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 m i a i an effective divisor. Fixing some additional data at each a i ∈ Σ Boalch constructs a smooth affine variety called wild character variety . M Betti = 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 m i a i an effective divisor. Fixing some additional data at each a i ∈ Σ Boalch constructs a smooth affine variety called wild character variety . M Betti = Moduli space of monodromy/Stokes data on Σ . There are various intriguing conjectures on the cohomology of M Betti by [Hausel-Letellier-Rodriguez-Villegas], [Hausel-Mereb-Wong], [de Cataldo-Hausel-Migliorini], in particular a conjectural formula for its mixed Hodge polynomial � h p , q ; i ( M Betti ) x p y q t i . MH ( M Betti ; x , y , t ) = c p , q , i 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 M DR of meromorphic connections on Σ. The (wild) Riemann-Hilbert correspondence defines a biholomorphism ν : M DR → M Betti . 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 M DR of meromorphic connections on Σ. The (wild) Riemann-Hilbert correspondence defines a biholomorphism ν : M DR → M Betti . If Σ = P 1 there is an open subvariety M ∗ DR ⊂ M DR , 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 M DR of meromorphic connections on Σ. The (wild) Riemann-Hilbert correspondence defines a biholomorphism ν : M DR → M Betti . If Σ = P 1 there is an open subvariety M ∗ DR ⊂ M DR , the open De Rham space defined by considering only connections on the trivial bundle. Conjecture For Σ = P 1 the Riemann-Hilbert map ν induces an isomorphism DR , C ) ∼ H ∗ c ( M ∗ = PH ∗ c ( M Betti , 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 = GL n and T ⊂ G for its standard maximal torus, g = Lie( G ), t = Lie( T ). For m ∈ N we also define G m = G ( C [[ z ]] / z m ) and g m = g ( C [[ z ]] / z m ). Via the trace-residue pairing we get an identification m ∼ g ∨ = z − m g m . Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 4 / 14

Formal types We fix n ∈ N and write G = GL n and T ⊂ G for its standard maximal torus, g = Lie( G ), t = Lie( T ). For m ∈ N we also define G m = G ( C [[ z ]] / z m ) and g m = g ( C [[ z ]] / z m ). Via the trace-residue pairing we get an identification m ∼ g ∨ = z − m g m . A formal type of order m ≥ 1 is a matrix of meromorphic 1-forms dz dz C = C m z m + · · · + C 1 z . . . with C i ∈ t . If m ≥ 2 we require further that C m 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 i =1 m i a i on P 1 and C = ( C 1 , . . . , C d ) a Fix an effective divisor D = � d tuple of formal types C i of order m i at a i . Then we define the open De Rham space  Meromorphic connections on the trivial bundle     of rank n on P 1 with poles along D  � M ∗ DR ( C ) = ∼ hol formally equivalent to C i at a i .     Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 5 / 14

De Rham spaces i =1 m i a i on P 1 and C = ( C 1 , . . . , C d ) a Fix an effective divisor D = � d tuple of formal types C i of order m i at a i . Then we define the open De Rham space  Meromorphic connections on the trivial bundle     of rank n on P 1 with poles along D  � M ∗ DR ( C ) = ∼ hol formally equivalent to C i at a i .     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 O C for its G m -coadjoint orbit and π : O C → g ∨ for the projection. Then d � O C i → g ∨ µ : i =1 � ( X 1 , . . . , X d ) �→ π ( X i ) 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 O C for its G m -coadjoint orbit and π : O C → g ∨ for the projection. Then d � O C i → g ∨ µ : i =1 � ( X 1 , . . . , X d ) �→ π ( X i ) is a moment map for simultaneous conjugation by G . Proposition (Boalch) For C generic we have DR ( C ) ∼ M ∗ = µ − 1 (0) // G , where // denotes the affine GIT quotient. Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 6 / 14

Example i =1 a i is a reduced divisor. Then C i ∈ g ∨ and Assume D = � d π : O C i → g ∨ is just the inclusion. Hence we have � � � � M ∗ � � ( X 1 , . . . , X d ) ∈ O C i DR ( C ) = X i = 0 G . � � Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 7 / 14

Example i =1 a i is a reduced divisor. Then C i ∈ g ∨ and Assume D = � d π : O C i → g ∨ is just the inclusion. Hence we have � � � � M ∗ � � ( X 1 , . . . , X d ) ∈ O C i DR ( C ) = X i = 0 G . � � For the corresponding character variety let O C i ⊂ G be the conjugacy class containing exp(2 π √− 1 C i ). Then � � � � � � M Betti ( C ) = ( g 1 , . . . , g d ) ∈ g i = 1 G . O C i � � Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 7 / 14

Example i =1 a i is a reduced divisor. Then C i ∈ g ∨ and Assume D = � d π : O C i → g ∨ is just the inclusion. Hence we have � � � � M ∗ � � ( X 1 , . . . , X d ) ∈ O C i DR ( C ) = X i = 0 G . � � For the corresponding character variety let O C i ⊂ G be the conjugacy class containing exp(2 π √− 1 C i ). Then � � � � � � M Betti ( C ) = ( g 1 , . . . , g d ) ∈ g i = 1 G . O C i � � 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 ∗ � ( M ∗ ( − 1) i h p , q ; i DR ) x p y q , DR ; x , y ) = c p , q , i agrees with the conjectural pure part (p = q and p + q = i) of MH ( M Betti ; 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 ∗ � ( M ∗ ( − 1) i h p , q ; i DR ) x p y q , DR ; x , y ) = c p , q , i agrees with the conjectural pure part (p = q and p + q = i) of MH ( M Betti ; x , y , t ) This gives numerical evidence for the purity conjecture and the formula for MH ( M Betti ; 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 ∗ � ( M ∗ ( − 1) i h p , q ; i DR ) x p y q , DR ; x , y ) = c p , q , i agrees with the conjectural pure part (p = q and p + q = i) of MH ( M Betti ; x , y , t ) This gives numerical evidence for the purity conjecture and the formula for MH ( M Betti ; 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 ( F q ) | = f ( q ) for ’sufficiently many’ finite fields F q . Then E ( X ; x , y ) = f ( xy ) . Dimitri Wyss (Paris 6) Riemann-Hilbert Correspondences 2018 February 7, 2018 9 / 14