(joint work with Atsushi Takahashi) Liverpool, June 21, 2012

Transcription

1 (joint work with Atsushi Takahashi) Institut für Algebraische Geometrie Leibniz Universität Hannover Liverpool, June 21, 2012

2 Wall 75, C.T.C. Wall: Kodaira and an extension of. Compositio Math. 56, 3 77 (1985).

3 Wall 75, C.T.C. Wall: Kodaira and an extension of. Compositio Math. 56, 3 77 (1985). C.T.C. Wall: A note on symmetry of. Bull. London Math. Soc. 12, (1980)

4 Classification of f (x) = f (x 1,..., x n ) complex polynomial with f (0) = 0 and isolated ( singularity at 0 C n ), i.e. grad f (x) = f x 1 (x),..., f x n (x) 0 for x 0, x < ε. X := f 1 (0) hypersurface singularity V. I. Arnold (1972, 1973, 1975): 0-modal (simple) : ADE unimodal : simple elliptic Tp,q,r : f (x, y, z) = x p + y q + z r + axyz, a C, 1 p + 1 q + 1 r < 1 (cusp ) 14 exceptional bimodal 8 bimodal series 14 exceptional

5 (1) 14 exceptional unimodal related to Schwarz triangular groups Γ(α 1, α 2, α 3 ) PSL(2; R) Dol(X ) = (α 1, α 2, α 3 ), π α 1, π α 2, π α 3 angles of hyperbolic triangle Gab(X ) = (γ 1, γ 2, γ 3 ), Coxeter-Dynkin diagram : X X Dol(X ) = Gab(X ) Gab(X ) = Dol(X )

6 (2) Name Dol(X ) Gab(X ) Dual E 12 2, 3, 7 2, 3, 7 E 12 E 13 2, 4, 5 2, 3, 8 Z 11 E 14 3, 3, 4 2, 3, 9 Q 10 Z 11 2, 3, 8 2, 4, 5 E 13 Z 12 2, 4, 6 2, 4, 6 Z 12 Z 13 3, 3, 5 2, 4, 7 Q 11 Q 10 2, 3, 9 3, 3, 4 E 14 Q 11 2, 4, 7 3, 3, 5 Z 13 Q 12 3, 3, 6 3, 3, 6 Q 12 W 12 2, 5, 5 2, 5, 5 W 12 W 13 3, 4, 4 2, 5, 6 S 11 S 11 2, 5, 6 3, 4, 4 W 13 S 12 3, 4, 5 3, 4, 5 S 12 U 12 4, 4, 4 4, 4, 4 U 12

7 E.-Wall extension Wall (1983): Classification of unimodal isolated of complete intersections (ICIS) 8 bimodal series 8 triangle ICIS in C 4 quasihomogeneous heads related to quadrilateral groups Γ[α 1, α 2, α 3, α 4 ] Series Head Dol(X ) Gab(X ) Dual J 3,k J 3,0 2, 2, 2, 3 2, 3, 10 J 9 Z 1,k Z 1,0 2, 2, 2, 4 2, 4, 8 J 10 Q 2,k Q 2,0 2, 2, 2, 5 3, 3, 7 J 11 W 1,k W 1,0 2, 3, 2, 3 2, 6, 6 K 10 W 1,k 2, 2, 3, 3 2, 5, 7 L 10 S 1,k S 1,0 2, 3, 2, 4 3, 5, 5 K 11 S 1,k 2, 2, 3, 4 3, 4, 6 L 11 U 1,k U 1,0 2, 3, 3, 3 4, 4, 5 M 11

8 (1) A quasihomogeneous polynomial f in n variables is invertible : f (x 1,..., x n ) = n n a i x E ij j i=1 j=1 for some coefficients a i C and for a matrix E = (E ij ) with non-negative integer entries and with det E Ex.: f (x, y, z) = x 6 y + y 3 + z 2, E = For simplicity: a i = 1 for i = 1,..., n, det E > 0. An invertible quasihomogeneous polynomial f is non-degenerate if it has an isolated singularity at 0 C n.

9 (2) f is quasihomogeneous, i.e. there exist weights w 1,..., w n Q such that f (λ w 1 x 1,..., λ wn x n ) = λf (x 1,..., x n ) for all λ C. Weights (w 1,..., w n ) defined by w 1 1 E. =. 1 w n Kreuzer-Skarke: A non-degenerate invertible polynomial f is a (Thom-Sebastiani) sum of x p1 1 x 2 + x p2 2 x x pm 1 m 1 x m + xm pm (chain type; m 1); x p 1 1 x 2 + x p2 2 x x pm 1 m 1 x m + xm pm x 1 (loop type; m 2).

10 Berglund-Hübsch transpose The Berglund-Hübsch transpose f T is f T (x 1,..., x n ) = n n a i i=1 j=1 x E ji j Ex.: E T = 1 3 0, f T (x, y, z) = x 6 + xy 3 + z

11 Group of diagonal symmetries G f of f { } G f = (λ 1,..., λ n ) (C ) n f (λ 1 x 1,..., λ n x n ) = f (x 1,..., x n ) finite group g 0 = (e 2πiw 1,..., e 2πiwn ) G f exponential grading operator, G 0 := g 0 G f. Berglund-Henningson: G G f subgroup (G T ) T = G G T G T 0 G T := Hom(G f /G, C ) f = {1} = G f T SL n(c) dual group

12 General assumption: n = 3, f (x, y, z) non-degenerate invertible polynomial such that f T (x, y, z) is also non-degenerate, both have singularity at 0 Aim: [ET, Compositio Math. 147 (2011)] (f, G f ) (f T, {1}) (G f = G 0 ) [ET, arxiv: , Int. Math. Res. Not.] Generalization: G 0 G G f {1} G T G T 0 (f, G) (f T, G T ) E.-Wall extension

13 Assumption: G 0 G G f Consider quotient stack {1} G Ĝ C 1 C (f,g) := [ ] f 1 (0)\{0} /Ĝ Deligne Mumford stack (smooth projective curve with finite number of isotropic points) g (f,g) := genus [C (f,g) ]

14 Definition : A (f,g) = (α 1,..., α r ) orders of isotropy groups of G Theorem G = G f g (f,g) = 0, r 3. A (f,gf ) = (α 1, α 2, α 3 ), α i order of isotropy of point P i. Notation: u v := (u,..., u) }{{} v times Theorem H i G f minimal subgroup with G H i, Stab(P i ) H i, i = 1, 2, 3. Then ( α ) A (f,g) = i H i /G G f /H i, i = 1, 2, 3, where one omits numbers equal to 1.

15 H p,q st (C (f,g)) Chen-Ruan orbifold cohomology Definition e st (C (f,g) ) := ( 1) p q dim C H p,q st p,q Q 0 stringy Euler number Proposition e st (C (f,g) ) = 2 2g (f,g) + (C (f,g)). r (α i 1) i=1

16 Assumption: {1} G G f SL 3 (C) For simplicity: f not simple or simple elliptic g G order r g = diag(e 2πia 1/r, e 2πia 2/r, e 2πia 2/r ) with 0 a i < r. age(g) := 1 r (a 1 + a 2 + a 3 ) Z j G := {g G age(g) = 1, g fixes only 0} Theorem f (x, y, z) xyz F (x, y, z) = x γ 1 +y γ 2 +z γ 3 axyz, a C, cusp singularity of type T γ 1,γ 2,γ 3

17 Definition of the pair (f, {1}): Γ (f,{1}) := (γ 1, γ 2, γ 3 ) Proposition Above coordinate change is G-equivariant. In particular, F G-invariant. Definition K i G maximal subgroup fixing i-th coordinate. ( γ ) Γ (f,g) = (γ 1,..., γ s ) := i G/K i K i, i = 1, 2, 3, where one omits numbers equal to 1. of the pair (f, G).

18 f (x 1,..., x n ), f : C n C, X f := f 1 (1) Milnor fibre mixed Hodge structure on H n 1 (X f, C) (Steenbrink) with automorphism c : H n 1 (X f, C) H n 1 (X f, C) given by monodromy, c = c ss c unip, H n 1 (X f, C) λ eigenspace of c ss for eigenvalue λ H p,q f := 0 p + q n Gr p F H n 1 (X f, C) 1 p + q = n, p Z Gr [p] F H n 1 (X f, C) e 2πip p + q = n, p / Z. {q Q H p,q f 0} of f. φ(f ; t) := q Q (t e 2πiq ) dim C H p,q f characteristic polynomial µ f = deg φ(f ; t) Milnor number

19 G-equivariant Action of G G-equivariant version Wall: G-equivariant Milnor number µ (f,g) G-equivariant G-equivariant characteristic polynomial φ (f,g) (t)

20 of a cusp singularity Now F (x, y, z) = x γ 1 + y γ 2 + z γ 3 axyz cusp singularity : { 1, 1 γ 1 + 1, 2 γ 1 + 1,..., γ 1 1 γ 1 + 1, 1 γ 2 + 1, 2 γ 2 + 1,...,..., γ 2 1 γ 2 + 1, 1 γ 3 + 1, 2 γ 3 φ (F,{1}) (t) = (t 1) 2 + 1,..., γ 3 1 γ 3 3 i=1 t γ i 1 t 1 } + 1, 2. G-equivariant characteristic polynomial and Milnor number: φ (F,G) (t) = (t 1) 2 2j G µ (F,G) = 2 2j G + s i=1 t γ i 1 t 1 s (γ i 1) i=1

21 Theorem A (f,gf ) = Γ (f T,{1}), A (f T,G f T ) = Γ (f,{1}). Corollary (G 0 = G f ). Theorem G 0 G G f, f T (x, y, z) xyz F (x, y, z) A (f,g) = Γ (f T,G T ), e st (C (f,g) ) = µ (F,G T ), g (f,g) = j G T Proof. K i = H T i for a suitable ordering of the isotropic points P 1, P 2, P 3.

22 F (x, y, z) f T (x, y, z) xyz cusp singularity Var (F,G T ) := Theorem = p,q Q s ( 1) p+q ( q 3 2) 2 h p,q (F, G T ) γ i 1 i=1 k=1 µ (F,G T ) = 2 2j G T + χ (F,G T ) := 2 2j G T + ( k 1 ) 2 γ i 2 s (γ i 1) i=1 s i=1 ( ) 1 1 γ i Var (F,G T ) = 1 12 µ (F,G T ) χ (F,G T ).

23 C = C (f,g) orbifold curve, smooth projective curve of genus j G T, with isotropic points of orders γ 1,..., γ s µ (F,G T ) = e st (C) stringy Euler number χ (F,G T ) = deg c 1 (C) orbifold Euler characteristic Compare with [Libgober-Wood, Borisov]: Theorem Let X be a smooth compact Kähler manifold of dimension n. Then ( ( 1) p+q q n ) 2 h p,q (X ) 2 p,q Z = 1 12 n χ(x ) X c 1 (X ) c n 1 (X )

24 Spectra of orbifold LG- f (x 1,..., x n ) invertible polynomial, G G f SL n (C) ĉ := n 2 Theorem n w i. Var (f,g) = ( ( 1) p+q q n ) 2 h p,q (f, G) = ĉ µ (f,g). p,q Q [ET, arxiv: ] i=1

25 14 exceptional unimodal Name α 1, α 2, α 3 f γ 1, γ 2, γ 3 Dual E 12 2, 3, 7 x 2 + y 3 + z 7 2, 3, 7 E 12 E 13 2, 4, 5 x 2 + y 3 + yz 5 2, 3, 8 Z 11 E 14 3, 3, 4 x 3 + y 2 + yz 4 2, 3, 9 Q 10 Z 11 2, 3, 8 x 2 + zy 3 + z 5 2, 4, 5 E 13 Z 12 2, 4, 6 x 2 + zy 3 + yz 4 2, 4, 6 Z 12 Z 13 3, 3, 5 x 2 + xy 3 + yz 3 2, 4, 7 Q 11 Q 10 2, 3, 9 x 3 + zy 2 + z 4 3, 3, 4 E 14 Q 11 2, 4, 7 x 2 y + y 3 z + z 3 3, 3, 5 Z 13 Q 12 3, 3, 6 x 3 + zy 2 + yz 3 3, 3, 6 Q 12 W 12 2, 5, 5 x 5 + y 2 + yz 2 2, 5, 5 W 12 W 13 3, 4, 4 x 2 + xy 2 + yz 4 2, 5, 6 S 11 S 11 2, 5, 6 x 2 y + y 2 z + z 4 3, 4, 4 W 13 S 12 3, 4, 5 x 3 y + y 2 z + z 2 x 3, 4, 5 S 12 U 12 4, 4, 4 x 4 + zy 2 + yz 2 4, 4, 4 U 12

26 E.-Wall extension of Bimodal series versus ICIS in C 4 A (f,g0 ) f Γ (f,{1}) BH-dual J 3,0 2, 2, 2, 3 x 6 y + y 3 + z 2 2, 3, 10 Z 13 Z 1,0 2, 2, 2, 4 x 5 y + xy 3 + z 2 2, 4, 8 Z 1,0 Q 2,0 2, 2, 2, 5 x 4 y + y 3 + xz 2 3, 3, 7 Z 17 W 1,0 2, 2, 3, 3 x 6 + y 2 + yz 2 2, 6, 6 W 1,0 S 1,0 2, 2, 3, 4 x 5 + xy 2 + yz 2 3, 5, 5 W 17 U 1,0 2, 3, 3, 3 x 3 + xy 2 + yz 3 3, 4, 6 U 1,0 A (f T,G f T ) f T Γ (f T,G T 0 ) ICIS Z 13 2, 3, 10 x 6 + xy 3 + z 2 2, 2, 2, 3 J 9 Z 1,0 2, 4, 8 x 5 y + xy 3 + z 2 2, 2, 2, 4 J 10 Z 17 3, 3, 7 x 4 z + xy 3 + z 2 2, 2, 2, 5 J 11 W 1,0 2, 6, 6 x 6 + y 2 z + z 2 2, 2, 3, 3 K 10 W 17 3, 5, 5 x 5 y + y 2 z + z 2 2, 2, 3, 4 K 11 U 1,0 3, 4, 6 x 3 y + y 2 z + z 3 2, 3, 3, 3 M 11

27 Example f of table, G = G 0 G f index 2. G0 T = Z/2Z acting on C 3 by Invariant polynomials: (x, y, z) ( x, y, z) W := y 2, X := x 2, Y := xy,, Z := z { XW Y 2 = 0 f T (W, X, Y, Z) = 0 } yields equations of ICIS in C 4 in five cases. Example f (x, y, z) = x 6 y + y 3 + z 2, f T (x, y, z) = x 6 + xy 3 + z 2 = X 3 + YW + Z 2

28 Equations of ICIS f T { (f T ) 1 (0)/G0 T Dual J 3,0 x 6 + xy 3 + z 2 XW Y 2 } X 3 + YW + Z 2 J 9 { Z 1,0 x 5 y + xy 3 + z 2 XW Y 2 } X 2 Y + YW + Z 2 J 10 { Q 2,0 x 4 z + xy 3 + z 2 XW Y 2 } X 2 Z + YW + Z 2 J 11 { W 1,0 x 6 + y 2 z + z 2 XW Y 2 } X 3 + WZ + Z 2 K 10 { S 1,0 x 5 y + y 2 z + z 2 XW Y 2 } X 2 Y + WZ + Z 2 K 11

29 Geometric interpretation of? and for n > 3? Study with symmetries!

30 Thank you!