Homological Mirror Symmetry for Blowups of CP 2 Denis Auroux (MIT) - PowerPoint PPT Presentation

Homological Mirror Symmetry for Blowups of CP 2 Denis Auroux (MIT) (joint work with L. Katzarkov, D. Orlov) (after ideas of Kontsevich, Seidel, Hori, Vafa, . . . ) See: math.AG/0404281, math.AG/0506166

Mirror Symmetry Complex manifolds: ( X, J ) locally ≃ ( C n , i ) Look at complex analytic cycles + holom. vector bundles, or better: coherent sheaves Intersection theory = Morphisms and extensions of sheaves. Symplectic manifolds: ( Y, ω ) locally ≃ ( R 2 n , � dx i ∧ dy i ) Look at Lagrangian submanifolds (+ flat unitary bundles): L n ⊂ Y 2 n with ω | L = 0 (locally ≃ R n ⊂ R 2 n ; in dim R 2, any embedded curve!) Intersection theory (with quantum corrections) = Floer homology (discard intersections that cancel by Hamiltonian isotopy) Mirror symmetry: D-branes = boundary conditions for open strings. Homological mirror symmetry (Kontsevich): at the level of derived categories, A-branes = Lagrangian submanifolds, B-branes = coherent sheaves. 1

HMS Conjecture: Calabi-Yau case 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 manifold. D b = bounded derived category: Objects = complexes 0 → · · · → E i d i → E i +1 → · · · → 0. Morphisms = morphisms of complexes (up to homotopy, + inverses of quasi-isoms) F ( Y ) = Fukaya A ∞ -category of ( Y, ω ). Roughly: Objects = (some) Lagrangian submanifolds (+ flat unitary bundles) Morphisms: Hom( L, L ′ ) = CF ∗ ( L, L ′ ) = C | L ∩ L ′ | if L ⋔ L ′ . (or � Hom( E p , E ′ p )) (Floer complex, graded by Maslov index) with: differential d = m 1 ; product m 2 (composition; only associative up to homotopy); and higher products ( m k ) k ≥ 3 (related by A ∞ -equations). 2

Fukaya categories Hom( L, L ′ ) = CF ∗ ( L, L ′ ) = C | L ∩ L ′ | if L ⋔ L ′ . Hom( E p , E ′ (or: � p )) p ∈ L ∩ L ′ • Differential d = m 1 : Hom( L 0 , L 1 ) → Hom( L 0 , L 1 )[1] � � D 2 u ∗ ω ) � m 1 ( p ) , q � = ± exp( − L 1 u ∈M ( p,q ) p q counts pseudo-holomorphic maps Y L 0 (in dim R 2: immersed discs with convex corners) 2 D • Product m 2 : Hom( L 0 , L 1 ) ⊗ Hom( L 1 , L 2 ) → Hom( L 0 , L 2 ) q L 1 L 2 � m 2 ( p, q ) , r � counts pseudo-holomorphic maps 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 ] p k � m k ( p 1 , . . . , p k ) , q � counts pseudo-holomorphic maps L L k 1 Y p q L 0 1 D 2 3

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

Categories of Lagrangian vanishing cycles L i ⊂ Σ 0 Lagrangian sphere = vanishing cycle associated to γ i Seidel: Lag ( W, { γ i } ) finite, directed A ∞ -category. L i Y Objects: L 1 , . . . , L r . Σ 0  CF ∗ ( L i , L j ) = C | L i ∩ L j | if i < j  w   Hom( L i , L j ) = γ C · Id if i = j 1 λ 1 λ 0  0 if i > j  C  γ λ r r 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 ] is trivial unless i 0 < · · · < i k . • m k counts discs in Σ 0 with boundary in � L i , with coefficients ± exp( − � D 2 u ∗ ω ). • in our case π 2 (Σ 0 ) = 0, π 2 (Σ 0 , L i ) = 0, so no bubbling. Remarks: • � L 1 , . . . , L r � = exceptional collection generating D b Lag . • objects also represent Lefschetz thimbles (Lagrangian discs bounded by L i , fibering above γ i ) Theorem. (Seidel) Changing { γ i } affects Lag ( W, { γ i } ) by mutations; D b Lag ( W ) depends only on W : ( Y, ω ) → C . 5

Example 1: weighted projective planes (Auroux-Katzarkov-Orlov, math.AG/0404281; 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 ) has an exceptional collection O , O (1) , . . . , O ( 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 )) ≃ deg. ( 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., the N crit. pts. are an orbit; complex conjugation. We choose ω invariant under Z /N and complex conj. ( ⇒ [ ω ] = 0 exact) Theorem. D b Lag ( W ) ≃ D b Coh ( X ) . (this should extend to weighted projective spaces in all dimensions; for technical reasons we only have a partial argument when dim C ≥ 3). 6

Non-commutative deformations Y = { x a y b z c = 1 } ⊂ ( C ∗ ) 3 , W = x + y + z , X = CP 2 ( a, b, c ); Theorem. If ω is exact, then D b Lag ( W ) ≃ D b Coh ( X ) . Can deform Lag ( W ) by changing [ ω ] (and introducing a B -field). ( S 1 × S 1 = generator of H 2 ( Y, Z ) ≃ Z ) � Choose t ∈ C , and take S 1 × S 1 [ B + iω ] = t → deformed category D b Lag ( W ) t . This corresponds to a non-commutative deformation X t of X : deform weighted polynomial algebra C [ x, y, z ] to with µ a 1 µ b 2 µ c 3 = e it yz = µ 1 zy, zx = µ 2 xz, xy = µ 3 yx, Theorem. ∀ t ∈ C , D b Lag ( W ) t ≃ D b Coh ( X ) t . 7

Example 2: Del Pezzo surfaces (Auroux-Katzarkov-Orlov, math.AG/0506166) X = CP 2 blown up at k ≤ 9 points, − K X ample (or more generally, nef). D b Coh ( X ) has an exceptional collection O , π ∗ T P 2 ( − 1) , π ∗ O P 2 (1) , O E 1 , . . . , O E k > > 2 > ❍ ❍ ❍ ❍ ✟ ✟ ✟ ✟ ✉ ✉ ✉ ✉ 3 3 O O E i T (-1) O (1) Compositions encode coordinates of blown up points. For generic blowups, Hom( O E i , O E j ) = 0. Infinitely close blowups give pairs of morphisms in deg. 0 and 1 (recover O C (-2-curve) as a cone). Mirror: mirror to CP 2 compactifies to M = resolution of { XY Z = T 3 } ⊂ CP 3 , with elliptic fibration W = T − 1 ( X + Y + Z ) : M → C ∪ {∞} . W is Morse, with 3 crit. pts. in {| W | < ∞} ; fiber at infinity has 9 components. Mirror to X = deform ( M, W ) to bring k of the crit. pts. over ∞ into finite part. Get an elliptic fibration over {| W k | < ∞} : W k : M k → C , with 3 + k sing. fibers. (symplectic form to be specified later) Theorem. For suitable choice of [ B + iω ] , D Lag ( W k ) ≃ D b Coh ( X k ) . 8

The vanishing cycles of W k x 1 ✲ ✲ r ¯ x ◗◗◗◗◗◗◗◗◗◗◗◗ r ✑ ❆ ✁ ✁ ✁ ✁ ✑ ❆ ✁ ✁ ✁ ✁ ❆ ❆ ✑ ✁ ✁ ✑ ✁ ✁ ❆ ❆ L 0 + ✁ ✁ ✑ ✑ r − ✁ ✁ ❆ ❆ ✁ ✁ z 0 ✑ ✑ ✁ ✁ ❆ ❆ ✁ ✁ ¯ ✑ ✁ ✁ y r y 1 L 1 ❆ ✑ ✁ ✁ ❆ r ✁ ✁ ❆ ❆ ✑ ✁ ✁ ✑ ✁ ✁ ❆ ❆ ✁ ✁ ✑ ✑ ◗ ✁ ✁ L 2 + ❆ ❆ ✁ ✁ ◗◗◗◗◗◗◗◗◗◗◗◗ r − r x 0 x 0 ✑ ✑ ✁ ✁ ❆ ❆ ✁ ✁ ✑ ❯ ✁ ✁ ✑ ❯ ❆ ❆ ✁ ✁ r z 1 ✁ ✁ ✑ ❆ ❆ ✁ ✁ ✑ r ¯ L 3+ j ✁ ✁ z ❆ ❆ ✁ ✁ ✑ ✑ ✁ ✁ ❆ ❆ ✁ ✁ ✁ ✁ ✑ + ✑ ❆ r − ❆ ✁ ✁ ✁ ✁ ✑ ✑ y 0 ❆ ❆ ✁ ✁ ✁ ✁ ✑ ❆ ❆ ❆ ✑ ✁ ✁ ◗ ✲ ✲ ✁ ✁ ❆ r r ¯ x x 1 Symplectic deformation parameters: [ B + iω ] ∈ H 2 ( M k , C ) : → cubic curve CP 2 ⊃ E ≃ C / ( Z + τ Z ) • Area of fiber: τ = 1 � Σ ( B + iω ) ← 2 π (all blowups are at points of E ; think of E as zero set of β ∈ H 0 (Λ 2 T ).) • Area of C ( ∂C = L 0 + L 1 + L 2 ) : t = 1 � C ( B + iω ) ← → σ ∈ Pic 0 ( E ) 2 π (same parameter as in Example 1; commutative deformations correspond to t = 0; takes values in C / ( Z + τ Z ).) 9