The Riemann existence theorem d k d , P ( z, z ) = a k ( z ) a - PDF document

1 2 The Riemann existence theorem � d � k d � , P ( z, ∂ z ) = a k ( z ) a k ∈ C [ z ] , a d �≡ 0 Irregular Hodge theory d z 0 S = { z | a d ( z ) = 0 } sing. set (assumed � = ∅ ) Claude Sabbah Associated linear system Centre de Math´ ematiques Laurent Schwartz     ´ Ecole polytechnique, CNRS, Universit´ e Paris-Saclay u 1 u 1 d Palaiseau, France . .     . . A ( z ) ∈ End( C ( z ) d ) . . ( ∗ )  = A ( z )  ,   d z u d u d Programme SISYPH ANR-13-IS01-0001-01/02 � Monodromy representation of the solution vectors by analytic continuation ρ : π 1 ( C � S, z o ) − → GL d ( C ) Irregular Hodge theory – p. 1/23 Irregular Hodge theory – p. 2/23 3 4 Rigid irreducible representations The Riemann existence theorem Assume ρ is irreducible : � � � − 1 ) ⇒ ( T s ∈ GL d ( C )) s ∈ S (and T ∞ := ρ ⇐ s T s cannot put all T s in a upper block-triang. form simultaneously Conversely , any ρ (any finite S ) comes from a and rigid : system ( ∗ ) s.t., ∀ s ∈ S ∪ ∞ , ∃ formal merom. gauge transf. → at most simple pole (i.e., reg. sing. ): then ρ ′ ∼ ρ if T ′ s ∼ T s ∀ s ∈ S ∪ ∞ , � s.t. � ∃ M ( z − s ) ∈ GL d C ( ( z − s ) ) and assume ∀ s ∈ S ∪ ∞ , � � M − 1 AM + M − 1 M ′ ( z − s ) · ∈ End( C [ [ z − s ] ]) . z ∀ λ eigenvalue of T s , | λ | = 1 Proof: Near s ∈ S , this amounts to finding ⇒ More structure on the solution to the Riemann C s ∈ End( C d ) s.t. T s = e − 2 πiC s . Then existence th. A ( z ) := C s / ( z − s ) has monodromy T s around s . Globalization: non-explicit procedure. Irregular Hodge theory – p. 3/23 Irregular Hodge theory – p. 4/23 Variations of pol. Hodge structure 5 Variations of pol. Hodge structure 6 T HEOREM (Deligne 1987, Simpson 1990): T HEOREM (Deligne 1987, Simpson 1990): ∃ ! var. of polarized Hodge structure (wt. = 0 ) adapted to ρ ∃ ! var. of polarized Hodge structure (wt. = 0 ) adapted to ρ G z : pos. def. Herm. d × d matrix, C ∞ w.r.t. z ∈ C� S ⇒ Numbers f p = rk F p H z attached to ρ . ∀ z ∈ C� S : Hodge decomp. Moreover (Griffiths), ⊥ � C d = = H p p H p H − p z , z z s ∈ S ] d = O ( C � S ) d C [ z, ( z − s ) − 1 G -mod. growth . z : C ∞ & possibly not hol. but z �→ H p z �→ F p H z := � p ′ � p H p ′ z holomorphic and � d � · F p H z ⊂ F p − 1 H z d z + A G z s. t. � � G z | H p := ( − 1) p G | H p , then ∂ z � G z · � G − 1 = t A ( z ) . z Irregular Hodge theory – p. 5/23 Irregular Hodge theory – p. 6/23 Hypergeom. differential eqns 7 Confluent hypergeom. diff. eqns 8 � 0 � α 1 � · · · � α d < 1 , Given α i � = β j ∀ i, j . � � � � d ′ z d d z d � � 0 � β 1 � · · · � β d < 1 , P ( z, ∂ z ) := d z − α i − z d z − β j i =1 j =1 � � � � � d z d � d z d P ( z, ∂ z ) := d z − α i − z d z − β j with d ′ < d ⇒ S = 0 and 0 is an irreg. sing. i =1 j =1 ( ∞ = reg. sing). S = { 0 , 1 } . Beukers & Heckman: ρ is irreducible rigid , with Riemann existence th. breaks down for irreg. sing. λ = e − 2 π i α or e 2 π i β . Need Stokes data to reconstruct the differential eqn Set ℓ j = # { i | α i � β j } − j from sols. T HEOREM (R. Fedorov, 2015): � Riemann-Hilbert-Birkhoff correspondence. f p = # { j | ℓ j � p } mixed: F 1 = 0 , F 0 = O ( C � S ) d ⇒ unitary conn. unmixed: 0 = F d ⊂ · · · ⊂ F 0 = O ( C � S ) d . Irregular Hodge theory – p. 7/23 Irregular Hodge theory – p. 8/23

Confluent hypergeom. diff. eqns 9 10 Harmonic metrics Given: � � � � a diff. system d � d ′ z d � d z d d z + A ( z ) , A ( z ) ∈ End( C ( z ) d ) , P ( z, ∂ z ) := d z − α i − z d z − β j i =1 j =1 pole set = S ⊂ C . with d ′ < d . G z : any pos. def. Herm. mtrx, C ∞ w.r.t. z ∈ C� S . G z d × d , C ∞ w.r.t. z , s.t. Same condition on α, β ’s ⇒ irreducible and rigid : Then ∃ ! A ′ G z , A ′′ irreducible : Cannot split P ( z, ∂ z ) = P 1 ( z, ∂ z ) · P 2 ( z, ∂ z ) in C ( z ) � ∂ z � with ∂ z G z = t A ′ G z · G z + G z · A ′′ G z deg P 1 , deg P 2 � 1 . (compatibility with G ) ∂ z G z = t A ′′ G z · G z + G z · A ′ rigid : Any other linear diff. syst. (sings at S ∪ ∞ ) G z which is gauge-equiv. over C ( ) at each − A ′′ = ( A − A ′ ) ∗ . ( z − s ) G z G z � �� � � �� � s ∈ S ∪ ∞ to the given system is gauge-equiv. θ ′′ θ ′ over C ( z ) to the given system. z z G is harmonic w.r.t. A if But: Cannot find a var. of pol. Hodge struct. s.t. the sol. to R-H-B exist. th. given by O ( C � S ) d G -mod. growth . ∂ z θ ′ z + [ θ ′ z , θ ′∗ z ] = 0 Irregular Hodge theory – p. 9/23 Irregular Hodge theory – p. 10/23 11 12 The irregular Hodge filtration Harmonic metrics T HEOREM (Simpson 1990, CS 1998, Biquard-Boalch Deligne (2007): 2004, T. Mochizuki 2011): “The analogy between vector bundles with integrable If A is irreducible , ∃ ! harmonic metric G w.r.t. A s.t. connection having irregular singularities at infinity on a Coefs of Char θ ′ have mod. growth at S ∪ ∞ , complex algebraic variety U and ℓ -adic sheaves with wild C [ z, (( z − s ) − 1 ) s ∈ S ] d = ( O ( C � S ) d ) G -mod. growth . ramification at infinity on an algebraic variety of characteristic p , leads one to ask how such a vector bundle with integrable connection can be part of a system of E.g., the Hodge metric of a var. pol. Hodge structure realizations analogous to what furnishes a family of is harmonic w.r.t. the reg. sing. conn. A . motives parametrized by U ... If A is irreg. , what about rigid irreducible A ? In the ‘motivic’ case, any de Rham cohomology group has Answer in the last slide of the talk. a natural Hodge filtration. Can we hope for one on dR ( U, ∇ ) for some classes of ( V, ∇ ) with irregular H i singularities?” Irregular Hodge theory – p. 11/23 Irregular Hodge theory – p. 12/23 The irregular Hodge filtration 13 The irregular Hodge filtration 14 “The reader may ask for the usefulness of a “Hodge Ex.: U = C ∗ , f : z �→ − z , ∇ = d + d f + α d z/z filtration” not giving rise to a Hodge structure. I hope that it C [ z, z − 1 ] · d z ∇ forces bounds to p -adic valuations of Frobenius C [ z, z − 1 ] H 1 dR ( U, ∇ ) z eigenvalues. That the cohomology of ‘ e − z z α ’ ( 0 < α < 1 ) has Hodge degree 1 − α is anlogous to formulas giving e − z z α ≀ ≀ e z z − α ≀ the p -adic valuation of Gauss sums.” � � C [ z, z − 1 ] · e − z z α d z e − z z α d z d C [ z, z − 1 ] e − z z α C · z z � ∞ e − z z α d z period: z = Γ( α ) 0 ? dR ( U, ∇ ) . ⇒ [ e − z z α d z/z ] ∈ F 1 − α H 1 Irregular Hodge theory – p. 13/23 Irregular Hodge theory – p. 14/23 The Hodge filtration in dim � 1 15 Twisted de Rham cohomology 16 Setting: Setting: U : smooth cplx quasi-proj. var. (e.g. U = ( C ∗ ) n ). Choose (according to Hironaka) any X such that U : smth cplx quasi-proj. var., f : U → C alg. fnct. X : smooth cplx proj. variety, D : reduced divisor with normal crossings in X dR ( U, d + d f ) : Twisted de Rham cohomology H k locally, D = { x 1 · · · x ℓ = 0 } Cohomology of the alg. de Rham cplx. E.g. U = C n : U = X � D . 0 → C [ x ] → � i C [ x ]d x i → · · · → � T HEOREM (Deligne 1972): i C [ x ]d � x i → C [ x ]d x → 0 H k ( U, C ) ≃ H k � � • X, (Ω X (log D ) , d) and ∀ p , ( E 1 -degeneration ) → � i ( g ′ x i + gf ′ g ( x ) �− x i )d x i H k � � → H k � � X, σ � p (Ω � � � • X (log D ) , d) − X, (Ω • X (log D ) , d) � i ( − 1) i − 1 (( h i ) ′ x i + h i f ′ i h i d � x i �− → x i ) d x is injective , its image defining the Hodge filtration F p H k ( U, C ) . � Mixed Hodge structure on H k ( U, C ) . Irregular Hodge theory – p. 15/23 Irregular Hodge theory – p. 16/23