theorem 1.1 impliesthat wid( C ) ≥ l X j =1 j . The above lower bound is better than the lower bound in [DW, Re-mark 3.17] . Acknowledgment.
I would like to thank Michael Wemyss andWill Donovan for valuable comments on the manuscript. I wouldalso like to thank Tom Bridgeland for checking a misprint of his pa-per [Bri02], and allowing me to correct it in Appendix B. This work issupported by World Premier International Research Center Initiative(WPI initiative), MEXT, Japan. This work is also supported by Grant-in Aid for Scientific Research grant (No. 26287002) from the Ministryof Education, Culture, Sports, Science and Technology, Japan.2.
Preliminary
Let X be a smooth quasi-projectivecomplex 3-fold. By definition, a flopping contraction is a birationalmorphism f : X → Y (2)which is isomorphic in codimension one, Y has only Gorenstein sin-gularities and the relative Picard number of f equals to one. In whatfollows, we always assume that the exceptional locus C of f is isomor-phic to P , and set p := f ( C ) ∈ Y. We say that C ⊂ X is ( a, b ) curve if N C/X is isomorphic to O C ( a ) ⊕O C ( b ). It is well-known that ( a, b ) is either one of the following:( a, b ) = ( − , − , (0 , − , (1 , − . We denote by l the length of O f − ( p ) at the generic point of C , where f − ( p ) is the scheme theoretic fiber of f at p . Then we have l ∈ { , , , , , } and l = 1 if and only if C is not a (1 , − l = 1, then we have b O Y,p ∼ = C [[ x, y, z, w ]] / ( x + y + z + w k )(3)for some k ∈ Z ≥ . The number k is called width of C in [Rei]. YUKINOBU TODA
Contraction algebras.
In the setting of Subsection 2.1, we set R = b O Y,p , and take the following completion of (2) b f : b X := X × Y Spec R → b Y := Spec R. (4)Then there is a line bundle L on b X such that deg( L| C ) = 1. We definethe vector bundle N on b X to be the extension0 → L − → N → O ⊕ r b X → H ( b X, L − ). We set U := O b X ⊕N , N := b f ∗ N and A := End b X ( U ) ∼ = End R ( R ⊕ N ) . By Van den Bergh [dB04, Section 3.2.8], we have a derived equivalence R Hom b X ( U , − ) : D b Coh( b X ) ∼ → D b mod A (5)whose inverse is given by − L ⊗ A U . Here mod A is the category offinitely generated right A -modules. Definition 2.1. ([DW, Definition 2.11])
The contraction algebra A con is defined to be A/I con , where I con is the two sided ideal of A consistingof morphisms R ⊕ N → R ⊕ N factoring through a member of add( R ) .Here add( R ) is the set of summands of finite sums of R . By [DW, Proposition 2.12], the algebra A con is finite dimensional. Remark 2.2.
The algebra A con is commutative if and only if C isnot a (1 , − -curve (cf. [DW, Theorem 3.15] ). In this case, A con isisomorphic to C [ t ] / ( t k ) , where k is the width of C which appears in(3). See [DW, Example 3.12] . The contraction algebra A con coincides with the universal algebrawhich represents the non-commutative deformation functor of O C ( − O C ( − : Art → Sets . (6)Here Art is the category of finite dimensional C -algebras Γ with someadditional conditions, and the functor (6) assigns each Γ to the setof isomorphism classes of flat deformation of O C ( −
1) to Coh( O X ⊗ C Γ). We refer [DW, Section 2] for details of the functor (6). SinceA con represents the functor (6), there is the universal non-commutativedeformation of O C ( − E ∈
Coh( O X ⊗ C A con ) . (7)Let A abcon be the abelization of A con . The algebra A abcon is a commutativeArtinian local C -algebra, which represents the commutative deforma-tion functor cDef O C ( − : cArt → Sets . (8) ON-COMMUTATIVE WIDTH AND GOPAKUMAR-VAFA INVARIANTS 5
Here cArt is the category of commutative Artinian local C -algebras,and the functor (8) is the restriction of the functor (6) to cArt . Werefer [DW, Section 3] for details of the above representabilities.2.3. Flop equivalences.
The contraction algebra A con plays an im-portant role in describing Bridgeland-Chen’s flop-flop autoequivalence.Let us consider the flop diagram of (2) X f φ X † f † Y. (9)By [Bri02] and [Che02], we have the derived equivalenceΦ O X × Y X † X → X † : D b Coh( X ) ∼ → D b Coh( X † ) . (10)Here we use the notation in Appendix A for the Fourier-Mukai functors.Composing (10) twice, we obtain the autoequivalenceΦ O X × Y X † X † → X ◦ Φ O X × Y X † X → X † : D b Coh( X ) ∼ → D b Coh( X ) . (11)The result of [DW, Proposition 7.18] shows that (11) has an inverseisomorphic to the non-commutative twist functor T E associated to theuniversal object (7). Namely T E is the autoequivalence of D b Coh( X )which fits into the distinguished triangle R Hom( E , F ) L ⊗ A con E → F → T E ( F )(12)for any F ∈ D b Coh( X ). If C is a ( − , − T E coin-cides with Seidel-Thomas twist [ST01] along O C ( − C is a (0 , − T E coincides with the author’s generalized twist [Tod07] .The kernel object of the equivalence T E is given byCone (cid:18) R Hom A ( A con , A ) L ⊗ A op ⊗ A ( U ∨ ⊠ U ) → O ∆ X (cid:19) . Here ∆ X ⊂ X × X is the diagonal (cf. [DW, Lemma 6.16]). Lemma 2.3.
The object R Hom A ( A con , A ) L ⊗ A op ⊗ A ( U ∨ ⊠ U ) is isomor-phic to F [ − for F ∈
Coh( X × X ) satisfying the following: there is afiltration F ⊂ F ⊂ · · · ⊂ F dim A con = F such that each subquotient F j / F j − is isomorphic to O C ( − ⊠ O C ( − . In [Tod07], it was stated that T E is isomorphic to (11), but it was wrong: thecorrect statement is T E is an inverse of (11). We explain details in Appendix B. YUKINOBU TODA
Proof.
By the definition of Art in [DW, Definition 2.1], there is a C -algebra homomorphism A con → C such that its kernel n ⊂ A con is nilpotent. The ideal n ⊂ A con is two-sided, and A con / n is a onedimensional A op ⊗ A -module. We have the filtration of A op ⊗ A -modules0 = n m ⊂ n m − ⊂ · · · ⊂ n ⊂ A con for some m > n i / n i +1 is an A con / n -module. Since A con / n = C , the object n i / n i +1 is a finite direct sumof A con / n . Therefore it is enough to show that R Hom A ( A con / n , A ) L ⊗ A op ⊗ A ( U ∨ ⊠ U )(13) ∼ = O C ( − ⊠ O C ( − − . Let S ∈ mod A be the object given by S := R Hom b X ( U , O C ( − . Note that we have dim C S = 1. The object S is the unique simple A con -module (cf. [DW, Section 2.3]), hence A con / n viewed as a right A con -module is isomorphic to S . On the other hand, the vector bundle U ∨ ⊠ U on b X × b X is a tilting vector bundle. Hence we have a derivedequivalence R Hom b X × b X ( U ∨ ⊠ U , − ) : D b Coh( b X × b X ) ∼ → D b mod( A op ⊗ A )with inverse given by − L ⊗ A op ⊗ A ( U ∨ ⊠ U ). Let D be the dualiz-ing functor R H om b X ( − , O b X ) on D b Coh( b X ). We have D ( O C ( − ∼ = O C ( − − R Hom b X × b X ( U ∨ ⊠ U , O C ( − ⊠ O C ( − − ∼ = R Hom b X × b X ( D ( U ) ⊠ U , D ( O C ( − ⊠ O C ( − ∼ = R Hom b X ( O C ( − , U ) ⊗ C R Hom b X ( U , O C ( − ∼ = R Hom A ( S, A ) ⊗ C S ∼ = R Hom A ( A con / n , A ) . Therefore we obtain the desired isomorphism (13). (cid:3)
Genus zero Gopakumar-Vafa invariants.
The genus zero GVinvariants defined in [Kat08] count one dimensional stable sheaves F on Calabi-Yau 3-folds satisfying χ ( F ) = 1. In the setting of Subsec-tion 2.1, the variety X may not be Calabi-Yau, but so in a neighbor-hood of C . Since C is rigid in X , we can define the genus zero GVinvariant with curve class j [ C ] on X as well. Indeed in [Kat08], thegenus zero GV invariants of X are shown to coincide with the multi-plicities of the Hilbert scheme of X at some subschemes supported on C . Let p ∈ H ⊂ Y be a general hypersurface, and H ⊂ X its proper ON-COMMUTATIVE WIDTH AND GOPAKUMAR-VAFA INVARIANTS 7 transform. Then we have C ⊂ H . Let I ⊂ O H be the ideal sheaf of C .For j ≥
1, we have the subscheme C ( j ) ⊂ X given by O C ( j ) = ( O H /I j ) /Q where Q is the maximum zero dimensional subsheaf of O H /I j . LetHilb( X ) be the Hilbert scheme parameterizing closed subschemes in X . If 1 ≤ j ≤ l , it is shown in [BKL01, Section 2.1], that C ( j ) is theisolated point in Hilb( X ), and the following number is defined: Definition 2.4.
For ≤ j ≤ l , we define n j ∈ Z ≥ to be n j : = dim C O Hilb( X ) ,C ( j ) . By convention, we define n j = 0 for j > l . Since O Hilb( X ) ,C ( j ) is a finitely generated Artinian C -algebra, the num-ber n j is well-defined. If l = 1, the number n equals to the width k in(3) as indicated in [BKL01, Section 1]. In general, Katz [Kat08] showsthat n j coincides with the genus zero GV invariant of X with curveclass j [ C ].The number n j also appears in the context of deformations in thefollowing way. By [BKL01, Section 2.1], there exists a flat deformationof (4) X g Y T (14)where T is a Zariski open neighborhood of 0 ∈ A such that g : X →Y is isomorphic to b f in (4), and g t : X t → Y t for t ∈ T \ { } isa flopping contraction whose exceptional locus is a disjoint union of( − , − X t , Y t are the fibers of X → T , Y → T at t ∈ T respectively. Then the number n j coincides with the number of g t -exceptional ( − , − C ′ ⊂ X t for t = 0 whose curve class equalsto j [ C ], i.e. for any line bundle L on X , we havedeg( L| C ′ ) = j deg( L| C )(15)where we regard C as a curve on the central fiber of X → T . In whatfollows, we write the exceptional locus of g t for t = 0 as C j,k ⊂ X t , ≤ j ≤ l, ≤ k ≤ n j where C j,k is a ( − , − j [ C ].3. Proof of Theorem 1.1
Proof.
The identity cwid( C ) = n is almost obvious from the defi-nitions of both sides. Indeed since A abcon represents the commutativedeformation functor (8), the scheme Spec A abcon is the component ofthe moduli scheme of one dimensional stable sheaves on X containing YUKINOBU TODA O C ( − L in Subsection 2.2, the schemeSpec A abcon is isomorphic to the component of the moduli scheme of sta-ble sheaves containing O C , which defines the invariant n . By theproof of [Kat08, Proposition 3.3], the degree of the virtual fundamen-tal cycle of Spec A abcon coincides with the dimension of A abcon . Thereforecwid( C ) = n holds.We show the identity of wid( C ). The morphism g in (14) is a flop-ping contraction, and the argument of [Che02, Section 6] shows that g admits a flop X g ψ X † g † Y such that we have the derived equivalenceΦ O X×Y X†
X →X † : D b Coh( X ) ∼ → D b Coh( X † ) . By composing the above equivalence twice, we obtain the autoequiva-lence Φ O X×Y X† X † →X ◦ Φ O X×Y X†
X →X † : D b Coh( X ) ∼ → D b Coh( X ) . (16)Let Ψ be an inverse of the equivalence (16), and P ∈ D b Coh(
X × T X )the kernel object of Ψ. By [Che02, Lemma 6.1], for each t ∈ T , wehave the commutative diagram D b Coh( X ) Ψ L i ∗ t D b Coh( X ) L i ∗ t D b Coh( X t ) Ψ t D b Coh( X t ) . Here i t : X t ֒ → X is the inclusion, and Ψ t is the Fourier-Mukai functorwith kernel P t := L j ∗ t P , where j t is the inclusion j t := ( i t × i t ) : X t × X t ֒ → X × T X . The functor Ψ t is an equivalence, and it has an inverse given by thecomposition (cf. [Che02, Corollary 4.5])Φ O X t ×Y t X† t X † t →X t ◦ Φ O X t ×Y t X† t X t →X † t : D b Coh( X t ) ∼ → D b Coh( X t ) . (17)Therefore by [DW, Proposition 7.18], the equivalence Ψ is isomorphicto the non-commutative twist functor T E in (12). By the uniqueness ofFourier-Mukai kernels in Lemma A.1 below, we have P ∼ = Cone (cid:0) F [ − → O ∆ X (cid:1) . (18)Here F is a sheaf F on X × X given in Lemma 2.3, restricted to b X × b X . ON-COMMUTATIVE WIDTH AND GOPAKUMAR-VAFA INVARIANTS 9
For t = 0, the birational map X t X † t is the composition of flops at( − , − C j,k for 1 ≤ j ≤ l , 1 ≤ k ≤ n j . Hence the equivalenceΨ t for t = 0 is isomorphic to the compositions of all the spherical twistsalong O C j,k ( −
1) for 1 ≤ j ≤ l , 1 ≤ k ≤ n j . Therefore using Lemma A.1again, we have P t ∼ = Cone (cid:0) F t [ − → O ∆ X t (cid:1) (19)where F t is a sheaf on X t × X t defined by F t := l M j =1 n j M k =1 O C j,k ( − ⊠ O C j,k ( − . (20) Lemma 3.1.
We have H i ( P ) = 0 for i = 0 , .Proof. For any t ∈ T , we have the distinguished triangle P → P → j t ∗ P t . By (18) and (19), we have H i ( P t ) = 0 for any t ∈ T and i = 0 ,
1. Bytaking the long exact sequence of cohomologies of the above triangle,we obtain j ∗ t H i ( P ) = 0 for any t ∈ T and i = 0 ,
1. Therefore we have H i ( P ) = 0 for i = 0 , (cid:3) Lemma 3.2.
We have H ( P ) ∼ = O ∆ X and H ( P ) is flat over T . Fur-thermore we have j ∗ t H ( P ) ∼ = F t for any t ∈ T .Proof. By Lemma 3.1, we have the distinguished triangle in D b Coh(
X × T X ) H ( P ) → P → H ( P )[ − . Applying L j ∗ t , we obtain the distinguished triangle in D b Coh( X t × X t ) L j ∗ t H ( P ) → P t → L j ∗ t H ( P )[ − . By taking the long exact sequence of cohomologies, we have L j ∗ t H ( P ) ∼ = j ∗ t H ( P ) , F t ∼ = j ∗ t H ( P )and the exact sequence0 → j ∗ t H ( P ) → O ∆ X t → H − ( L j ∗ t H ( P )) → . (21)Below we denote by ∆ : X → X × T X , ∆ t : X t → X t × X t the di-agonal morphisms, and C the exceptional locus of g : X → Y . Theisomorphism F t ∼ = j ∗ t H ( P ) implies that H ( P ) is supported on C × C ,hence H − ( L j ∗ t H ( P )) is supported on C t × C t . The exact sequence(21) also implies that H − ( L j ∗ t H ( P )) is supported on ∆ X t , hence on∆ X t ∩ ( C t × C t ) = ∆ t ( C t ). It follows that H ( P ) is written as ∆ ∗ I fora rank one torsion free sheaf I on X , and the exact sequence (21) isgiven by ∆ t ∗ of the exact sequence of the following form0 → i ∗ t I → O X t → O C ′ t → for some subscheme C ′ t ⊂ X t supported on C t . Also by the genericflatness, there is a non-empty Zariski open subset U ⊂ T such that H − ( L j ∗ t H ( P )) = 0 for all t ∈ U . This implies that C ′ t = ∅ for all t ∈ U , hence I is isomorphic to O X away from C t for t ∈ T \ U . Bytaking the double dual of I , we obtain the exact sequence0 → I → O X → O C ′ → C ′ is supported on C t for t ∈ T \ U . If C ′ = ∅ , then j ∗ t H ( P ) ∼ =∆ t ∗ i ∗ t I contains the non-zero sheaf ∆ t ∗ H − L i ∗ t ( O C ′ ) for some t ∈ T \ U supported on C t , which contradicts to (21). Therefore C ′ = ∅ and H ( P ) ∼ = O ∆ X holds.Now in the sequence (22), we have i ∗ t I ∼ = O X t for any t ∈ T , hence C ′ t = ∅ as C ′ t has codimension bigger than or equal to two in X t . Thisimplies that H − ( L j ∗ t H ( P )) = 0 for any t ∈ T , hence H ( P ) is flatover T . (cid:3) By the above lemma, the sheaf F t for t = 0 is a flat deformationof F . Since they have compact supports, F and F t have the sameHilbert polynomials. It follows that, for a g -ample line bundle L on X with d := deg( L| C ) >
0, we have the equality χ ( F ⊗ ( L ⊠ L )) = χ ( F t ⊗ ( L ⊠ L )) . (23)By Lemma 2.3 and the Riemann-Roch theorem, we have χ ( F ⊗ ( L ⊠ L )) = dim C A con · χ ( O C ( − ⊗ L ) = dim C A con · d . By the definition of F t for t = 0 in (20), we have χ ( F t ⊗ ( L ⊠ L )) = l X j =1 n j X k =1 χ ( O C j,k ( − ⊗ L ) = l X j =1 j · n j · d . Here we have used the relation (15) for C ′ = C j,k . Since d >
0, theequality (23) implies the desired equality for wid( C ). (cid:3) Appendix A. Uniqueness of Fourier-Mukai kernels
Let Y be a quasi-projective complex variety, or a spectrum of acompletion of a finitely generated C -algebra at some maximum ideal.Suppose that f i : X i → Y are projective morphisms for i = 1 ,
2, and X i are regular schemes. Given an object P ∈ D b Coh( X × X )supported on X × Y X , we have the Fourier-Mukai functorΦ P X → X : D b Coh( X ) → D b Coh( X ) ON-COMMUTATIVE WIDTH AND GOPAKUMAR-VAFA INVARIANTS 11 defined by Φ P X → X ( − ) := R p ∗ ( L p ∗ ( − ) L ⊗ P )where p i : X × X → X i is the projection. The above functor preservescoherence since p | Supp( P ) is projective. For another regular scheme X ,a projective morphism f : X → Y and an object Q ∈ D b Coh( X × X )supported on X × Y X , we haveΦ Q X → X ◦ Φ P X → X ∼ = Φ Q◦P X → X where Q ◦ P is defined by (cf. [Che02, Proposition 2.3])
Q ◦ P := R p ∗ ( p ∗ P L ⊗ p ∗ Q ) . Here p ij : X × X × X → X i × X j is the projection.If Y = Spec C and Φ P X → X is an equivalence, then Orlov [Orl97]showed that the kernel object P is unique up to an isomorphism, i.e.Φ P X → X ∼ = Φ Q X → X implies P ∼ = Q . It should be well-known that thesame claim holds without Y = Spec C assumption, but as the authorcannot find a reference we include a proof here. Lemma A.1.
For P , Q ∈ D b Coh( X × X ) supported on X × Y X ,suppose that the following conditions hold: • We have an isomorphism of functors Φ P X → X ∼ = Φ Q X → X . • The functors Φ P X → X , Φ Q X → X are equivalences.Then we have P ∼ = Q .Proof. Let Q ∗ be the object of D b Coh( X × X ) given by Q ∗ := R H om X × X ( Q , O X × X ) ⊗ p ∗ ω X [dim X ] . By the Grothendieck duality, the functor Φ Q ∗ X → X is the right adjointof Φ Q X → X , hence an inverse of it. We haveΦ Q ∗ X → X ◦ Φ P X → X ∼ = Φ Q ∗ ◦P X → X and it is isomorphic to the identity functor. Then Φ Q ∗ ◦P X → X sends O x to O x for any x ∈ X , and O X to O X . Applying the argumentof [Huy06, Corollary 5.23], it follows that Q ∗ ◦ P ∼ = O ∆ X . Similarlywe have Q ◦ Q ∗ ∼ = O ∆ X . We obtain P ∼ = O ∆ X ◦ P ∼ = Q ◦ Q ∗ ◦ P ∼ = Q ◦ O ∆ X ∼ = Q as desired. (cid:3) Appendix B. Correction on flop-flop autoequivalence
In this occasion, I would correct a wrong statement in [Tod07, Sec-tion 3] on the description of flop-flop autoequivalence. Let us considerthe equivalenceΦ O X × Y X † X † → X ◦ Φ O X × Y X † X → X † : D b Coh( X ) ∼ → D b Coh( X )(24)associated to the flop diagram (9). In [Tod07, Theorem 3.1], it wasstated that if C is either a ( − , − , − T E . However this turns out to be wrong: the correct statementis that the equivalence (24) is an inverse of T E . Indeed the statementin [Tod07, Section 3] that the equivalenceΦ O X × Y X † X → X † : D b Coh( X ) ∼ → D b Coh( X † ) . (25)takes O C ( − O C † ( −
1) was wrong: it should be corrected that(25) takes O C ( −
1) to O C † ( − T E with T − E in theproof of [Tod07, Theorem 3.1], we obtain the statement that (11) isisomorphic to T − E .We explain why the above statement in [Tod07, Section 3] was wrong.In [Tod07, Section 3], I referred [Tod08, Ver 1, Lemma 5.1], whichin turn referred [Bri02, (4.8)] that the equivalence (25) induces theequivalence − Per(
X/Y ) ∼ → Per( X † /Y ) . (26)(Here we have used the fact that the equivalence (25) coincides with theequivalence Φ given in [Bri02, Section 4] by [Che02]). However (26) wasnot correct: it should be corrected that (25) induces the equivalence Per(
X/Y ) ∼ → − Per( X † /Y ) . (27)Indeed let C X ⊂ Coh( X ) be the category of sheaves F with R f ∗ F = 0.Then [Bri02, (4.5)] shows that (25) takes C X to C X † [1]. On the otherhand, as p Per(
X/Y ) is the gluing of Coh( Y ) and C X [ − p ] (not C X [ p ])by the definition, the equivalence (25) should reduce the perversityone. After correcting (26) as (27), the argument of [Tod08, Ver 1,Lemma 5.1] shows that (25) takes O C ( −
1) to O C † ( − References [BKL01] J. Bryan, S. Katz, and N. C. Leung,
Multiple covers and integrality conjec-ture for rational curves on Calabi-Yau threefolds , J. Algebraic Geom. (2001), 549–568.[Bri02] T. Bridgeland, Flops and derived categories , Invent. Math (2002),613–632. In the notation of [Bri02, (4.8)], the equivalence p Per(
W/X ) ∼ = p +1 Per(
Y /X )should be corrected as p Per(
W/X ) ∼ = p − Per(
Y /X ) ON-COMMUTATIVE WIDTH AND GOPAKUMAR-VAFA INVARIANTS 13 [Che02] J-C. Chen,
Flops and equivalences of derived categories for three-folds withonly Gorenstein singularities , J. Differential. Geom (2002), 227–261.[dB04] M. Van den Bergh, Three dimensional flops and noncommutative rings ,Duke Math. J. (2004), 423–455.[DW] W. Donovan and M. Wemyss,
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Kavli Institute for the Physics and Mathematics of the Universe,University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan.