Vector Fields Liftable Over Stable Maps

Transcription

1 Vector Liftable Over Stable Maps Joint with Daniel Littlestone School of Mathematics University of Leeds Leeds United Kingdom 1st Workshop on Singularities in Generic Geometry and, Valencia, Spain 29

2 Outline

3 The motivation for this study originated in generic geometry but the results have been applied in wider situations. Suppose that we have a map from a surface to R 3. The generic situation is that the singularities of the map are Whitney cross-caps (umbrellas) or transverse crossings of two or three planes.

4 The (Whitney) cross-cap ϕ : (K 2, ) (K 3, ) is given by ϕ(x, y) = (x, y 2, xy). Figure 1: The Whitney Cross-Cap. corank 1, i.e., the rank of the Jacobian drops by at most 1 at the singular point, Example 2.9 (i). Let f : R n R be given by f(x) =±x 2 1 ± x 2 2 ± ± x 2 n. Th f is stable, a Morsei.e., singularity. small perturbations It is well-knowngive that(up Morse to proved that a funct with diffeomorphism) a critical point suchthe thatsame the second map. differential is non-degenerate (equ alently, the square matrix of second derivatives is non-singular) is equival

5 We know that we can study manifolds by studying functions, such as the height function and distance squared function (as described in the Izumiya-Ruas mini-course). We would like to do the same on spaces given as the image of generic maps. For surfaces this means studying functions on the cross-cap. Janet West in her 1995 PhD thesis began this classification and she published a paper on it with Bill Bruce: Bruce and West, Functions on a crosscap, Mathematical Proceedings of the Cambridge Philosophical Society (1998)

6 To classify functions on the cross-cap they needed to make diffeomorphisms of the ambient space such that the cross-cap was preserved. To generate these diffeomorphisms they needed to know the vector fields tangent to the image. In the complex case these are equivalent to the liftable vector fields.

7 Definition Let f be a smooth mapping f : (K n, ) (K p, ). A vector field ξ on (K p, ) is liftable over f if there is a vector field η on (K n, ) such that df η = ξ f. That is, the following diagram commutes df T (K n, ) T (K p, ) η ξ (K n, ) f (K p, ). In these circumstances η is called lowerable.

8 Example For the cross-cap ϕ(v 1, y) = (v 1, y 2, v 1 y) = (V 1, W 1, W 2 ) the following are liftable vector fields: W 2 V 1 W 1, V 1 2W 1, 2W 2 V 2 1 and V 1 2W 1 2W 2 The corresponding lowerable vector fields are (respectively) ( ) ( ) ( ) ( ) v1 y v1 v1,, and. y v 1 y. These vector fields generate the module of liftable vector fields.

9 Example Taking the second vector field in the list we have ( ) 1 ( ) v1 dϕ = 2y v1 y y y v 1 v 1 V 1 = 2y 2 = 2W 1 ϕ.

10 cross-cap Definition For k 2 the minimal cross-cap mapping of multiplicity k is the map ϕ k : (K 2k 2, ) (K 2k 1, ) given by ϕ k (u 1,..., u k 2, v 1,..., v k 1, y) ( ) k 2 k 1 = u 1,..., u k 2, v 1,..., v k 1, y k + u i y i, v i y i i=1 i=1 We shall label the coordinates of the target U 1,..., U k 2, V 1,..., V k 1, W 1 and W 2, respectively. Example The Whitney cross-cap is ϕ 2 (v 1, y) = (v 1, y 2, v 1 y).

11 Daniel Littlestone s task: Find liftables for ϕ k. Holland and Mond 1999: For K = C the module of liftable vector fields is generated by 3k 2 liftables. Thus image of ϕ k is not a free divisor. Has k 1 extra generators. Compare with n p case. Used Singular for small k. Guessed form of lowerables. Then generalized form for liftables and lowerables. Used dϕ k η = ξ ϕ k.

12 Euler vector field One vector field was easy. The Euler vector field: (k 1)U 1 (k 2)U 2. 2U k 2 ξ e = (k 1)V 1 (k 2)V 2. V k 1 kw 1 kw 2 The other 3k 3 vector fields come in three families. We need some way of describing these liftables in a compact form.

13 ξj f = A f 1,j. A f k 2,j B f 1,j. B f k 1,j C f 1,j C f 2,j U 1. U k 2 V 1. V k 1 W 1 W 2 f = 1, 2, 3 denotes the family. j = 1, 2, 3,..., k 1 denotes which member of the family.

14 First Family Theorem (Littlestone) For 1 j k 1 the vector field given by the following components is liftable over ϕ k : A 1 i,j = (k i)(k j)u i U j, 1 i k 2, Xi 1 ix Bi,j 1 = k U i+j r V r k U r V i+j r (i 1)(k j)u j V i r=1 r=1 + kv i+j W 1 ku i+j W 2, 1 i k 1, C 1 1,j = k(k j)u j W 1, C 1 2,j = kv j W 1 + (k j)u j W 2. We set U k 1 = V k =, U k = 1 and U i = V i = for i < and i > k.

15 Second family Theorem (Littlestone) For 1 j k 1 the vector field given by the following components is liftable over ϕ k : A 2 i,j = k(k + i j + 1)U k+i j+1 W 1 + k r=1 j(i + 1)U i+1 U k j, 1 i k 2, B 2 i,j = k(k + i j + 1)V k+i j+1 W 1 + k k ix (k + i j 2r + 1)U r U k+i j r+1 ix (k + i j r + 1)U r V k+i j r+1 r=1 ix ru k+i j r+1 V r j(i + 1)U k j V i+1, 1 i k 1, r=1 C 2 1,j = k(k j + 1)U k j+1 W 1 + ju 1 U k j, C 2 2,j = k(k j + 1)V k j+1 W 1 + jv 1 U k j.

16 Third family Theorem (Littlestone) For 1 j k 1 the vector field given by the following components is liftable over ϕ k : A 3 i,j = k(k + i j + 1)U k+i j+1 W 2 + k k ix (k + i j r + 1)U k+i j r+1 V r r=1 ix ru r V k+i j r+1 k(i + 1)U i+1 V k j, 1 i k 2, r=1 B 3 i,j = k(k + i j + 1)V k+i j+1 W 2 + k r=1 k(i + 1)V i+1 V k j, 1 i k 1, ix (k + i j 2r + 1)V r V k+i j r+1 C 3 1,j = k(k j + 1)U k j+1 W 2 + ku 1 V k j C 3 2,j = k(k j + 1)V k j+1 W 2 + kv 1 V k j.

17 The proofs are just long calculations involving writing down a lowerable and showing that dϕ k η = ξ ϕ k.

18 An example ϕ 3 : (K 4, ) (K 5, ): (Variables: U 1, V 1, V 2, W 1, W 2 ) 4U1 2 3U 1 V 1 + 3V 2 W 1 3U 2 W 2 ξ1 1 = 3U 2 V 1 3(U 1 V 2 + U 2 V 1 ) 2U 1 V 2 + 3V 3 W 1 3U 3 W 2 6U 1 W 1. 3V 1 W 1 + 2U 1 W 2 However, recall U 2 = V 3 = and U 3 = 1. 4U1 2 3U 1 V 1 + 3V 2 W 1 ξ1 1 = 5U 1 V 2 3W 2 6U 1 W 1 3V 1 W 1 + 2U 1 W 2

19 Liftable vector fields for k = 3 ξ1 1 = 4U1 2 3U 1 V 1 + 3V 2 W 1 5U 1 V 2 3W 2 6U 1 W 1 3V 1 W 1 + 2U 1 W 2 ξ1 2 = ξ1 3 = 6U 1 3V 1 6V 2 9W 1 1 C A 9V 1 6V 2 2 9W 2 + 3U 1 V 2 3V 1 V 2 1 C A 1 C A ξ2 2 = ξ e = ξ2 1 = ξ2 3 = 2U 1 2V 1 V 2 3W 1 3W 2 3U 1 V 2 3W 2 3V 1 3V 2 W 1 9W 1 2U 1 V 2 3V 1 2U1 2 6V 2 W 1 + 2U 1 V 1 1 C A. 1 C A 9W 2 3U 1 V 2 3V 1 V 2 3U 1 V 1 6V 2 W 2 + 3V C A 1 C A

20 They generate Derlog Theorem (H ) The module of vector fields liftable over ϕ k : (C 2k 2, ) (C 2k 1, ) is generated by ξ e, ξ 1 j, ξ 2 j, ξ 3 j, 1 j k 1. The module of liftable vector fields is also known as Derlog(V ) where V is the image (discriminant) of the map.

21 Sketch of Proof Sketch of proof: The proof uses a monomial ordering and a division algorithm for modules. Take monomial order so that W 1 and W 2 are leading. E.g. Reverse lexicographic order. Order module of vector fields with Term over position. For any liftable vector field ξ divide by our vector fields so that have remainder r. This is liftable. Division algorithm says that remainder has no W 1 s and W 2 s in most positions. Show no lowerable exists for the remainder unless the remainder was.

22 A conjecture The image of ϕ k is a hypersurface, V k. Suppose that a reduced defining equation for this is d k : (K 2k 1, ) (K, ). Conjecture: ξ f j (d k) = for f = 1, 2, 3 and j = 1,..., k 1. True for k 7 by using Singular. Important because then Derlog(V k ) = ξ e ξ 1 j, ξ 2 j, ξ 3 j k 1 j=1. Using Derlog(V k ) we can calculate the A e -codimension of maps generated from ϕ k. Using ξj 1, ξj 2, ξj 3 k 1 j=1 we can calculate the number of vanishing cycles associated to these maps.

23 : Classification of A e -codimension 1 maps. Use V K-codimension of linear function (not K V -codimension of an immersion). Tells us what generic means. More general classification of corank 1 maps. C 3 to C 4. Topology Calculate number of vanishing cycles for examples. A proof of the Mond conjecture for corank 1 maps? f finitely A-determined, A e-codim(f ) µ I =number of vanishing cycles.

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