THE MONODROMY GROUP
CONTENTS ANALYTIC FUNCTIONS AND MORSE THEORY THEOREM ABOUT MONODROMY MORSE LEMMA THE MORSE THEORY NORMAL FORMS OF FUNCTIONS TOUGERON THEOREM DEFORMATIONS PROOFS OF THEOREMS 2 3 AND 2 4 CLASSI?CATION OF SINGULARITIES ALGEBRAIC TOPOLOGY OF MANIFOLDS HOMOLOGY AND COHOMOLOGY HOMOTOPY THEORY TOPOLOGY AND MONODROMY OF FUNCTIONS TOPOLOGY OF A NON-SINGULAR LEVEL PICARD-LEFSCHETZ FORMULA ROOT SYSTEMS AND COXETER GROUPS BIFURCATIONAL DIAGRAMS RESOLUTION AND NORMALIZATION INTEGRALS ALONG VANISHING CYCLES ANALYTIC PROPERTIES OF INTEGRALS SINGULARITIES AND BRANCHING OF INTEGRALS PICARD–FUCHS EQUATIONS GAUSS–MANIN CONNECTION OSCILLATING INTEGRALS VECTOR FIELDS AND ABELIAN INTEGRALS PHASE PORTRAITS OF VECTOR FIELDS METHOD OF ABELIAN INTEGRALS QUADRATIC CENTERS AND BAUTIN’S THEOREM HODGE STRUCTURES AND PERIOD MAP HODGE STRUCTURE ON ALGEBRAIC MANIFOLDS HYPERCOHOMOLOGIES AND SPECTRAL SEQUENCES MIXED HODGE STRUCTURES MIXED HODGE STRUCTURES AND MONODROMY PERIOD MAPPING IN ALGEBRAIC GEOMETRY LINEAR DI?ERENTIAL SYSTEMS INTRODUCTION REGULAR SINGULARITIES IRREGULAR SINGULARITIES GLOBAL THEORY OF LINEAR EQUATIONS RIEMANN–HILBERT PROBLEM THE BOLIBRUCH EXAMPLE ISOMONODROMIC DEFORMATIONS RELATION WITH QUANTUM FIELD THEORY HOLOMORPHIC FOLIATIONS LOCAL THEORY FOLIATIONS AND COMPLEX STRUCTURES RESOLUTION FOR VECTOR FIELDS ONE-DIMENSIONAL ANALYTIC DI?EOMORPHISMS THE ECALLE APPROACH MARTINET–RAMIS MODULI NORMAL FORMS FOR RESONANT SADDLES THEOREMS OF BRIUNO AND YOCCOZ 10 HOLOMORPHIC FOLIATIONS GLOBAL ASPECTS ALGEBRAIC LEAVES MONODROMY OF THE LEAF AT IN?NITY GROUPS OF ANALYTIC DI?EOMORPHISMS THE ZIGLIN THEORY 11 THE GALOIS THEORY PICARD–VESSIOT EXTENSIONS TOPOLOGICAL GALOIS THEORY 12 HYPERGEOMETRIC FUNCTIONS THE GAUSS HYPERGEOMETRIC EQUATION THE PICARD–DELIGNE–MOSTOW THEORY MULTIPLE HYPERGEOMETRIC INTEGRALS CHAPTER THEOREM ABOUT MONODROMY 1 1 DE?NITION OF THE ANALYTIC ELEMENT AND OF THE RIEMANN SURFACE BY AN ANALYTIC ELEMENT WE MEAN A PAIR (D, F), WHERE D = DA ?C IS A DISC WITH CENTER AT A AND F = FA IS A HOLOMORPHIC FUNCTION ON D, SUCH THAT THE TAYLOR SERIES OF F AT A IS CONVERGENT IN DA WE SAY THAT THE ANALYTIC ELEMENT (DA, FA) HAS PROLONGATION TO AN ELEMENT (DB, FB) ALONG A PATH ? ?C IF ? CAN BE COVERED BY DOMAINS OF ANALYTIC ELEMENTS SUCH THAT THE CORRESPONDING FUNCTIONS AGREE AT THE ADJACENT INTERSECTIONS THE SUM OF ANALYTIC ELEMENTS OBTAINED FROM PROLONGATIONS OF (D, F) FORMS THE RIEMANN SURFACE OF THE (GENERALLY MULTIVALUED) HOLOMORPHIC FUNCTION F FIGURE 1 2 THEOREM ABOUT MONODROMY IF THE PATHS ?1, ?2 JOINING THE POINTS A AND B ARE HOMOTOPICALLY EQUIVALENT IN A DOMAIN WHERE THE FUNCTION F IS (LOCALLY) ANALYTIC, THEN THE PROLONGATIONS OF (DA, F) TO (DB,I, FI) ALONG ?I’S ARE THE SAME IT MEANS THAT F1 ?F2 AT DB,1 ?DB,2 CHAPTER 1 ANALYTIC FUNCTIONS AND MORSE THEORY PROOF LET ?S BE A 1-PARAMETER FAMILY OF PATHS JOINING A WITH B AND REALIZING THE HOMOTOPY BETWEEN ?I THEIR UNION SPREADS OVER SOME COMPACT DOMAIN E WE COVER THIS DOMAIN BY ANALYTIC ELEMENTS, STARTING FROM (DA, F), WHICH AGREE AT INTERSECTIONS IN THIS WAY WE OBTAIN THE RIEMANN SURFACE OF F OVER E IT IS CLEAR THAT THIS RIEMANN SURFACE IS DI?EOMORPHIC TO E THUS F IS SINGLE-VALUED ON E ? 1 3 REMARK THE READER CAN SEE THAT THE THEOREM ABOUT MONODROMY BEARS A TOPOLOGICAL CHARACTER; IT INFORMS US ABOUT COVERINGS ONE CAN FORMULATE IT IN THE FOLLOWING WAY: LET ? : Y ?X BE A COVERING OF TOPOLOGICAL SPACES AND LET ?I, I = 1, 2, BE TWO PATHS IN X JOINING THE POINTS A AND B IF THE PATHS ARE HOMOTOPICALLY EQUIVALENT, THEN THE TWO MAPS ??1(A) ???1(B), DE?NED BY LIFTS OF THE PATHS ?I TO Y , COINCIDE IF F IS A MULTIVALUED FUNCTION ON U ?C AND M IS ITS RIEMANN SURFACE, THEN ONE HAS THE SINGLE-VALUED FUNCTION ˜F ON M, M ˜ F? C ? ? ?F U = U FIGURE WE CAN PROLONG ? TO THE MAP ON C, ?(X) = X2 THEN WE SAY THAT ? IS A RAMI?ED COVERING; WITH ONE RAMI?CATION POINT X = 0 §2 MORSE LEMMA MORSE LEMMA LET U ?C BE A DOMAIN CONTAINING 0 AND LET F : U ?C BE A HOLOMORPHIC FUNCTION 1 4 DE?NITION WE SAY THAT THE POINT 0 IS CRITICAL FOR F I?F ?(0) = 0 THE CRITICAL POINT 0 IS CALLED NON-DEGENERATE I?F ??(0) ?= 0 THE VALUE F(0) IS CALLED THE CRITICAL VALUE OF F THE MORSE LEMMA IN ONE DIMENSION LET 0 BE A NON-DEGENERATE CRITICAL POINT OF F THEN THERE EXISTS A LOCAL HOLOMORPHIC CHANGE X = ?(Y), X(0) = 0 SUCH THAT F(?(Y)) = F(0) + Y2 PROOF WE CAN ASSUME THAT F(0) = 0 THE HADAMARD LEMMA IF F(0) = 0, THEN F(X) = XG(X) WITH SOME ANALYTIC FUNC- TION G FIGURE PROOF WE HAVE F(X) = F(X) ?F(0) = 0 [ D DTF(TX)]DT = X 0 F ?(TX)DT ? CHAPTER 1 ANALYTIC FUNCTIONS AND MORSE THEORY WE HAVE G(0) = F ?(0) = 0 WE APPLY THE HADAMARD LEMMA AGAIN AND WE OBTAIN G(X) = XH(X) AND F(X) = X2H(X), WHERE H(0) = 1 HAS TWO BRANCHING POINTS Y1 = ±?C WHEN THE VARIABLE Y1 VARIES, TURNING ONCE AROUND ONE BRANCHING POINT, THEN WE ARRIVE AT THE OTHER SHEET OF THE RIEMANN SURFACE WHEN Y1 RUNS AROUND BOTH RAMI?CATIONS, THEN WE ARRIVE AT THE SAME PLACE IN ORDER TO GET THE RIEMANN SURFACE OF Y2(Y1) WE TAKE TWO COPIES OF THE COMPLEX PLANE CUT ALONG THE SEGMENT JOINING THE RAMI?CATION POINTS WE PUT ONE SHEET ABOVE ANOTHER, TURN THE UPPER SHEET AROUND THE LINE PASSING THROUGH THE BRANCHING POINTS AND GLUE THE TWO SHEETS ALONG THE CUTS (SEE FIGURE 5) WE OBTAIN A SURFACE DI?EOMORPHIC TO AN IN?NITE CYLINDER THE IMAGE OF THE CUT FORMS A CLOSED CURVE ?; IT IS A CYCLE GENERATING THE HOMOLOGY GROUP OF THIS SURFACE IN DIMENSION 1 §2 MORSE LEMMA FIGURE 2, U1V1 + U2V2 = 0 THE LATTER EQUATION MEANS THAT THE VECTORS U AND V ARE ORTHOGONAL, ?U, V?= 0 THE DI?EOMORPHISM IS (U, V) ? U/ C + |V|2, V (THE ?RST COMPONENT LIES IN S1, THE SECOND COMPONENT LIES IN THE LINEAR SUBSPACE OF R2 ORTHOGONAL TO THE ?RST COMPONENT, I E TANGENT TO S1) R ?{G = R2} WITH RADII R ?[0, 1] THIS IS THE BALL DN FROM PROPOSITION 1 8(B) ALSO IT IS NOT DI?CULT TO CONSTRUCT THE DEFORMATION RETRACTION AS IN THE THESIS OF PROPOSITION 1 8(B) ? FIGURE 1 10 DE?NITION THE CYCLE ?IS CALLED THE VANISHING CYCLE PROOF OF THE MORSE LEMMA HERE I PRESENT THE PROOF SUGGESTED TO ME BY T MASZCZYK FIRSTLY WE NEED A MULTIDIMENSIONAL VERSION OF THE HADAMARD LEMMA THE HADAMARD LEMMA IF F(X), X ?(CN, 0) IS A GERM OF AN ANALYTIC FUNCTION SUCH THAT F(0) = 0, THEN F(X) = XIGI(X) WITH ANALYTIC FUNCTIONS GI LET 0 BE A NON-DEGENERATE CRITICAL POINT OF THE FUNCTION F WE CAN ASSUME THAT F(0) = 0 WE APPLY THE HADAMARD LEMMA TWO TIMES, TO F AND TO THE GI’S, AND §3 THE MORSE THEORY OBTAIN F(X) = HIJ(X)XIXJ WHERE HIJ ARE ANALYTIC FUNCTIONS APPLYING THE CHANGE (HIJ) ?( THE MORSE THEORY CONSIDER THE TANGENT BUNDLE TO THE SPHERE TSN LET ?= {(X, V) : V = 0} BE ITS ZERO SECTION REPRESENTING A CLOSED N-DIMENSIONAL CYCLE IS IT POSSIBLE TO PERTURB SLIGHTLY THE CYCLE ?, TO SOME CYCLE ?1 IN SUCH A WAY THAT ???1 = ?? IN ORDER TO STUDY THIS PROBLEM WE REFORMULATE IT WE CAN CONSIDER THE CYCLE ?AS A MAP SN ?TSN, ?: X ?(X, 0) ITS PERTURBATION ALSO WILL BE A MAP FROM SN TO ITS TANGENT BUNDLE OF THE FORM ?1 : X ?(Y(X), V(X)), WHERE SUPX{|Y(X) ?X| + |V(X)|} < ? IT IS CLEAR THAT WE CAN ASSUME THAT Y(X) ?X IN SUCH CASE WE HAVE A VECTOR ?ELD {V(X)} ON THE SPHERE AND THE PREVIOUS PROBLEM FOLLOWS: CAN WE COMB THE SPHERE? (IS THERE A VECTOR ?ELD ON SN NON-VANISHING AT ANY POINT?) IF N = 1, THEN THE VECTOR ?ELD (X1, X2) ?(X2, ?X1)PROVIDES AN EXAMPLE GEN- ERALLY, IF N IS ODD, THEN THE ANSWER IS POSITIVE: V(X1, X2, X3, X4, , X2K?1, X2K) = (X2, ?X1, X4, ?X3, , X2K, ?X2K?1) IF N IS EVEN, THEN THE ANSWER IS NEGATIVE TO SHOW THIS WE NEED SOME NEW NOTIONS LET M BE A REAL N–DIMENSIONAL MANIFOLD AND LET {V(X), X ?M} BE A VECTOR ?ELD ON IT, V(X) ?TXM 1 11 DE?NITION A POINT X0 ?M IS CALLED A SINGULAR (OR CRITICAL OR EQUILIBRIUM) POINT I?V(X0) = 0 ASSUME THAT X0 IS AN ISOLATED SINGULAR POINT TAKE A SMALL SPHERE S(X0, ?) AROUND X0 OF RADIUS ? AND CONSIDER THE MAP S(X0, ?) ?X ? ??V(X) |V(X)| ?SN?1 THE DEGREE OF THE MAP ? IS CALLED THE INDEX OF THE SINGULAR POINT X0 AND IS DENOTED BY IX0V CHAPTER 1 ANALYTIC FUNCTIONS AND MORSE THEORY FIGURE ±1, WHERE THE SIGN IS +, WHEN DF(XI) PRESERVES THE ORIENTATION AND IS ?, IF IT REVERSES THE ORIENTATION IF THE MAP F IS NOT REGULAR, THEN WE APPROXIMATE IT BY A REGULAR MAP F? AND PUT DEG F = DEG F? EXAMPLES FOR THE VECTOR ?ELD ?X = X, ?Y = ?Y (OR X ? ?X ?Y ? ?Y) THE INDEX AT 0 IS ?1 FOR THE VECTOR ?ELD ?X = X + Y, ?Y = ?X + Y THE INDEX IS 1 THE EXAMPLES OF VECTOR ?ELDS WITH INDEX 2 AND ?2 PROVIDE THE ?ELDS ?Z = Z2 AND ?Z = ¯Z2 RESPECTIVELY (WRITTEN USING THE COMPLEX VARIABLE Z = X + IY) GENERALLY, FOR A LINEAR VECTOR ?ELD, GIVEN BY THE NON-SINGULAR MATRIX A, THE INDEX AT 0 IS SIGNDET A CONSIDER THE VECTOR ?ELD ?Z = 1 IN C ?R2 IT PROLONGS ITSELF TO A VECTOR ?ELD IN THE RIEMANN SPHERE WITH INDEX AT ?EQUAL TO 2 INDEED, IN THE VARIABLE ? = 1/Z WE GET ?? = ??2, WHICH IS (UP TO SIGN) THE SAME AS THE VECTOR ?ELD FROM ONE OF THE PREVIOUS EXAMPLES §3 THE MORSE THEORY FOR THE VECTOR ?ELD ?Z = Z THE INDEX AT Z = 0 IS 1 AND THE INDEX AT Z = ?IS ALSO 1 THUS THE SUM OF INDICES IS THE SAME FOR BOTH VECTOR ?ELDS AND IS EQUAL TO 2 FIGURE 1 14 THE POINCARɖHOPF THEOREM IF THE VECTOR ?ELD V(X) HAS ONLY ISOLATED SINGULAR POINTS X1, , XR, THEN IXJV = ?(M), WHERE ?(M) IS THE EULER CHARACTERISTIC OF THE MANIFOLD M 1 15 REMARK THE EULER CHARACTERISTIC IS CALCULATED AS FOLLOWS WE TAKE A PARTITION OF THE MANIFOLD M INTO CELLS (OR SIMPLICES) AND WE OBTAIN A COMPLEX WHICH CONSISTS OF M0 0-DIMENSIONAL CELLS, M1 1-DIMENSIONAL CELLS ETC THEN WE DE?NE ?(M) = M0 ?M1 + M2 ?M3 + ± MN PROOF OF THEOREM 1 14 WE SKETCH THE PROOF OF THE POINCARɖHOPF THEOREM BASED ON MORSE’S LEMMA LET X BE THE SPACE OF ALL DI?ERENTIABLE VECTOR ?ELDS (EQUIPPED WITH SOME NATURAL TOPOLOGY) LET X0 ?X CONSIST OF VECTOR ?ELDS WITH ONLY NON-DEGENERATE SINGULAR POINTS, I E SUCH THAT DET DV(XI) ?= 0 NOTE THAT THEN IXIV =SIGNDET DV(XI) THE SET X0 IS OPEN AND DENSE IN X (SEE REMARK 1 12 ABOVE) THE FUNCTION V ? (INDICES) IS LOCALLY CONSTANT AT X0 IT IS ENOUGH TO SHOW THAT IT IS THE SAME AT THE BOUNDARY OF X0 LET V ?X \ X0 BE A VECTOR ?ELD WITH DEGENERATE CRITICAL POINTS BUT ISOLATED AND WITH ?NITE INDICES WE TAKE SOME SMALL PERTURBATION V? ?X0 OF V THE ?ELD V? HAS ONLY NON-DEGENERATE CRITICAL POINTS, WHERE SOME OF THEM MAY COALESCE AS V? ?V FROM FIGURE 8(B) IT IS SEEN THAT THE SUMS OF INDICES OF V AND OF ITS PERTURBATION ARE THE SAME; (THE INDEX CALCULATED ALONG THE OUTER CYCLE IS EQUAL TO THE SUM OF INDICES CALCULATED ALONG THE INNER CIRCLES) THEREFORE IT REMAINS TO CALCULATE THE SUM OF INDICES OF SOME PARTICULAR VECTOR ?ELD FROM X0 CHAPTER 1 ANALYTIC FUNCTIONS AND MORSE THEORY 1 16 DE?NITION A TWICE DI?ERENTIABLE FUNCTION F : M ?R WHICH HAS ONLY NON- DEGENERATE CRITICAL POINTS AND DI?ERENT CRITICAL VALUES IS CALLED THE MORSE FUNCTION ASSUME THAT M IS A RIEMANNIAN MANIFOLD, I E IT IS EQUIPPED WITH A RIEMANNIAN METRIC ? , ?X (IF M IS COMPACT AND SMOOTH, THEN USING THE PARTITION OF UNITY CAN ALWAYS CONSTRUCT SUCH A METRIC ) THIS METRIC TENSOR DE?NES THE ISOMORPHISM TXM ?W ?? , W??T ? XM IF F IS A FUNCTION ON M, THEN APPLYING THE INVERSE OF THIS ISOMORPHISM TO DF(X) ?T ? XM WE OBTAIN THE GRADIENT VECTOR ?ELD V(X) =GRADF(X) = ?F(X) IN LOCAL COORDINATES WITH THE EUCLIDEAN METRIC, WE HAVE ?XI = ?F/?XI AND DV = (?2F/?XI?XJ) IN PARTICULAR, THE INDEX OF THE GRADIENT VECTOR ?ELD AT A CRITICAL POINT X0 IS SIGNDET(?2F/?XI?XJ) IN THE CASE OF GENERAL METRIC ? , ?X = (A(X) , ) WE HAVE ?F = A?1?F/?X, AND THE SAME FORMULA FOR INDEX HOLDS FIGURE J = {XJ} OF THE PROMISED CELL COMPLEX IF K = N ?1, THEN WE OBSERVE THE FOLLOWING BIFURCATION LOCALLY THE SETS {F ?C}, C < F(0) CONSIST OF TWO PIECES; THEY ARE DI?EOMORPHIC TO DN×S0, WHERE DK DENOTES §3 THE MORSE THEORY J = S0 ?{F < F(0) ??} GENERALLY, NEAR ANY CRITICAL POINT OF MORSE INDEX I = N ?K THE SETS {F ?C}, C < F(0) ARE DI?EOMORPHIC TO DN?I+1 ×SI?1 ?SI?1 THE BIFURCATION IS EQUIVALENT TO ADJOINING TO THIS SET THE HANDLE DN?I × DI ?DI SO WE ADD TO OUR COMPLEX AN I-DIMENSIONAL CELL ?I L GLUED ALONG THE BOUNDARY TO THE (I ?1)-DIMENSIONAL SKELETON (SEE FIGURE 9) OF COURSE, THE EULER CHARACTERISTIC OF M IS EQUAL TO THE NUMBER OF CELLS ASSOCIATED WITH CRITICAL POINTS OF INDEX 0 MINUS THE NUMBER OF CELLS ASSOCIATED WITH POINTS OF INDEX 1 PLUS, ETC THIS COMPLETES THE PROOF OF THE POINCARɖHOPF THEOREM ? 1 17 THE SELF-INTERSECTION OF THE CYCLE ? WE HAVE ?(SN) = 0 IF N IS ODD, AND = IF N IS EVEN BY THE POINCARɖHOPF THEOREM THIS MEANS THAT THE ODD-DIMENSIONAL SPHERES CAN HAVE EMPTY INTERSECTIONS WITH THEIR DEFORMATIONS IN THEIR TANGENT SPACES AND THE EVEN-DIMENSIONAL SPHERES DO NOT HAVE THIS PROPERTY THE SUM OF INDICES OF A VECTOR ?ELD ON SN CAN BE TREATED AS THE INDEX OF SELF- INTERSECTION OF THIS SPHERE IN ITS TANGENT BUNDLE WE PROVED THAT THIS NUMBER IS EQUAL TO (?, ?) = 1 + (?1)N WE SHALL USE THESE FACTS IN THE SEQUEL FIGURE