Homological Mirror Symmetry for Fano Surfaces Denis Auroux (joint - PDF document

Homological Mirror Symmetry for Fano Surfaces Denis Auroux (joint work with L. Katzarkov, D. Orlov) (after ideas of Kontsevich, Seidel, Hori, Vafa, . . . )

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Mirror Symmetry Complex manifolds: ( X, J ) locally ≃ ( C n , i ) Look at complex analytic subvarieties + holom. vector bun- dles, or better: coherent sheaves (cokernels of morphisms of holom. bundles with finite resolution) Intersection theory = Morphisms and extensions of sheaves. Symplectic manifolds: ( Y, ω ) locally ≃ ( R 2 n , � dx i ∧ dy i ) (in dim R 2, any orientable surface!) Look at Lagrangian submanifolds: L n ⊂ Y 2 n with ω | L = 0 (locally ≃ R n ⊂ R 2 n ) (in dim R 2, all embedded curves!) Intersection theory = Floer homology (discard intersections that cancel by Hamiltonian isotopy) Mirror symmetry: Duality between type II A and II B string theories. D-branes = boundary conditions for open strings. Homological mirror symmetry (Kontsevich, ...): A-branes = Lagrangian submanifolds, B-branes = coherent sheaves. only in a weaker sense: derived categories. 1

Homological Mirror Symmetry Conjecture: Calabi-Yau case Roughly: X, Y Calabi-Yau ( c 1 = 0) mirror pair ⇒ D b Coh ( X ) ≃ D F ( Y ) ≃ D b Coh ( Y ) D F ( X ) Coh ( X ) = category of coherent sheaves on X complex mfld. D b = bounded derived category Objects = complexes 0 → · · · → E i d i → E i +1 → · · · → 0. � morphisms of complexes Morphisms = +formal inverses of quasi-isoms F ( Y ) = Fukaya A ∞ -category of ( Y, ω ). Roughly: Objects = (some) Lagrangian submanifolds (+flat bundles) Morphisms: Hom( L, L ′ ) = CF ∗ ( L, L ′ ) = C | L ∩ L ′ | if L ⋔ L ′ . (Floer complex, graded by Maslov index) • Differential d = m 1 : Hom( L 0 , L 1 ) → Hom( L 0 , L 1 )[1] • Product m 2 : Hom( L 0 , L 1 ) ⊗ Hom( L 1 , L 2 ) → Hom( L 0 , L 2 ) (associative up to homotopy) • Higher products m k : Hom( L 0 , L 1 ) ⊗· · ·⊗ Hom( L k − 1 , L k ) → Hom( L 0 , L k )[2 − k ] (related by A ∞ -equations) 2

Fukaya categories F ( Y ) = Fukaya A ∞ -category of ( Y, ω ). Objects = (some) Lagrangian submanifolds (+flat bundles) Morphisms: Hom( L, L ′ ) = CF ∗ ( L, L ′ ) = C | L ∩ L ′ | if L ⋔ L ′ . (Floer complex, graded by Maslov index) • Differential d = m 1 : Hom( L 0 , L 1 ) → Hom( L 0 , L 1 )[1] � m 1 ( p ) , q � counts pseudo-holomorphic maps (in dim R 2, same as immersed discs with convex corners) L 1 p q Y L 0 2 D • Product m 2 : Hom( L 0 , L 1 ) ⊗ Hom( L 1 , L 2 ) → Hom( L 0 , L 2 ) � m 2 ( p, q ) , r � counts pseudo-holomorphic maps q L 1 L 2 p r Y L 0 2 D • Higher products m k : Hom( L 0 , L 1 ) ⊗ · · · ⊗ Hom( L k − 1 , L k ) → Hom( L 0 , L k )[2 − k ] � m k ( p 1 , . . . , p k ) , q � counts pseudo-holomorphic maps p k L L k 1 Y p q L 0 1 D 2 3

Homological Mirror Symmetry Conjecture: Fano case M.S. X Fano ( c 1 ( TX ) > 0) ← → “Landau-Ginzburg model” � Y (non-compact) manifold w : Y → C “superpotential” D b Coh ( X ) ≃ D F ( w ) D F ( X ) ≃ D Sing ( w ) D F ( w ) (Lagrangians) and D Sing ( w ) (sheaves) = symplectic and complex geometries of singularities of w . If w : Y → C Lefschetz fibration (isolated non-deg. crit. pts): L i ⊂ Σ 0 Lagrangian sphere = vanishing cycle associated to γ i L i Y (collapses to crit. pt. by // transport) Σ 0 Seidel: F ( w, { γ i } ) w finite, directed A ∞ -category. γ 1 λ 1 λ 0 C Objects: L 1 , . . . , L r . γ λ r r  CF ∗ ( L i , L j ) = C | L i ∩ L j | if i < j   Hom( L i , L j ) = C · Id if i = j  0 if i > j  Products: ( m k ) k ≥ 1 = Floer theory for Lagrangians ⊂ Σ 0 . 4

Fukaya-Seidel categories L i ⊂ Σ 0 Lagrangian sphere = vanishing cycle associated to γ i L i Y (collapses to crit. pt. by // transport) Σ 0 Seidel: F ( w, { γ i } ) w finite, directed A ∞ -category. γ 1 λ 1 λ 0 C Objects: L 1 , . . . , L r . γ λ r r  CF ∗ ( L i , L j ) = C | L i ∩ L j | if i < j   Hom( L i , L j ) = C · Id if i = j  0 if i > j  Products: ( m k ) k ≥ 1 = Floer theory for Lagrangians ⊂ Σ 0 . m k : Hom( L i 0 , L i 1 ) ⊗ · · · ⊗ Hom( L i k − 1 , L i k ) → Hom( L i 0 , L i k )[2 − k ] – trivial unless i 0 < · · · < i k – count discs in Σ 0 w/ boundary in � L i (Floer theory) Remarks: • � L 1 , . . . , L r � = exceptional collection generating D F . • objects are also Lefschetz thimbles (discs bounded by L i ) • in our case, no technical issues such as bubbling etc. D 2 u ∗ ω ) � • coefficient ring: R = C , count w/ coef. ± exp( − Theorem. (Seidel) Changing { γ i } affects F ( w, { γ i } ) by mutations; D F ( w ) depends only on w : ( Y, ω ) → C . 5

Example: weighted projective planes (cf. work of Seidel on CP 2 ) X = CP 2 ( a, b, c ) = ( C 3 − { 0 } ) / ( x, y, z ) ∼ ( t a x, t b y, t c z ) (Fano orbifold). D b Coh ( X ) generated by exceptional collection O X , O X (1) , . . . , O X ( N − 1) ( N = a + b + c ) (Homogeneous coords. x, y, z are sections of O ( a ) , O ( b ) , O ( c )) Hom( O ( i ) , O ( j )) ≃ degree ( j − i ) part of symmetric algebra C [ x, y, z ] (degs. a, b, c ) All in degree 0 (no Ext’s); composition = obvious. Mirror: Y = { x a y b z c = 1 } ⊂ ( C ∗ ) 3 , w = x + y + z . ( Y ≃ ( C ∗ ) 2 if gcd ( a, b, c ) = 1) Z /N ( N = a + b + c ) acts by diagonal mult.; complex conjugation. We choose ω invariant under Z /N and complex conj. ( ⇒ [ ω ] = 0 exact) Theorem. D F ( w ) ≃ D b Coh ( X ) (should also work in higher dimensions...) 6

Non-commutative deformations X = CP 2 ( a, b, c ); Y = { x a y b z c = 1 } ⊂ ( C ∗ ) 3 , w = x + y + z , ω invariant under Z /N and complex conj. ( ⇒ exact): Theorem. D F ( w ) ≃ D b Coh ( X ) Can deform FS ( w ) by changing [ ω ] (& introducing a B -field). � Choose τ ∈ C , and take S 1 × S 1 [ ω + iB ] = τ ( S 1 × S 1 generates H 2 ( Y, Z ) ≃ Z ) (keeping Z /N -invariance) → deformed category D F ( w ) τ . ⇐ ⇒ non-commutative deformation X τ of X : deform polynomial algebra C [ x, y, z ] to yz = µ 1 zy, zx = µ 2 xz, xy = µ 3 yx, with µ a 1 µ b 2 µ c 3 = e − τ Theorem. ∀ τ ∈ C , D F ( w ) τ ≃ D b Coh ( X ) τ . 7

Outline of argument Y = { x a y b z c = 1 } ⊂ ( C ∗ ) 3 , w = x + y + z : crit w = { λ ∈ C , λ a + b + c = ( a + b + c ) a + b + c } = { λ j , 0 ≤ j < N } a a b b c c λ 0 ∈ R + , λ j = λ 0 exp( − 2 πi j a + b + c ) Reference fiber: Σ 0 = w − 1 (0); arcs γ j = straight lines. ⇒ vanishing cycles L j ⊂ Σ 0 . If ω is Z N -invariant, then L j = exp( − 2 πi j a + b + c ) · L 0 . Visualize L j and intersections via projection π x : Σ 0 → C ∗ . ( b + c -fold branched covering, with a + b + c branch points) L 4 r r L 5 r L 3 L 1 r L 2 r ❝ ❝ r L 6 L 2 r L 0 r L 0 L 1 r r ( a, b, c ) = (4 , 2 , 1) ( a, b, c ) = (1 , 1 , 1) ⇒ Description of F ( w, { γ j } ): • Objects: L j , 0 ≤ j < N . i<j CF ∗ ( L i , L j ) = free module of rank 3 N , generators • � x i ∈ CF ∗ ( L i , L i + a ), x i ∈ CF ∗ ( L i , L i + b + c ), ¯ y i ∈ CF ∗ ( L i , L i + b ), y i ∈ CF ∗ ( L i , L i + a + c ), ¯ z i ∈ CF ∗ ( L i , L i + c ), z i ∈ CF ∗ ( L i , L i + a + b ). ¯ 8

Outline of argument Description of F ( w, { γ j } ): • Objects: L j , 0 ≤ j < N . i<j CF ∗ ( L i , L j ) = free module of rank 3 N , generators • � x i ∈ CF ∗ ( L i , L i + a ), x i ∈ CF ∗ ( L i , L i + b + c ), ¯ y i ∈ CF ∗ ( L i , L i + b ), y i ∈ CF ∗ ( L i , L i + a + c ), ¯ z i ∈ CF ∗ ( L i , L i + c ), z i ∈ CF ∗ ( L i , L i + a + b ). ¯ • for suitable graded Lagrangian lifts of L j , deg( x i , y i , z i ) = 1 , deg(¯ x i , ¯ y i , ¯ z i ) = 2 . • m k = 0 for k � = 2. • only non-zero compositions: m 2 ( x i , z i + a ) = α ′ ¯ m 2 ( x i , y i + a ) = α ¯ z i , y i , m 2 ( y i , x i + b ) = α ′ ¯ m 2 ( y i , z i + b ) = α ¯ x i , z i , m 2 ( z i , y i + c ) = α ′ ¯ m 2 ( z i , x i + c ) = α ¯ y i , x i . If [ ω ] = 0 then α = α ′ ( ⇒ exterior algebra), in general α 1 � � � α ′ = exp − S 1 × S 1 ω + iB . a + b + c Then pass to dual exceptional collection by “full mutation” (change { γ j } to { γ ′ j } with base point at infinity) ⇒ exterior algebra becomes truncated symmetric algebra. 9