Categories and Filtrations

Transcription

1 Categories and Filtrations Ludmil Katzarkov University of Miami July 10, / 1

2 Overview Navier-Stokes problems and Denaturation of DNA Burnside problem Uniformization 2 / 1

3 Navier-Stokes For V - velocity, P - pressure: V t + (V )V + 1 P = ν V + f (x) P V =0 This produces traveling waves. L. D. Landau da dt =aµ + b A 2 + h.o.t. 3 / 1

4 Denaturation of DNA double stranded single stranded denaturation Temperature T This is described by a one-dimensional lattice and the following lattice ODE. d 2 u n dt 2 + W (u n ) = V (u n+1 u n ) V (u n u n 1 ) W - on-site potential V - interaction potential 4 / 1

5 PDE Method of Central Manifolds ODE 1 d 2 y J 2 dx 2 = V (y(x + 1) y(x)) V (y(x) y(x 1)) x = n t 2 u n (x) = y 5 / 1

6 Recall the method of central manifolds. { Central manifolds of equilibrium points = orbits } neither attraction of stable manifold nor repulsion of unstable manifold Central manifold is given by the linearization of eigenvalues λ i with Reλ i = 0 or λ i = 0. λ i = 0 slow manifold spanned by eigenvectors 6 / 1

7 central long term dynamics manifold t = Recall If we have a dynamical system dx dt = f (x) linearization dx dt = Ax A 0 - eigenvectors with λ = 0 A 0 tangent to slow manifold 7 / 1

8 Burnside theory Recall that Burnside group is defined as F n is a free group with n generators. B(n, d) = {F n x d = 1, x F n } B(n, 2) = Z n 2 B(n, 3) = finite with order 3 C, where C depends on the nilpotency class B(n, 4) = finite (Sanov) B(n, 5) =? (n 2) B(n, 6) = finite (M. Hall) 8 / 1

9 Question Can we find n 2 and d so that B(n, d) is infinite? Theorem (Adian, Novikov) infinite B(n, d), where d is odd and d Theorem (Olshanski) Let Γ be a hyperbolic group. Then Γ(d) is infinite for d >> 0. 9 / 1

10 Example Γ = π 1 (C), C - Riemann surface Theorem (Zelmanov) B(n, d) is not residually finite if infinite. Question (Zelmanov) Can we find n 1 > n 2 so that B(n 1, m) is infinite and B(n 2, m) is finite? 10 / 1

11 Uniformization Recall: Theorem (Riemann) Let X be a one-dim smooth projective variety. Then X = C, P 1, D. If dim C X = 2, then X = C C, P 2, D D,... Question (Shafarevich) X is hol. convex for X a smooth projective variety. Recall: Definition A complex space M is hol. convex if sequence of q 1,..., q n without a limit point, a hol. function unbounded on it. 11 / 1

12 Example Y Y 1 S Y 2 Y t Let Im(H 1 (Y 1, Z) H 1 (S, Z)) = 0 Im(H 1 (Y 2, Z) H 1 (S, Z)) = 0 and Im(H 1 (Y, Z) H 1 (S, Z)) = q 1 q2 Ỹ S is not holomorphically convex. 12 / 1

13 But strictness of MHS implies Im(H 1 (Y 1, Z) H 1 (S, Z)) = 0 Im(H 1 (Y 2, Z) H 1 (S, Z)) = 0 and Im(H 1 (Y, Z) H 1 (S, Z)) = 0 13 / 1

14 Theorem (K) H 1 (S) π Malcer 1 (S) Let S be a smooth projective variety and π 1 (S) nilpotent. Then S is holomorphically convex. Theorem (EKPR) H 1 (S, Z) π Malcer 1 (S) Let X be a smooth projective variety and π 1 (X) GL(n, C). Then X is holomorphically convex. Remark This technique could lead to π 1 (X) residually finite. 14 / 1

15 Uniformization Burnside problem Γ n - vanishing cycle S n : 1 C Base change Γ - vanishing cycle n : 1 Y S 1 π 1 (Y )/ <Γ> π 1 (S ) π 1 (C) 1 C 15 / 1

16 Theorem (Zelmanov) π 1 (S ) is not residually finite. g 1 g 1 g g 1 < g If Γ g1 (m) is finite but Γ g (m) is infinite. S is not holomorphically convex. 16 / 1

17 Question (Zelmanov) related Question (Shafarevich) 17 / 1

18 Uniformization and Central Manifolds Hodge theory of central manifolds Vector or Higgs bundles V 1 V 2 F V 3 V 4 Example Solutions of H t = ΛF + x center manifold ODE 18 / 1

19 In fact: Theorem (Haiden, Katzarkov, Kontsevich,Pandit) This ODE is connected with a quiver: hi ϕ i V 1 V 2 V 3 V 4 J i hi 1 dh i dt = α:i j α:i j hi 1 ϕ + α h j ϕ α ϕ α hj 1 ϕ + α h i 19 / 1

20 Theorem (HKKP) The asymptotics of ODE define a filtration on the central manifold Z. The dimension of Z is from d = to k k2 d. The asymptotics are The define a filtration F t. Rt + R log(t) + + R log(log( (t))). 20 / 1

21 Properties 1 F satisfies Griffiths transversality. 2 F satisfies functoriality. 3 F satisfies strictness. 4 The monodromy action on Z is semi-simple. 21 / 1

22 Theorem (KP) Monge-Ampère Equation ODE. P(O + E) C X X t metric h 1 h 2 quiver as before 22 / 1

23 Semistable Semistable Kähler metric on total space T (E) V 1 V 2 V 3 V 4 E V 1 h 1 h 2 ext V 2 metrics with very small vectors V 1 + V 2 23 / 1

24 stable V 1 T (V 1 ) KE ext V 2 T (V 2 ) KE quiver of semistable metrics quiver of KE metrics 24 / 1

25 Recall we have a HMS. V 1 V 2 V 3 V 4 HMS V / 1

26 π 1 (Y i ) Fuk FS (Y i ) = MF Mean curvature flow central manifold with all the properties: 1 Functoriality 2 Strictness 3 Semi-stability Expectation 1, 2, 3 imply the Shafarevich conjecture. 26 / 1

27 Central manifold and Zelmanov conjecture We have the following: Theorem (HKKP) Consider R n as C n /S 1n and V = grad f. Let f be a convex function defined by where u α, u R n. Then f = C α < u α, V > + < u, V >, V = V 0 t + V 1 log(t) + + V n log(log( (t))) + O(1). 27 / 1

28 Remark For more complicated functions, we get different solutions. Theorem (HKKP) In the above situation, we have a central manifold Z, with 1 dim Z n. Theorem (HKKP) In the case G/K, we have the same for the gradient equation. 28 / 1

29 Conjecture The same holds for CAT(W ). If this is proven, we can directly work with Burnside groups. If B(n 1, m) does not have central manifold, B(n 2, m) does not have either. No jumps in cardinality. 29 / 1

30 We have a sequence (**): PDE ODE Z - central manifold CATEGORY Example Donaldson-YM equation Mean curvature equation Monge-Ampère equation 30 / 1

31 Definition PDE satisfying (**) is called categorical. Question Find a sufficient condition so that PDE is categorical. This gives additional structures on solutions e.g. satisfying: Transversality Functoriality Semi-simplicity 31 / 1

32 Recall: Classical SL 2 nilpotent orbit theorem says that for f : D U, a nilpotent orbit O s.t. O f (D) O approximates f (D). 32 / 1

33 Consider a spherical functor S. Let D b (Q) be a quiver category and ḣ the flow over metrized objects. Conjecture The superposition of ḣ and the flow of S define a new filtration. 33 / 1

34 Example 1 D b (A 1 ) = {V }, N-PL. This is the standard Hodge filtration. Example 2 S-localization. We get a bifurcation diagram for ḣ. Example 3 Families of matrix factorizations. 34 / 1

35 Multi SL 2 -Nilpotent Orbit Teorem We have the following parallel: V 1 V 2 V 3 V 4 A n N S A n A n A n monodromy invariant under monodromy spherical functors 35 / 1

36 We have the classical SL 2 -Nilpotent Orbit Theorem: f : D U N Approximation We also have: N 1 N 2 N 1 + N 2 36 / 1

37 We combine two flows: 1 The flow given by Nahm s equation - NE. 2 The Quiver equation - QE. J i hi 1 dh i dt = α:i j hi 1 ϕ + α h j ϕ α α:i j ϕ(ne, QE) ϕ α hj 1 ϕ + α h i NE QE Conjecture We get a new refined filtration. 37 / 1

38 In terms of stability conditions, we get: Γ,h e h e (f ) dzdh a k z k b n log(z)c m (log(log( (h)))) k,n,m m Conjecture The refined filtration depends on Ospec D b (C, A n ). 38 / 1

39 We give a geometric example. C C N QE Proposition We have a refined filtration on D b (A n, C). 39 / 1

40 In general, we have: 1 A flow C which defines ODE (associated with a quiver). 2 Several (spherical) functors S 1, S 2,..., S n. This defines: A) A refined filtration. B) A new Futaki type of invariant minimizing the refined filtration. 40 / 1

41 Example 1 Mean curvature + 2 spherical functors S 1 S 2 FS 1 FS 2 S 2 /Γ n localization We get a new filtration with strictness and functoriality. The initial flow is: ċ = d Arg(Ω c ) ω 1 41 / 1

42 Example 2 Calabi Flow M 1 M 2 M k S - test configuration spherical functor Flow: ω = ω Ric(ω). Refined Futaki invariant for refined filtration. 42 / 1

43 Example 3 V 1 V i V n W j S 1 degenerations S 2 Flow: Yang Mills Higgs - Semi-stable degenerations - Functoriality - Strictness 43 / 1

44 The End 44 / 1