Arithmetic aspects of open de Rham spaces Dimitri Wyss Universit e - - PowerPoint PPT Presentation

Arithmetic aspects of open de Rham spaces

Dimitri Wyss

Universit´ e de Pierre et Marie Curie dimitri.wyss@imj-prg.fr

February 7, 2018

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 1 / 14

Motivation: Wild character varieties

Let Σ be a smooth projective curve over C and D = d

i=1 miai an

effective divisor. Fixing some additional data at each ai ∈ Σ Boalch constructs a smooth affine variety called wild character variety. MBetti = Moduli space of monodromy/Stokes data on Σ.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 2 / 14

Motivation: Wild character varieties

Let Σ be a smooth projective curve over C and D = d

i=1 miai an

effective divisor. Fixing some additional data at each ai ∈ Σ Boalch constructs a smooth affine variety called wild character variety. MBetti = Moduli space of monodromy/Stokes data on Σ. There are various intriguing conjectures on the cohomology of MBetti by [Hausel-Letellier-Rodriguez-Villegas], [Hausel-Mereb-Wong], [de Cataldo-Hausel-Migliorini], in particular a conjectural formula for its mixed Hodge polynomial MH(MBetti; x, y, t) =

• p,q,i

hp,q;i

c

(MBetti)xpyqti.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 2 / 14

Motivation: Purity conjecture

From the same initial data one can construct the moduli space MDR

• f meromorphic connections on Σ. The (wild) Riemann-Hilbert

correspondence defines a biholomorphism ν : MDR → MBetti.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 3 / 14

Motivation: Purity conjecture

From the same initial data one can construct the moduli space MDR

• f meromorphic connections on Σ. The (wild) Riemann-Hilbert

correspondence defines a biholomorphism ν : MDR → MBetti. If Σ = P1 there is an open subvariety M∗

DR ⊂ MDR, the open De

Rham space defined by considering only connections on the trivial bundle.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 3 / 14

Motivation: Purity conjecture

From the same initial data one can construct the moduli space MDR

• f meromorphic connections on Σ. The (wild) Riemann-Hilbert

correspondence defines a biholomorphism ν : MDR → MBetti. If Σ = P1 there is an open subvariety M∗

DR ⊂ MDR, the open De

Rham space defined by considering only connections on the trivial bundle.

Conjecture

For Σ = P1 the Riemann-Hilbert map ν induces an isomorphism H∗

c (M∗ DR, C) ∼

= PH∗

c (MBetti, C),

where PH∗

c denotes the pure part of H∗ c .

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 3 / 14

Formal types

We fix n ∈ N and write G = GLn and T ⊂ G for its standard maximal torus, g = Lie(G), t = Lie(T). For m ∈ N we also define Gm = G(C[[z]]/zm) and gm = g(C[[z]]/zm). Via the trace-residue pairing we get an identification g∨

m ∼

= z−mgm.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 4 / 14

Formal types

We fix n ∈ N and write G = GLn and T ⊂ G for its standard maximal torus, g = Lie(G), t = Lie(T). For m ∈ N we also define Gm = G(C[[z]]/zm) and gm = g(C[[z]]/zm). Via the trace-residue pairing we get an identification g∨

m ∼

= z−mgm. A formal type of order m ≥ 1 is a matrix of meromorphic 1-forms C = Cm dz zm + · · · + C1 dz z . . . with Ci ∈ t. If m ≥ 2 we require further that Cm is regular. In particular C defines an element in g∨

m.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 4 / 14

De Rham spaces

Fix an effective divisor D = d

i=1 miai on P1 and C = (C 1, . . . , C d) a

tuple of formal types C i of order mi at ai. Then we define the open De Rham space M∗

DR(C) =

     Meromorphic connections on the trivial bundle

• f rank n on P1 with poles along D

formally equivalent to C i at ai.     

• ∼hol

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 5 / 14

De Rham spaces

Fix an effective divisor D = d

i=1 miai on P1 and C = (C 1, . . . , C d) a

tuple of formal types C i of order mi at ai. Then we define the open De Rham space M∗

DR(C) =

     Meromorphic connections on the trivial bundle

• f rank n on P1 with poles along D

formally equivalent to C i at ai.     

• ∼hol

We will always assume that C is generic (an open condition on the C i

1’s), in which case M∗ DR(C) is smooth.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 5 / 14

De Rham spaces

A formal type C of order m defines an element in g∨

m, we write OC

for its Gm-coadjoint orbit and π : OC → g∨ for the projection. Then µ :

d

• i=1

OC i → g∨ (X1, . . . , Xd) →

• π(Xi)

is a moment map for simultaneous conjugation by G.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 6 / 14

De Rham spaces

A formal type C of order m defines an element in g∨

m, we write OC

for its Gm-coadjoint orbit and π : OC → g∨ for the projection. Then µ :

d

• i=1

OC i → g∨ (X1, . . . , Xd) →

• π(Xi)

is a moment map for simultaneous conjugation by G.

Proposition (Boalch)

For C generic we have M∗

DR(C) ∼

= µ−1(0)//G, where // denotes the affine GIT quotient.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 6 / 14

Example

Assume D = d

i=1 ai is a reduced divisor. Then C i ∈ g∨ and

π : OC i → g∨ is just the inclusion. Hence we have M∗

DR(C) =

• (X1, . . . , Xd) ∈
• OC i
• Xi = 0

G.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 7 / 14

Example

Assume D = d

i=1 ai is a reduced divisor. Then C i ∈ g∨ and

π : OC i → g∨ is just the inclusion. Hence we have M∗

DR(C) =

• (X1, . . . , Xd) ∈
• OC i
• Xi = 0

G. For the corresponding character variety let OC i ⊂ G be the conjugacy class containing exp(2π√−1C i). Then MBetti(C) =

• (g1, . . . , gd) ∈
• OC i
• gi = 1

G.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 7 / 14

Example

Assume D = d

i=1 ai is a reduced divisor. Then C i ∈ g∨ and

π : OC i → g∨ is just the inclusion. Hence we have M∗

DR(C) =

• (X1, . . . , Xd) ∈
• OC i
• Xi = 0

G. For the corresponding character variety let OC i ⊂ G be the conjugacy class containing exp(2π√−1C i). Then MBetti(C) =

• (g1, . . . , gd) ∈
• OC i
• gi = 1

G. This case was considered by [Hausel-Letellier-Rodriguez-Villegas], hence we will always assume that at least one pole has order ≥ 2.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 7 / 14

Main Result (jt with T. Hausel and M. Wong)

Theorem (Hausel-Wong-W.)

The E-polynomial of M∗

DR

E(M∗

DR; x, y) =

• p,q,i

(−1)ihp,q;i

c

(M∗

DR)xpyq,

agrees with the conjectural pure part (p = q and p + q = i) of MH(MBetti; x, y, t)

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 8 / 14

Main Result (jt with T. Hausel and M. Wong)

Theorem (Hausel-Wong-W.)

The E-polynomial of M∗

DR

E(M∗

DR; x, y) =

• p,q,i

(−1)ihp,q;i

c

(M∗

DR)xpyq,

agrees with the conjectural pure part (p = q and p + q = i) of MH(MBetti; x, y, t) This gives numerical evidence for the purity conjecture and the formula for MH(MBetti; x, y, t) proposed by [Hausel-Mereb-Wong].

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 8 / 14

Main Result (jt with T. Hausel and M. Wong)

Theorem (Hausel-Wong-W.)

The E-polynomial of M∗

DR

E(M∗

DR; x, y) =

• p,q,i

(−1)ihp,q;i

c

(M∗

DR)xpyq,

agrees with the conjectural pure part (p = q and p + q = i) of MH(MBetti; x, y, t) This gives numerical evidence for the purity conjecture and the formula for MH(MBetti; x, y, t) proposed by [Hausel-Mereb-Wong]. We conjecture H∗

c (M∗ DR) to be pure and Hodge-Tate and hence

E(M∗

DR; x, y) has positive coefficients.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 8 / 14

E-polynomials and finite fields

We can determine E(M∗

DR; x, y) using arithmetics of finite fields.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 9 / 14

E-polynomials and finite fields

We can determine E(M∗

DR; x, y) using arithmetics of finite fields.

Theorem (Katz)

Let X be a complex variety and f ∈ Z[t] such that |X(Fq)| = f (q) for ’sufficiently many’ finite fields Fq. Then E(X; x, y) = f (xy).

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 9 / 14

E-polynomials and finite fields

We can determine E(M∗

DR; x, y) using arithmetics of finite fields.

Theorem (Katz)

Let X be a complex variety and f ∈ Z[t] such that |X(Fq)| = f (q) for ’sufficiently many’ finite fields Fq. Then E(X; x, y) = f (xy). Clearly M∗

DR is defined over Fq as soon as the C i’s are and in this

case (assuming C generic) we have |M∗

DR(Fq)| = |µ−1(0)(Fq)|

|PGLn(Fq)| .

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 9 / 14

Fourier transform and convolution

We fix a non-trivial additive character Ψ : Fq → C×. Let V be a Fq-vector space and f : V → C a function. Then the Fourier transform F(f ) : V ∨ → C is defined as F(f )(y) =

• x∈V

f (x)Ψ(y, x), for y ∈ V ∨

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 10 / 14

Fourier transform and convolution

We fix a non-trivial additive character Ψ : Fq → C×. Let V be a Fq-vector space and f : V → C a function. Then the Fourier transform F(f ) : V ∨ → C is defined as F(f )(y) =

• x∈V

f (x)Ψ(y, x), for y ∈ V ∨ For f , g : V → C we define their convolution f ∗ g by f ∗ g(x′) =

• x∈V

f (x)g(x′ − x), for x′ ∈ V .

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 10 / 14

Fourier transform and convolution

We fix a non-trivial additive character Ψ : Fq → C×. Let V be a Fq-vector space and f : V → C a function. Then the Fourier transform F(f ) : V ∨ → C is defined as F(f )(y) =

• x∈V

f (x)Ψ(y, x), for y ∈ V ∨ For f , g : V → C we define their convolution f ∗ g by f ∗ g(x′) =

• x∈V

f (x)g(x′ − x), for x′ ∈ V . F(F(f ))(x) = |V |f (−x)

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 10 / 14

Fourier transform and convolution

We fix a non-trivial additive character Ψ : Fq → C×. Let V be a Fq-vector space and f : V → C a function. Then the Fourier transform F(f ) : V ∨ → C is defined as F(f )(y) =

• x∈V

f (x)Ψ(y, x), for y ∈ V ∨ For f , g : V → C we define their convolution f ∗ g by f ∗ g(x′) =

• x∈V

f (x)g(x′ − x), for x′ ∈ V . F(F(f ))(x) = |V |f (−x) F(f ∗ g) = F(f )F(g)

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 10 / 14

Fourier transform and convolution

For a formal type C we define #C : g∨ → C by #C(Y ) = |π−1(Y )(Fq)|, where π : OC → g∨ denotes the projection.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 11 / 14

Fourier transform and convolution

For a formal type C we define #C : g∨ → C by #C(Y ) = |π−1(Y )(Fq)|, where π : OC → g∨ denotes the projection. The definition of convolution implies |µ−1(0)(Fq)| = #C 1 ∗ · · · ∗ #C d(0).

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 11 / 14

Fourier transform and convolution

For a formal type C we define #C : g∨ → C by #C(Y ) = |π−1(Y )(Fq)|, where π : OC → g∨ denotes the projection. The definition of convolution implies |µ−1(0)(Fq)| = #C 1 ∗ · · · ∗ #C d(0). And by Fourier inversion #C 1 ∗ · · · ∗ #C d(0) = |g∨(Fq)|−1F d

• i=1

F(#C i)

• (0).

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 11 / 14

Fourier transform and convolution

For a formal type C of order 1, #C is the characteristic function of the coadjont orbit OC ⊂ g∨. Its Fourier transform is known by work

• f Springer.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 12 / 14

Fourier transform and convolution

For a formal type C of order 1, #C is the characteristic function of the coadjont orbit OC ⊂ g∨. Its Fourier transform is known by work

• f Springer.

Theorem (Hausel-Wong-W.)

If C is of order m ≥ 2, F(#C) is supported on semi-simple elements whose eigenvalues are in Fq. For such an X of type λ we have F(#C)(X) = κ(n, m, λ, q)

• t∈t∩OX

Ψ(C1, t).

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 12 / 14

Fourier transform and convolution

For a formal type C of order 1, #C is the characteristic function of the coadjont orbit OC ⊂ g∨. Its Fourier transform is known by work

• f Springer.

Theorem (Hausel-Wong-W.)

If C is of order m ≥ 2, F(#C) is supported on semi-simple elements whose eigenvalues are in Fq. For such an X of type λ we have F(#C)(X) = κ(n, m, λ, q)

• t∈t∩OX

Ψ(C1, t).

Corollary

If N ⊂ g denotes the nilpotent cone we have F(#C) = F(#C1)F(1N )m−1. This implies the main theorem.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 12 / 14

Final remarks

The previous theorem also holds in the Grothendieck ring of varieties with exponentials, but there is no formula for order 1 poles.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 13 / 14

Final remarks

The previous theorem also holds in the Grothendieck ring of varieties with exponentials, but there is no formula for order 1 poles. As soon as there is at least one pole of order ≥ 2, we can give a closed formula for E(M∗

DR(C); x, y) and it follows that they are

always non-empty [Hiroe] and connected.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 13 / 14

Final remarks

The previous theorem also holds in the Grothendieck ring of varieties with exponentials, but there is no formula for order 1 poles. As soon as there is at least one pole of order ≥ 2, we can give a closed formula for E(M∗

DR(C); x, y) and it follows that they are

always non-empty [Hiroe] and connected. As we conjecture that M∗

DR(C) has pure cohomology, this also

determines the Betti-numbers.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 13 / 14

Final remarks

The previous theorem also holds in the Grothendieck ring of varieties with exponentials, but there is no formula for order 1 poles. As soon as there is at least one pole of order ≥ 2, we can give a closed formula for E(M∗

DR(C); x, y) and it follows that they are

always non-empty [Hiroe] and connected. As we conjecture that M∗

DR(C) has pure cohomology, this also

determines the Betti-numbers. We also give a quiver-like description of M∗

DR(C) in the spirit of

Crawley-Boevey, but the symmetry groups are GLn(C[[z]]/zm), hence non-reductive.

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 13 / 14

Thank you!

Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 14 / 14