p -adic actions on Fukaya categories and iterations of symplectomorphisms Yusuf Barı¸ s Kartal Princeton University August 4, 2020 Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 1 / 27
Overview Motivation and the main result 1 Local action on the Fukaya category 2 Fukaya category over p -adics and p -adic action 3 Proof of the Theorem 4 Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 2 / 27
Motivation: Bell’s theorem Theorem (J. Bell, 2005) Let X be an affine variety over a field of characteristic 0 and φ be an automorphism of X. Consider a subvariety Y ⊂ X and a point x ∈ X. Then the set { k ∈ N : φ k ( x ) ∈ Y } is a union of finitely many arithmetic progressions and finitely many other numbers. This theorem has versions for coherent sheaves as well, describing similar results for { k ∈ N : Tor ( F , ( φ k ) ∗ F ′ ) � = 0 } . It is valid for surfaces. Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 3 / 27
Symplectic analogues? Then one can ask if there is a symplectic analogue of this theorem. For instance: Conjecture (Seidel) Let L and L ′ be two Lagrangians in a symplectic manifold M with a symplectomorphism φ . Then the set { k ∈ N : φ k ( L ) is Floer theoretically isomorphic to L ′ } is a union of finitely many arithmetic progressions and finitely many other numbers. The conjecture is for isomorphisms up to twist by local systems. For the heuristic relation of Bell’s theorem to this conjecture, consider X = “moduli of Lagrangians”, x = L ∈ X , Y = { L ′ } ⊂ X . Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 4 / 27
Main result Theorem (K., 2020) Let M be a monotone symplectic manifold and φ be a symplectomorphism isotopic to identity. Given Lagrangians L , L ′ ⊂ M, the rank of HF ( L , φ k ( L ′ )) is constant in k ∈ Z , with finitely many exceptions. Assumptions: M is non-degenerate (hence, F ( M ; Λ) is finitely generated and smooth), integral (or rational) ∃ set { L i } of generators such that each L i is Bohr-Sommerfeld monotone ( and has minimal Maslov number 3) same assumptions on L and L ′ Remark Bohr-Sommerfeld monotonicity assumption on L and L ′ can be dropped Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 5 / 27
Explanation of terms and notation Λ = Q (( T R )) Novikov field with rational coefficients and real exponents F ( M ; Λ) Fukaya category spanned by { L i } Bohr-Sommerfeld monotone ⇒ ∃ only finitely many holomorphic curves with boundary components on L , L ′ and various L i Example M = a higher genus surface, non-separating curves have unique B-S monotone representative in their isotopy classes Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 6 / 27
Main tool Bell proves his theorem by interpolating the orbit { φ k ( x ) } by a p -adic analytic arc Analogous main tool for us: interpolate iterates of φ by a p -adic analytic action Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 7 / 27
Local action on F ( M , Λ) Let φ = φ 1 α , where α is a closed 1-form on M . Symplectomorphisms φ t α give rise to construction of F ( M , Λ)-bimodules M Λ α | T t (using quilted strips etc.). We construct this family by deforming the diagonal bimodule: Definition Let M Λ α | T 0 ( L i , L j ) = Λ � L i ∩ L j � . Define the structure maps via: � ± T E ( u ) . y ( x 1 , . . . x m | x | x ′ 1 , . . . x ′ n ) �→ u varies among the discs as in figure below: Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 8 / 27
Local action on F ( M , Λ) Definition Let M Λ α | T t ( L i , L j ) = Λ � L i ∩ L j � . Define the structure maps via: � ± T E ( u ) T t α ([ ∂ h u ]) . y ( x 1 , . . . x m | x | x ′ 1 , . . . x ′ n ) �→ u varies among the discs as in figure below: Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 9 / 27
Local action on F ( M , Λ) Lemma The family of bimodules M Λ α | T t behave like a “local group action”, i.e. for small t 1 , t 2 M Λ α | T t 2 ⊗ F ( M , Λ) M Λ α | T t 1 ≃ M Λ α | T t 1+ t 2 Proof. Write a map g of bimodules such that g k | 1 | n � ± T E ( u ) T t 1 α ([ ∂ 1 u ]) T t 2 α ([ ∂ 2 u ]) . y ( x 1 , . . . , x k | m 2 ⊗· · ·⊗ m 1 | x ′ 1 , . . . , x ′ n ) �− → where [ ∂ 1 u ], [ ∂ 2 u ] are as in the figure: Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 10 / 27
Local action on F ( M , Λ) Proof. Write a map g of bimodules such that g k | 1 | n � ± T E ( u ) T t 1 α ([ ∂ 1 u ]) T t 2 α ([ ∂ 2 u ]) . y ( x 1 , . . . , x k | m 2 ⊗· · ·⊗ m 1 | x ′ 1 , . . . , x ′ n ) �− → where [ ∂ 1 u ], [ ∂ 2 u ] are as in the figure (concatenated with fixed paths) g is a quasi-isomorphism at t 1 = t 2 = 0 ⇒ quasi-iso near (0 , 0) Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 11 / 27
Review of p -adics Let p > 2 be a prime. Recall: Z p = { m 0 + m 1 p + m 2 p 2 + . . . } , where m i ∈ { 0 , . . . , p − 1 } Z p = completion of Z with respect to norm | x | p := p − val p ( x ) Q p = field of fractions of Z p , normed field Upshot: One can do analytic geometry over Q p D 1 = closed unit disc = Z p Q p � t � = { � a i t i : a i ∈ Q p , | a i | p → 0 } = analytic functions on D 1 D p − n = closed disc of radius p − n = p n Z p Q p � t / p n � = analytic functions on D p − n Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 12 / 27
Some strange features of p -adic analytic disc 1 , 2 , 3 , · · · ∈ D 1 = unit disc Unit disc is an additive group (Strassman’s theorem) if f ( t ) ∈ Q p � t � has infinitely many 0’s, f ( t ) = 0 Coherent sheaves on D p − n are locally free outside finitely many points Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 13 / 27
Fukaya category over smaller fields and over Q p B-S monotone ⇒ the coefficients � ± T E ( u ) are finite Fukaya category is defined over Q ( T R ) E ( u ) ∈ ω M ( H 2 ( M , � L i ∪ L ∪ L ′ )) and the latter is a finitely generated additive subgroup of R Given finitely generated G ⊃ ω M ( H 2 ( M , � L i ∪ L ∪ L ′ )), and basis g 1 , . . . , g k , Fukaya category is defined over Q ( T G ) = Q ( T g 1 , . . . , T g k ) (denote it by F ( M , Q ( T G ))) Any embedding µ : Q ( T G ) → Q p defines a category F ( M , Q p ) Assume α ( H 1 ( M )) ⊂ G Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 14 / 27
Action on F ( M , Q p ) Want: p -adic family of bimodules Suggestion: Replace previous formula by � ± µ ( T E ( u ) ) µ ( T α ([ ∂ h u ]) ) t . y ( x 1 , . . . x m | x | x ′ 1 , . . . x ′ n ) �→ To define µ ( T α ([ ∂ h u ]) ) t ∈ Q p � t � , we need µ ( T α ([ ∂ h u ]) ) ≡ 1 ( mod p ) Definition (Poonen, Bell) Given v ∈ 1 + p Z p , define v t := � � t ( v − 1) i ∈ Q p � t � � i We can choose µ : Q ( T G ) → Q p such that µ ( T g ) ≡ 1 ( mod p ) Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 15 / 27
Action on F ( M , Q p ) Definition Let M Q p α ( L i , L j ) = ( Q p � t � ) � L i ∩ L j � . Define the structure maps via: � ± µ ( T E ( u ) ) µ ( T α ([ ∂ h u ]) ) t . y ( x 1 , . . . x m | x | x ′ 1 , . . . x ′ n ) �→ (finite sum). u varies among the discs as in figure below: Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 16 / 27
Action on F ( M , Q p ) Proposition M Q p also behaves like a “local group action”, i.e. α M Q p α | t = t 2 ⊗ F ( M , Q p ) M Q p α | t = t 1 ≃ M Q p α | t = t 1 + t 2 for small t 1 , t 2 ∈ Z p . Observe: In Z p , t 1 , t 2 are small iff t 1 , t 2 ∈ D p − n = p n Z p , for some n ≫ 0. As p n Z p is a group, one has an analytic p n Z p -action. Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 17 / 27
Relations of two (local) actions Let K = Q ( T fg : g ∈ G , f ∈ Z ( p ) ) ⊂ Λ, where Z ( p ) is the set of rationals with denominator not divisible by p (also Z ( p ) = Q ∩ Z p ). Extend µ : Q ( T G ) → Q p to K → Q p via µ ( T fg ) = µ ( T g ) f . We can define bimodules M K α | T f for f ∈ Z ( p ) over F ( M , K ) satisfying M K α | T f turns into M Λ α | T f under base change along K → Λ α | T f turns into M Q p M K α | t = f under base change along µ : K → Q p Corollary For f 1 , f 2 ∈ p n Z ( p ) (i.e. p-adically small) M K α | T f 2 ⊗ F ( M , Λ) M K α | T f 1 ≃ M K α | T f 1+ f 2 M Λ α | T f 2 ⊗ F ( M , Λ) M Λ α | T f 1 ≃ M Λ α | T f 1+ f 2 Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 18 / 27
Relations of two (local) actions Remark We know M Λ α | T f is “geometric” for small f , i.e. it corresponds to action of φ f α . By the corollary, this holds for any f ∈ p n Z ( p ) = p n Z p ∩ Q Yusuf Barı¸ s Kartal (Princeton) p-adic action on Fukaya categories August 4, 2020 19 / 27
Recommend
More recommend
