Topics in Geometry: Mirror Symmetry

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1 MIT OpenCourseWare Topcs n Geometry: Mrror Symmetry Sprng 2009 For normaton about ctng these materals or our Terms o Use, vst:

2 MIRROR SYMMETRY: LECTURE 17 DENIS AUROUX 1. Coherent Sheaves on a Complex Manold (contd.) We now recall the ollowng dentons rom category theory. Denton 1. An addtve category s one n whch Hom(A, B) are abelan groups, composton s dstrbutve, and there s a drect sum and a zero object 0. An abelan category s an addtve category s.t. every morphsm has a kernel and cokernel, e.g. a kernel o : A B s a morphsm K A s.t. g : C A actors through K unquely g = 0. One can dene complexes n an addtve category, but one needs to be n an abelan category to have notons o exact sequences and cohomology. Recall that, gven chan complexes C, D, a chan map : C D s a collecton o maps C D commutng wth δ. Gven two such maps = { }, g = {g }, we call them homotopc there s a map h : A B[ 1] (B shted down by 1) s.t. b = d B h + hd A,.e. d 1 A 1 d d A +1 A+1 A h h (1) 1 g 1 g +1 g +1 d 1 d d +1 B 1 B B +1 A chan map s a quas-somorphsm the nduced maps on cohomology are somorphsms. Ths s stronger than H (C ) = H (D ). For A an abelan category, the category o bounded chan complexes s the derental graded category whose objects are bounded chan complexes n A and whose morphsms are pre-homomorphsms o complexes Hom k (A, B ) = Hom A (A, B +k ): t s equpped wth a derental δ where (2) Hom k (A, B ) = δ() = d B + ( 1) k+1 d A Hom k+1 (A, B ) Chan maps are precsely the elements o Ker (δ : Hom 0 Hom 1 ), and the nullhomotopc maps are elements o m (δ : Hom 1 Hom 0 ), so H 0 Hom(A, B) gves the space o chan maps up to homotopy. Denton 2. For A an abelan category, the bounded derved category D b (A) s the trangulated category whose objects are bounded chan complexes n A and 1

3 2 DENIS AUROUX whose morphsms are gven by chan maps up to homotopy localzng w.r.t. quassomorphsms. That s, quas-somorphsms are ormally nverted; or any quassomorphsm s, we add a morphsm s 1. More precsely, Hom D (A)(A b, B ) = s {A A B}/ where s s a quas-somorphsm, s a chan map, and s homotopy equvalence. We smlarly dene the categores D + (A), D (A) o chan complexes bounded above/below. To explan the noton o trangulated category, recall the ollowng: In the category o topologcal spaces (or smplcal complexes), there are no kernels and cokernels. Gven a map, however, the mappng cone C = (X [0, 1]) Y/(x, 0) (x, 0), (x, 1) (x) acts as both smultaneously. There are natural maps : Y C (ncluson) and q : C ΣX (collapsng Y ), and we obtan a sequence o topologcal spaces (3) X Y ΣX C q wth compostons null-homotopc. Ths gves a long exact sequence o (4) H (X) H (Y ) H (C ) H (ΣX) = H 1 (X) H (ΣY ) = H 1 (Y ) I X, Y are smplcal complexes, a smplcal map, C dened analogously s a smplcal complex, wth -cells gven by cones on ( 1) cells ( o X and ) -cells o Y. The boundary map s gven by the matrx X 0. Y I A and B are complexes, a chan map, we dene ( C = A[1] ) B,.e. C = A +1 B δ. The boundary map s δ = A [1] 0. Note δ that, A, B are sngle objects, Cone( : A B) s just {0 A B 0}. We have natural chan maps B C (subcomplex) and q C A [1] (quotent complex). As beore, A [1] s quas-somorphc to Cone( : B C ). Fnally, n the derved category, the nverson o quas-somorphsms gves us exact trangles B (5) wth A B [1] C (6) H (A) H (B) H (C) H +1 (A)

4 MIRROR SYMMETRY: LECTURE 17 3 Denton 3. A trangulated category s an addtve category wth a sht unctor [1] and a set o dstngushed trangles satsyng varous axoms: d X, X X 0 X[1] s dstngushed, u X Y, there s a dstngushed trangle X Y Z X[1] (Z s called the mappng cone o ). The rotaton o any dstngushed trangle s dstngushed,.e. or X Y Z X[1] dstngused, Y Z X[1] Y [1] and Z X[1] Y [1] Z[1] are dstngushed. Gven a square (7) (8) X X Y Y there s a map between the mappng cones o, that makes everythng commute n the nduced map o dstngushed trangles X Y Z X[1] X Y Z X [1]. u v Gven a par o maps X Y Z, there are maps between the mappng cones C u, C v, C v u o u, v, and v u that make every commute n the nduced maps o dstngushed trangles. C u v C u [1] C v (9) [1] v u [1] X Z u v Y 1.1. Derved unctors. Let F : A B be a let exact unctor between abelan categores. R A s called an adapted class o objects or F R s stable under drect sums, or C an acyclc complex o objects n R, F (C ) s acyclc, and A A, R R s.t. 0 A R.

5 4 DENIS AUROUX For nstance, the set o njectve objects s such an adapted class. Let K + (R) be the homotopy category o complexes bounded below o objects n R. RF gves a composton D + (A) K + (R) F D + (B), where the rst map s nduced by resoluton by objects o R. The map D + (A) D + (B) s exact,.e. t maps exact trangles to exact trangles, and R F = H (RF ) Extensons. Let A, B A D b (A) be sngle object complexes concentrated n degree 0, so B[k] s conentrated n degree k. Proposton 1. Hom D b (A)(A, B[k]) = Ext k (A, B). A We can use ths to dene a product Ext k (A, B) Ext l (B, C) Ext k+l (A, C) as a composton A B[k] C[k + l] n A D b A A (A). Example. For k = 1, we have (10) 0 0 A 0 0 B 0 0 There are no chan maps, but we can nvert quas-somorphsms. I we have an g extenson 0 A B C 0 n A, we have chan maps 0 0 C 0 (11) 0 A B 0 d 0 A 0 0 gvng an element n Hom D b (A)(C, A[1]) = Ext 1 (C, A). There are two ways to understand the above proposton. Frst, A has enough njectves, take a resoluton o B by a complex I 0 I 1 quassomorphc to B: the chan maps rom A to I are, up to homotopy, somorphc to H k (Hom(A, I )) = Ext k (A, B). Second, we can check the denton o a derved unctor. Gven a short exact sequence 0 A B C 0 n A, we get an g w exact trangle A B C A[1] quas-somorphc to a dstngushed trangle wth Cone(). g g

6 MIRROR SYMMETRY: LECTURE 17 5 g h Proposton 2. For an exact trangle A B C A[1] and an object E, we have long exact sequences (12) Hom(E, A[]) Hom(E, B[]) g Hom(E, C[]) h Hom(E, A[ + 1]) h g Hom(A[ + 1], E) Hom(C[], E) Hom(B[], E) Hom(A[], E)