A proof of Kontsevich-Soibelman conjecture

A PR OOF OF K ONTSE VICH-SOI BELMAN CONJEC TURE ALEXANDER I. EFIMO V Abstra ct. It is w ell kno wn that ” F uk a ya category” is in fact an A ∞ - pre-category in sense of Kontsevic h and Soib elman [KS]. The reason is that in general t he morphism spaces are d efi ned only for transv ersal pairs of Lagrangians, and higher products are defin ed only for transversal sequences of Lagrangians. In [KS] it is conjectured that fo r an y graded comm ut ative ring k , quasi-equ iv alence cla sses of A ∞ - pre-categories o ver k are in bijectio n with quasi-equiv alence clas ses of A ∞ - categories o ver k with strict (or w eak) identit y morph isms. In this pap er we prov e th is conjecture for essentially small A ∞ - (pre-)categories, in the case when k is a fi eld. In particular, it follo ws that we can replace F uk ay a A ∞ - pre- category with a quasi-equiv alent actual A ∞ - category . W e also presen t natural construction of p re- triangulated env elop e in the framew ork of A ∞ - pre-categories. W e prove its inv ariance under q uasi-equiv alences. Contents 1. In tro d uction 2 2. Preliminaries on A ∞ -(pre-)categ ories 3 2.1. Non-unital A ∞ -algebras and A ∞ -categ ories 3 2.2. Identi ty morphisms 5 2.3. A ∞ -pre-catego ries 5 3. Main Theorem 7 3.1. F rom essen tially small to small 7 3.2. Minimal mo dels 8 3.3. Ho c hsh ild cohomology of graded pr e-categ ories 9 3.4. Main Lemma 10 3.5. A ∞ - structures on a graded p re-catego ry 13 3.6. Inv ariance Theorem 15 3.7. Pro of of Main Th eorem 16 4. Twisted complexes ov er A ∞ - pre-categorie s 17 References 20 The author w as partially sup p orted by the Mo ebius Contest F oundation for Y oung Scientists, and by th e NSh gran t 1983.200 9.1. 1 2 ALEXANDER I. EFIMOV 1. Introduction A r emark able construction of K. F uk a y a [F] asso ciates to a symplectic manifold a ( Z - or Z / 2- )graded A ∞ - pre-category in sen s e of Kontsevic h and Soib elman [KS]. Its ob- jects are Lagrangian submanifolds with some additional str uctures. Th is is not an actual A ∞ - catego ry since in general the morphism spaces are defined only for tr ansv ersal pairs of Lagrangians, and higher p ro ducts are defined only for tr ansv ersal sequen ces of L agrangians. F uk ay a’s construction is used in the categorical interpretation of mirror symmetry [K] for Calabi-Y au v arieties, and fu rther generalizations to F ano and general cases, the so-cal led homologica l mirror symmetry conjecture. F or the systematic exp osition of different versions of F uk a y a A ∞ - pre-categorie s, see [Se]. Ho w eve r, in order to p ro ve HMS conjecture at least in some sp ecial cases, one should fi r st replace F uk a ya A ∞ - pre-category with a (quasi-equiv alent ) actual A ∞ - catego ry . Clearly , eac h A ∞ - catego ry (with w eak ident ity morphisms ) ca n b e considered also as an A ∞ - pre- catego ry . Kontse vich and Soib elman [KS] form ulated the follo w ing n atural conjecture. Conjecture 1.1. ( [KS] )L et k b e a gr ade d c ommutative ring. Then quasi-e quivalenc e classes of A ∞ - pr e-c ate gories over k ar e in bij e ction with quasi-e quiv alenc e classes of A ∞ - c ate gories over k with strict (or we ak) identity morph isms. The main result of th is pap er is the follo wing theorem. Theorem 1.2. L et k b e a field. Then quasi-e quivalenc e classes of essential ly smal l A ∞ - pr e-c ate gories over k ar e in bije ction with qu asi-e quiv alenc e classes of essential ly smal l A ∞ - c ate gories over k with strict (or we ak) identity morph isms. W e deal with A ∞ - (pre-)categ ories o ver a field b ecause w e n eed to pass to m in imal A ∞ - (pre-)categ ories (i.e. with m 1 = 0 ). F urther, w e deal with essen tially small A ∞ - (pre- )categ ories for p urely set-theoretical reason: we need to consider Ho chshild cohomology of graded (pre-)catego ries. The second sub ject of the pap er is the natural construction of t wisted complexes in the framew ork of A ∞ - pre-categorie s. Here the main statemen t is the inv ariance of t wisted com- plexes u nder quasi-equiv alences (Prop osition 4.6). F or ord inary A ∞ - catego ries w e obtain standard pr e-triangulated env elop es. The pap er is organized as follo w s. In Section 2 we define A ∞ - catego ries, strict and weak id en tit y morphisms, quasi- equiv alences, and A ∞ - pre-categorie s, follo w ing [K S]. In Section 3 w e prov e Main Theorem 1.2 (Th eorem 3.2). Th e pro of go es as follo ws. In Subsection 3.1 we pass from essen tially sm all A ∞ - (pre-)categ ories to small ones. F urth er, in A PROOF OF KONTSEVICH-SOIBELMAN CONJECTURE 3 Subsection 3.2 we pass from s mall to small m inimal A ∞ - (pre-)categ ories. In S ubsection 3.3 w e define Hochshild cohomology of graded pre-categories. Roughly sp eaking, obstr u ctions to constructing, step by step, of A ∞ - structures and A ∞ - morphisms liv e in th ese cohomology spaces. In Subs ection 3.4 w e form u late and pro ve Main Lemma (Lemma 3.4 ) ab out in v ariance of Ho c hshild cohomology u nder equ iv alences of graded pre-catego r ies. In con trast to ordinary DG an d A ∞ - catego ries, this is non-trivial, and this is in fact the crucial p oin t in the pro of of Main Theorem. Here w e use the language of simp licial lo cal systems. In Subsection 3.5 w e in tro d u ce the sets of equiv alence classes of minimal A ∞ - structures on graded pr e-categ ories, and dev elop basic obstru ction theory for lifting A ∞ - structures and A ∞ - homotopies. In S ubsection 3.6 we apply Main Lemma to pr o v e th e inv ariance of the set of equiv alence classes of minimal A ∞ - structures on graded pr e-cate gories. Finally , in Subs ection 3.7 w e prov e Main Theorem u sing the in v ariance resu lt. In Section 4 w e present the construction of pre-triangulated env elop e for A ∞ - pre- catego ries o ve r arb itrary graded comm utativ e r in g. W e verify that it is w ell-defined and is in v arian t under quasi-equiv alences. Ac kno wledgments. I am grateful to D. Kaledin for his remarks. 2. Preliminaries on A ∞ -(pre-)ca te gories Fix some basic field k of arb itrary c haracteristic. 2.1. Non-unital A ∞ -algebras and A ∞ -categories. Let A = L i ∈ Z A i b e a Z (resp. Z / 2 )-graded v ector space. De n ote by A [ n ] its shift by n : A [ n ] i := A i + n . Definition 2.1. A structur e of a non-unital A ∞ - algebr a on A is given by de gr e e +1 c o derivation b : T + ( A [1]) → T + ( A [1]) , such that b 2 = 0 . Her e T + ( A [1]) = L n ≥ 1 A [1] ⊗ n is a c ofr e e tensor c o algebr a. The cod eriv ation b is uniquely determined b y its ”T a ylor co efficients” m n : A ⊗ n → A [2 − n ] , n ≥ 1 . Th e condition b 2 = 0 is equiv alen t to the series of quadratic relations (2.1) X i + j = n +2 X 0 ≤ l ≤ i − 1 ( − 1) ǫ m i ( a 0 , . . . , a l − 1 , m j ( a l , . . . , a l + j − 1 ) , a l + j , . . . , a n ) = 0 , where a m ∈ A, and ǫ = j P 0 ≤ s ≤ l − 1 deg( a s ) + l ( j − 1 ) + j ( i − 1) . In particular, for n = 0 , w e ha ve m 2 1 = 0 . Definition 2.2. An A ∞ - morphism of non-unital A ∞ - algebr as A → B is a morphism of the c orr esp onding non-c ounital DG c o algebr as T + ( A [1]) → T + ( B [1]) . 4 ALEXANDER I. EFIMOV Suc h an A ∞ - morphism is uniquely determined by its ”T a ylor co efficien ts” f n : A ⊗ n → B [1 − n ] , satisfying the follo wing system of equations: (2.2) X 1 ≤ l 1 1 (r esp. if it pr eserves we ak identity mor- phisms). Definition 2.10. A strictly (r esp. we akly) unital A ∞ - functor F : C → D b etwe en strictly (r esp. we akly) unital A ∞ - c ate gories i s c al le d a quasi-e quivalenc e if the induc e d fu nc tor H ( F ) : H ( C ) → H ( D ) is an e qu i valenc e of gr ade d c ate gories. Two A ∞ -c ate gories with we ak i dentity morphisms C and D ar e c al le d quasi-e qui valent if ther e exists a finite se quenc e of A ∞ - c ate gories with we ak identity morphisms C 0 = C , C 1 , . . . , C n = D such that for 0 ≤ i ≤ n − 1 ther e exists a quasi-e quivalenc e C i → C i +1 or vic e versa. The follo win g statement is we ll-known, see [L-H]. Prop osition 2.11. If we c onsider only A ∞ - c ate gories with strict identity morphisms and strictly unital quasi-e quivalenc es, then the r esulting q uasi-e qui valenc e classes ar e in bije ction with quasi-e qu ivalenc e classes of A ∞ - c ate gories with we ak identity morph isms. 2.3. A ∞ -pre-categories. No w we recall the definition of A ∞ - pre-categorie s whic h were originally defin ed in [KS]. W e start w ith the notion of a n on-unital A ∞ - pre-category . 6 ALEXANDER I. EFIMOV Definition 2.12. A non-unital Z (r e sp. Z / 2 )-gr ade d A ∞ - pr e-c ate gory C is the fol lowing data: a) A class of obje cts O b ( C ) , b) F or any n ≥ 1 a su b class C n tr ⊂ O b ( C ) n of tr ansversal se quenc e s. It is r e qu ir e d that C 1 tr = O b ( C ) . c) F or e ach p air ( X 1 , X 2 ) ∈ C 2 tr , a g r ade d ve ctor sp ac e Hom( X 1 , X 2 ) . d) F or a tr ansversal se quenc e of obje cts X 0 , . . . , X n , a map of gr ade d ve ctor sp ac es m n : N 0 ≤ i ≤ n − 1 Hom( X i , X i +1 ) → Hom ( X 0 , X n )[2 − n ] . It is r e qu ir e d that e ach subse quenc e ( X i 1 , X i 2 , . . . , X i l ) , 0 ≤ i 1 < i 2 < · · · < i l ≤ n of a tr ansversal se quenc e ( X 0 , . . . , X N ) is tr ansversal, and that the gr ade d ve ctor sp ac e L 0 ≤ i 0 0 for l = 0 . F urther, b y the definition of an A ∞ - pre-category , ther e exists an ob ject e Y ∈ O b ( C ) and a quasi-isomorphism e Y → Y 0 suc h that all sequences ( e Y , S j ) are transv ersal. Since F induces a bijection b et ween quasi-isomorphism classes in C and ob jects in D , w e h a v e that F ( e Y ) = F ( Y 0 ) . Th erefore, F ( e Y , S j ) = ( F ( Y 0 ) , F ( S j )) = g ∗ j ( T ) . Put (3.16) H ( a ) = p X j =1 λ j ( g j , ( e Y , S j )) . A PROOF OF KONTSEVICH-SOIBELMAN CONJECTURE 13 Then from the c losedness of a we im m ediately obtain that ∂ ( H ( a )) = a. This prov es Sublemma.  Lemma is prov ed.  3.5. A ∞ - structures on a graded pre-category. Definition 3.9. An A ∞ - structur e on a gr ade d pr e-c ate gory C is a c ol le ction of maps m n , n ≥ 3 , d eg( m n ) = 2 − n f or al l tr ansversal se quenc es, such that to gether with m 2 ( a, b ) = ab and m 1 = 0 they give a structur e of A ∞ - pr e-c ate gory on C . Two A ∞ - structur es m a nd m ′ on C ar e c al le d str ongly homo topic if ther e exists an A ∞ - morph ism F : ( C , m ) → ( C , m ′ ) with F ( X ) = X for X ∈ O b ( C ) , and f 1 = id . In this c ase F is c al le d a str ong homoto py b etwe en m and m ′ . F ormal collectio ns of maps f n : N 1 ≤ i ≤ n Hom( X i − 1 , X i ) → Hom ( X 0 , X n ) of degree 1 − n for all tran s v ersal sequences ( X 0 , . . . , X n ) ∈ C n +1 tr , n ≥ 1 , with f 1 = id , form a group G C . The pr o duct of f and g is giv en by the s ame formula as the comp osition of A ∞ - functors. This group acts on the set A ∞ ( C ) of A ∞ - structures on C . Namely , f ( m ) = m ′ iff f is an A ∞ - morphism from ( C , m ) to ( C , m ′ ) . T autologic ally , t wo A ∞ - structures are strongly homotopic iff they lie in the same orbit of this action. The follo wing Lemmas are well- kn o wn f or A ∞ - algebras, and the pro of is in f act straigh t- forw ard. W e omit the p r o of. Lemma 3.10. L e t ( m 3 , . . . , m n − 1 ) b e p artial ly define d A ∞ - structur e on a gr ade d pr e- c ate gory C , i.e. th e maps m ≤ n − 1 satisfy al l the r e quir e d e qu ations which do not c ontain m ≥ n . Write the first A ∞ - c onstr aint c ontaining m n in the form (3.17) ∂ ( m n ) = Φ , wher e ∂ is the Ho chshild differ ential and Φ = Φ( m 3 , . . . , m n − 1 ) is quadr atic expr ession. Then we always have ∂ (Φ) = 0 . Lemma 3.11. L et m and m ′ b e two A ∞ -structur es on a gr ade d pr e-c ate gory C , with m i = m ′ i for i < n . L et f : ( C , m ) → ( C , m ′ ) b e an A ∞ - morph ism with f 1 = id , and f i = 0 for 2 ≤ i ≤ n − 2 . Then m ′ i = m i for i ≤ n − 1 , and m ′ n = m n + ∂ ( f n − 1 ) . Lemma 3.12. L et m and m ′ b e two A ∞ -structur es on a gr ade d pr e-c ate gory C . Supp ose that ( f 1 = id , f 2 , . . . , f n − 1 ) is a p artial ly define d str ong homotop y b etwe en m and m ′ . i.e . the ma ps f ≤ n − 1 satisfy al l the r e qui r e d e quations which do not c ontain f ≥ n . Write the first A ∞ - c onstr aint c ontaining f n in the form (3.18) ∂ ( f n ) = Ψ , 14 ALEXANDER I. EFIMOV wher e ∂ is the Ho chshild differ ential and Ψ = Ψ( f 2 , . . . , f n − 1 ; m , m ′ ) is a p olynomial ex- pr ession. Then we always have ∂ (Ψ) = 0 . W e will also need the notion of homotop y b et w een t wo A ∞ - functors. First, let f , f ′ : A → B b e t wo A ∞ - morphisms of (p ossibly non-u n ital) A ∞ - algebras. W e ha ve the asso ciated morph isms o f DG coa lgebras f , f ′ : T + ( A [1]) → T + ( B [1]) . A homotop y b et ween f and f ′ is a map H : T + ( A [1]) → T + ( B [1]) satisfying the iden tities (3.19) ∆ ◦ H = ( f ⊗ H + H ⊗ f ′ ) ◦ ∆ , and (3.20) f − f ′ = b B ◦ H + H ◦ b A . An y map H satisfying (3.19) is uniqu ely determined by its comp onen ts h n : A ⊗ n → B , deg( h n ) = − n. Definition 3.13. L et F, F ′ : C → D b e A ∞ - functors b etwe en A ∞ - pr e-c ate gories, such that F ( X ) = F ′ ( X ) for e ach X ∈ O b ( C ) . A homotopy H b etwe en f and f ′ is a c ol le c- tion of maps h n : N 1 ≤ i ≤ n Hom C ( X i − 1 , X i ) → Hom D ( F ( X 0 ) , F ( X n )) of de g r e e − n, for al l tr ansversal se quenc es ( X 0 , . . . , X n ) , satisfying the fol lowing pr op erty. F or e ach tr ansversal se quenc e ( X 0 , . . . , X N ) ∈ C n +1 tr , we have that the maps h n define a homotopy b etwe en the r estricte d A ∞ - functors (3.21) F , F ′ : M ij ≥ 0 (deg( x i ) + k i ) k j + n P i =0 k i ( k i +1) 2 + n P i =1 ik i . Prop osition 4.4. The A ∞ - pr e-c ate gory C pr e − tr is wel l-define d. Pr o of. The only non-ob vious thing to c heck is the exte n s ion prop ert y . T o pro v e it, we need the follo wing lemma. Lemma 4.5. L e t D b e an A ∞ - pr e-c ate gory, and ( X 1 , . . . , X n , Y 1 , . . . , Y n ) a tr ansver- sal se quenc e in D . Supp ose that we ar e giv e n with q u asi-isomorphism s F i : X i → Y i , 1 ≤ i ≤ n. Then for e ach Maur er- Cartan solution β ∈ End + ( Y 1 , . . . , Y n ) (r esp. α ∈ End + ( X 1 , . . . , X n ) ) ther e exists a Maur er-Cartan solution α ∈ End + ( X 1 , . . . , X n ) (r esp. β ∈ End + ( Y 1 , . . . , Y n ) ), to gether with a q u asi-isomorphism G : (( X 1 , . . . , X n ) , α ) → (( Y 1 , . . . , Y n ) , β ) in D pr e − tr . Pr o of. Consider the follo wing A ∞ - algebras: A 1 = End + ( X 1 , . . . , X n ) , A 2 = End + ( Y 1 , . . . , Y n ) and B describ ed as follo ws. As a k - mo d ule, (4.8) B = A 1 ⊕ A 2 ⊕ M 1 ≤ i j and the comp onen ts f ii : X i → Y i are quasi-isomorphisms. Clearly , go o d quasi-isomorph ism s are p reserv ed by F ∗ . It follo ws from L emma 4.5 that for eac h qu asi-isomorphism f 1 : E 1 → E 2 in D pr e − tr 1 , th ere exist quasi-isomorphisms f 0 : E 0 → E 1 , f 2 : E 2 → E 3 suc h that the sequence ( E 0 , E 1 , E 2 , E 3 ) is transversal and m 2 ( f 1 , f 0 ) , m 2 ( f 2 , f 1 ) are homotopic to go o d quasi-isomorphisms. Th is pr o v es part 1) of Prop osition. 2) I f F is a quasi-isomorphism, then all morph isms f 1 : Hom D pr e − tr 1 ( E 1 , E 2 ) → Hom D 2 ( F ( E 1 ) , F ( E 2 )) ind uce quasi-isomorphisms on the sub q u otien ts with resp ect to the natural filtrations (4.11) F r Hom((( X 1 , . . . , X n ) , α ) , (( Y 1 , . . . , Y m ) , β )) = M j − i ≥ r Hom( X i , Y j ) . Essen tial su rjectivit y is im p lied b y Lemma 4.5, together w ith Lemma 4.2.  F or the ordinary A ∞ - catego ries, our constru ction giv es stand ard A ∞ - catego ries of t wisted complexes in tro d uced in [BK] for DG categories and generalized in [K] to A ∞ - catego ries. No w sup p ose th at k is again a field. By Prop osition 4.6 2), w e hav e that passing from an essen tially small A ∞ - pre-category to quasi-equiv alen t A ∞ - catego ry comm utes (up to quasi-equiv alence) with taking of t wisted complexes. Referen ces [BK] A.I. Bondal, M.M. Kapranov, Enh anced triangulated categories, Mat. Sb ., 181:5 (1990), 669683. [ELO2] A. Efimov, V . Lunts, D . Orlo v, Deformation t h eory of ob jects in homotopy and d erive d categories I I: pro-representabilit y of t he d eformation functor, arXiv:math/0702839 (preprint), to app ear in Adv. Math. A PROOF OF KONTSEVICH-SOIBELMAN CONJECTURE 21 [F] K. F u k ay a, Morse homotopy , A ∞ - category , and Floer h omologies , in Pro ceedings of GARC W ork- shop on Geometry and T op ology , S eoul N ational Universit y , 1993. [K] M. Kontsevic h, Homological algebra of mirror symmetry , in Proceedings of ICM’94, Zurich, Birkhauser, 1995, 120-139. [KS] M. Kontsevic h and Y. Soib elman, Homological mirror symmetry and torus fibrations, I n Symplectic geometry and mirror symmetry (Seoul, 2000), pages 203-263. W orld Sci. Pub lishing, River Edge, NJ, 2001. [L-H] K. Lef ` evre-Hasega wa, Sur les A ∞ - cat ´ egories, Th ` ese de do ctorat, U niversit ´ e Denis Diderot - Paris 7, 2003. [P] A. Poli shchuk, Extensions of homogeneous co ordinate rings to A ∞ -algebras, H omology Homotop y Appl. V olume 5, Number 1 ( 2003), 407-421. [Se] P . Seidel, F uk ay a categories an d Picard-Lefschetz theory , Europ ean Math. So c., 2008. Dep a r tment of M echanics and Ma the ma tics, Mosco w St a te University, Mosco w, Russia Independe nt U niversity of M osco w, Mosco w, Russia E-mail addr ess : efimov@mccme.ru