Pranav Pandit joint work with Fabian Haiden, Ludmil ... · Pranav Pandit joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich University of Vienna June 7, 2018 Pranav

Gradient flows, iterated logarithms, and semistability

Pranav Pandit

joint work with Fabian Haiden, Ludmil Katzarkov,and Maxim Kontsevich

University of Vienna

June 7, 2018

Pranav Pandit (U Vienna) Gradient flows and iterated logs June 7, 2018 1 / 16

This talk is based on:

- arXiv:1706.01073

- arXiv:1802.04123

Goals:

Describe the asymptotic behavior of the flows discussed in theprevious talk (in special cases)

Describe a canonical refinement of the Harder-Narasimhan filtration,and its relation to the asymptotic behavior of the flow

Category Fuk(X , ω) Rep(Q)

Object Lag upto isotopy (Evv , Tαα∈Arr(Q))

Metrized object Lagrangian (Ev , hv ), hv hermitian metric

Kahler data Ω P :=∑

zvprv +∑

[T ∗α,Tα]hol vol form zv ∈ H; v ∈ Q0

Flow F L = ArgΩL h−1h = ArgP

Mass M M(L) =∫L |Ω| M = |P|

Central charge Z (L) =∫L Ω Z =

∑zvχ(Ev )

Kahler potential dSC(f ) =∫L Ωf SC =

∑log det hv +

∑T ∗αTα

Harmonic metric Fixed points of F Fixed points of F/rescaling= Crit(SC) = Crit(SC)

= special Lagrangian

Amplitude sup / inf Arg Ω|L sup / inf Spec Arg P

DUY theorem ?? King’s theorem

Good features (some proven): (i) Mass, Amp decrease with flow; (ii) BPSinequality |Z | ≤ M and (iii) “properness” of mass.

Recall:

- There is a flow on the space of “metrized objects”.

- Fixed points of the flow on metrics upto rescaling are harmonicmetrics. The underlying objects should be polystable.

- The “speed” of the rescaling action gives the slope/phase of thepolystable object.

- For general objects, the flow should “decompose” the object into itspolystable constituents, equipped with harmonic metrics.

The Harder-Narasimhan filtration only decomposes an object intosemistable constituents.

Basic problem: Describe the decomposition (induced by the flow) of asemistable object into polystable pieces.

C stable ∞-category equipped with Bridgeland stability conditionCssθ θ∈R,Z : K0(C)→ C.

For each θ ∈ R, Cssθ is an Artinian abelian category equipped with ahomomorphism

X := exp(−iθ)Z : K0(Cssθ )→ R

which is positive on non-zero objects.

Natural filtrations

A Artinian abelian category; E ∈ E0 = E0 ⊂ E1 ⊂ E2 ⊂ . . .En = E filtration.

1 Socle: Ek is maximal containing Ek−1 such that Ek/Ek−1 issemisimple (one extreme).

2 Cosocle: Ek is minimal contained in Ek+1 such that Ek+1/Ek issemisimple (another extreme).

3 Let A be the category of pairs (V ,N) consisting of a vector space Vand a nilpotent endomorphism N. Then there is a unique filtrationlabelled by half-integers such that

Nk : Grk/2V → Gr−k/2V

is an isomorphism for all k. (Balanced; this is the weight/Lefschetzfiltration from mixed Hodge theory).

Natural filtrations

Balanced filtration

Theorem (Haiden-Katzarkov-Kontsevich-P.)

A Artinian abelian category, X : K0(A)→ R, positive on non-zero objects,E ∈ A. Then there exists a unique R-filtration

0 = E0 ⊂ E1 ⊂ . . . ⊂ En = E

labelled byλ1 < λ2 < . . . < λn

characterized by

1 Paracomplementedness: Ej/Ek−1 is semisimple whenever λj − λk < 1.

2 Balancing condition:∑

j λjX (Ej/Ej−1) = 0

3 For any Fj with Ej−1 ⊂ Fj ⊂ Ej such that Fj/Fk is semisimple forλj − λk ≤ 1, we have ∑

λjX (Fj/Ej−1) ≤ 0

Balanced filtration

0 = E0 ⊂ E1 ⊂ . . . ⊂ En = E

characterized by

Balanced filtration

0 = E0 ⊂ E1 ⊂ . . . ⊂ En = E

characterized by

Balanced filtration

0 = E0 ⊂ E1 ⊂ . . . ⊂ En = E

characterized by

Iterated balanced filtration and asymptotics

The last condition can be formulated as stability of the filtration FλEconsidered as an object in an auxillary abelian category

- Iterating the construction of the previous theorem gives a canonicalfiltration of E labelled by R∞ equipped with the lexicographic order.

- When A = Rep(Q) this filtration controls the asymptotic behavior ofthe gradient flow on the space of metrics

log |h(t)| = λ1logt + λ2loglogt + . . .+ λnlog (n)t + O(1)

on the (λ1, λ2, . . . , λn) piece of the filtration.

Meta-principle: The asymptotic dynamics of geometric flows (e.g., meancurvature flow, Yang-Mills flow) can be reduced to the finite dimensionalquiver situation using the theory of center manifolds.

Dynamical systems from quiver representations

Q = (Q0,Q1) quiver; Q0 vertices, Q1 arrows.

Quiver representation:Vertex i 7→ Ei vector space.Arrow α : i → j 7→ Tα : Ei → Ej operator.

Metrized quiver representation: hermitian metric hi on Ei

adjoint operator T ∗α : Ej → Ei .

Choosing “masses” (mi )i∈Q0 flow on the space of metrics.

mih−1i hi =

∑α:i→j

h−1i T ∗αhjTα −

∑α:k→i

Tαh−1j T ∗αhi

This is asymptotic to the previous flow when the central charge takesvalues in a ray (the positive reals).

X compact Riemann surface, Kahler form ω, E finite dimensionalholomorphic vector bundle on X , and h a hermitian metric on E . Considerthe flow:

h−1∂th = −2i(ΛF − λ)

There is a canonical filtration F kE =: Ek on E labelled by

βk ∈ Rt ⊕ R log t ⊕ R log log t ⊕ · · · ' R∞

such that

|| log h(t)|Ek|| = βk + O(1)

Ek/Ek−1 is a sum of stable bundles of slope µk given by

βk = 4π

(∫Xω

(µk − µ(E ))t + . . .

Iterated Logarithms from a dynamical system

log(1)(t) := log tlog(n)(t) := log(log(n−1) t)

x1 = e−x1 ⇒ x1 = log t

x2 = e−(x1+x2) ⇒ x2 = 1t e−x2 ⇔ d

d log t x2 = e−x2 ⇒ x2 = log(2) t

x3 = e−(x1+x2+x3) ⇔ dd log t x3 = e−(x2+x3) ⇒ x3 = log(3) t

...xn = e−(x1+x2+...xn) ⇔ d

d log t xn = e−(x2+...+xn) ⇒ xn = log(n) t

The original system of n differential equations reduces to an identicalsystem of n-1 equations in one less variable upon passing to logarithmictime x1 = s := log t. .

x1 = e−x1 ⇒ x1 = log t

x2 = e−(x1+x2) ⇒ x2 = 1t e−x2 ⇔ d

d log t x2 = e−x2 ⇒ x2 = log(2) t

...xn = e−(x1+x2+...xn) ⇔ d

x1 = e−x1 ⇒ x1 = log t

x2 = e−(x1+x2) ⇒ x2 = 1t e−x2

⇔ dd log t x2 = e−x2 ⇒ x2 = log(2) t

...xn = e−(x1+x2+...xn) ⇔ d

x1 = e−x1 ⇒ x1 = log t

x2 = e−(x1+x2) ⇒ x2 = 1t e−x2 ⇔ d

d log t x2 = e−x2

⇒ x2 = log(2) t

...xn = e−(x1+x2+...xn) ⇔ d

x1 = e−x1 ⇒ x1 = log t

x2 = e−(x1+x2) ⇒ x2 = 1t e−x2 ⇔ d

d log t x2 = e−x2 ⇒ x2 = log(2) t

...xn = e−(x1+x2+...xn) ⇔ d

x1 = e−x1 ⇒ x1 = log t

x2 = e−(x1+x2) ⇒ x2 = 1t e−x2 ⇔ d

d log t x2 = e−x2 ⇒ x2 = log(2) t

x3 = e−(x1+x2+x3) ⇔ dd log t x3 = e−(x2+x3)

⇒ x3 = log(3) t...xn = e−(x1+x2+...xn) ⇔ d

x1 = e−x1 ⇒ x1 = log t

x2 = e−(x1+x2) ⇒ x2 = 1t e−x2 ⇔ d

d log t x2 = e−x2 ⇒ x2 = log(2) t

...xn = e−(x1+x2+...xn)

⇔ dd log t xn = e−(x2+...+xn) ⇒ xn = log(n) t

x1 = e−x1 ⇒ x1 = log t

x2 = e−(x1+x2) ⇒ x2 = 1t e−x2 ⇔ d

d log t x2 = e−x2 ⇒ x2 = log(2) t

...xn = e−(x1+x2+...xn) ⇔ d

d log t xn = e−(x2+...+xn)

⇒ xn = log(n) t

x1 = e−x1 ⇒ x1 = log t

x2 = e−(x1+x2) ⇒ x2 = 1t e−x2 ⇔ d

d log t x2 = e−x2 ⇒ x2 = log(2) t

...xn = e−(x1+x2+...xn) ⇔ d

1 dim representations

Input:

1 Directed acyclic graph G = (G0,G1)

2 Masses (mi )i∈G0 ∈ RG0>0 metric

∑mi (dxi )

2 on RG0 .

3 Weights (cα)α∈G1 ∈ RG1>0 action functional S : RG0 → R

S(x) :=∑α:i→j

cαexj−xi

gradient flow of S with respect to metric

mi xi =∑α:i→j

cαexj−xi −∑α:k→i

cαexi−xk

Claim: The asymptotic behavior of these dynamical systems is controlledby iterated logarithms.

Input:

2 Masses (mi )i∈G0 ∈ RG0>0

metric∑

mi (dxi )2 on RG0 .

S(x) :=∑α:i→j

cαexj−xi

mi xi =∑α:i→j

cαexi−xk

Input:

∑mi (dxi )

2 on RG0 .

S(x) :=∑α:i→j

cαexj−xi

mi xi =∑α:i→j

cαexi−xk

Input:

∑mi (dxi )

2 on RG0 .

3 Weights (cα)α∈G1 ∈ RG1>0

action functional S : RG0 → R

S(x) :=∑α:i→j

cαexj−xi

mi xi =∑α:i→j

cαexi−xk

Input:

∑mi (dxi )

2 on RG0 .

S(x) :=∑α:i→j

cαexj−xi

mi xi =∑α:i→j

cαexi−xk

Input:

∑mi (dxi )

2 on RG0 .

S(x) :=∑α:i→j

cαexj−xi

mi xi =∑α:i→j

cαexi−xk

An example

m1• c // •m2

Gradient flow:

m1x1 = cex2−x1

m2x2 = −cex2−x1

Solution:

x1(t) =m2

m1 + m2log t + log c1

x2(t) = − m1

m1 + m2log t + log c2

Wherec2

c(m1 + m2)

General caseAnsatz: xi = vi log t + bi ; vi , bi ∈ R.

t=∑α:i→j

cαtvj−vi ebj−bi −∑α:k→i

cαtvi−vk ebi−bk

Interested in asymptotic behavior of xi upto O(1), so look for solutionsxi (t) of the differential equation correct upto terms in L1(R).

=⇒ only powers t≤−1 in RHS=⇒ vi − vj ≥ 1 if ∃α : i → j ⇔ (vi )i is an admissible grading (defn)

comparing coefficients=⇒ mivi =

∑α:i→j cαuα −

∑α:k→i cαuα, and uα = 0 if vi − vj > 1.

Here uα := ebj−bi ∈ R>0 for α : i → j .Key observation: uα’s are Lagrange multipliers for a convex optimizationproblem.

t=∑α:i→j

=⇒ only powers t≤−1 in RHS

=⇒ vi − vj ≥ 1 if ∃α : i → j ⇔ (vi )i is an admissible grading (defn)

t=∑α:i→j

Here uα := ebj−bi ∈ R>0 for α : i → j .

Key observation: uα’s are Lagrange multipliers for a convex optimizationproblem.

t=∑α:i→j

Balanced weight grading

C := set of admissible gradings ⊂ RG0 (convex body).M : C → R≥0; M((vi )i ) :=

∑i miv

2i convex function.

Proposition

1 (vi )i minimizes M.

2 ∃uα ≥ 0 α ∈ G1 satisfying

mivi =∑α:i→j

cαuα −∑α:k→i

cαuα

3 (i)∑

i mivi = 0 and (ii)∑

i∈E mivi ≤ 0 for all E ⊂ G0 such thati ∈ G0 and ∃α : i → j =⇒ j ∈ E “slope semistability”.

Unique grading satisying (1)-(3) is called the balanced weight grading.

∑i miv

2i convex function.

Proposition

mivi =∑α:i→j

cαuα

3 (i)∑

i mivi = 0

and (ii)∑

∑i miv

2i convex function.

Proposition

mivi =∑α:i→j

cαuα

3 (i)∑

i mivi = 0 and (ii)∑

Iterated balanced weight grading

There are walls in the parameter space of mi ’s (recall: (mi )i ∈ RG0>0

parametrize certain metrics on RG0) along which the some of the uα = 0for α : i → j with vi − vj = 1.Simplest example:

m1•u12

m3•u32

##m2• m4•

It turns out u32 = m2m3−m1m4∑i mi

Wall: m2m3 = m1m4..

Iterate procedure along a certain subgraph asymptotics governed byiterated logarithms along wall.

Iterated balanced weight grading

There are walls in the parameter space of mi ’s (recall: (mi )i ∈ RG0>0

parametrize certain metrics on RG0) along which the some of the uα = 0for α : i → j with vi − vj = 1.Simplest example:

m1•u12

m3•u32

##m2• m4•

It turns out u32 = m2m3−m1m4∑i mi

Wall: m2m3 = m1m4..

Iterate procedure along a certain subgraph asymptotics governed byiterated logarithms along wall.

Pranav Pandit joint work with Fabian Haiden, Ludmil ... · Pranav Pandit joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich University of Vienna June 7, 2018 Pranav

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