Moderation and Mediation in Interindividual Longitudinal Analysis

arXiv:1606.07973v1 [math.AG] 25 Jun 2016 Picard-Lefschetz Monodromy Groups of Quadratic Hypersurfaces Daodao Yang Abstract We study the topology of the space of affine hyperplanes L ⊂ Cn which are in general position with respect to a given generic quadratic hypersurface A, and calculate the monodromy action of the fundamental group of this space on the relative homology groups H∗ (Cn , A ∪ L) associated with such hyperplanes. 1 The statement of the problem and the relative homology group A is an non-degenerate quadratic hypersurface in Cn . For instance, A could be the set {(z1 , z2 , ...zn ) ∈ Cn z12 +z22 +...+zn2 = 1}. L is a complex hyperplane in Cn . n−1 By CP∞ we denote the “infinitely distant” part CPn \Cn of the projective closure of Cn . A is the closure of A in CPn . Non-degeneracy of A implies that A is n−1 n−1 is a transversally, and so A ∩ CP∞ smooth in CPn and intersects CP∞ n−1 non-degenerate quadric hypersurface in CP . ∨ Let CPn be the space of all hyperplanes in CPn . n−1 Definition 1. L is asymptotic for A ⊂ Cn if L ∩ CP∞ is tangent to A ∩ CPn−1 . ∞ L is not in general position with respect to A if either it is tangent to A at some point in Cn , or it is asymptotic for A. 1 In other words, L is in general position with respect to A if and only if n−1 its closure L ⊂ CPn is transversal to the (stratified) algebraic set A ∪ CP∞ . ∨ ∨ ∗ Notation. Denote by A and A the subsets in CPn consisting of all tangent and asymptotic hyperplanes of A, respectively; in addition, the point in ∨ CPn corresponding to the “infinitely distant” hyperplane also is by definition ∗ included into A. By the Thom’s isotopy lemma (see [2], [5]) the pairs of spaces (CPn , A ∪ ∨ ∨ ∗ n−1 ) form a locally trivial fiber bundle over the space CPn \ (A ∪ A) L ∪ CP∞ of planes L which are in general position with respect to A. Therefore the fundamental group of the latter space acts on all homology groups related n−1 ) by the monodromy, in particular on the with spaces (CPn , A ∪ L ∪ CP∞ n groups Hn (C , A ∪ L). The explicit calculation of this action is the main goal of this work; this is a sample result for a large family of similar problems concerning the hypersurfaces of higher degrees and/or non-generic ones. This action is important in the problems of integral geometry, when the integration contour is represented by a relative chain in Cn with boundary at A ∪ L, and integration n-form is holomorphic and has singularity at the infinity; see e.g. [5], Chapter III. We always assume that n ≥ 2, because otherwise the problem is trivial. 1.1 The representation space Proposition 1. If L is in general position with respect to A, then Hn (Cn , A ∪ L) ∼ = Z2 , = Hn−1 (A ∪ L) ∼ and Hi (Cn , A ∪ L) ∼ = 0 for all i 6= n (here H̃ means homology = H̃i−1 (A ∪ L) ∼ group reduced modulo a point). Proof. First, we have the long exact sequence for the pair (Cn , A ∪ L): ... → Hi (A∪L) → Hi (Cn ) → Hi (Cn , A∪L) → Hi−1 (A∪L) → Hi−1 (Cn ) → ... (1) 2 The homology groups of Cn coincide with these of a point. So Hi (Cn , A ∪ L) ∼ = H̃i−1 (A ∪ L) for any i. Second, Milnor theorem shows that A is homotopy(equivalent to S n−1 , and 0 for others; A∩L is homotopy equivalent to S n−2 . Thus Hk (A) = Z for k = 0 or n − 1, ( 0 for others; Hk (A ∩ L) = Z for k = 0 or n − 2. L is homeomorphic to Cn−1 , so Hk (L) = 0 for k ≥ 1. Third, we have the Mayer-Vietoris sequence for A and L: ... → Hn−1 (A ∩ L) → Hn−1 (A) ⊕ Hn−1 (L) → Hn−1 (A ∪ L) → Hn−2 (A ∩ L) → Hn−2 (A) ⊕ Hn−2 (L) → ... which in the case n > 2 is as follows: ... → 0 → Z ⊕ 0 → Hn−1 (A ∪ L) → Z → 0 ⊕ 0 → ... Therefore Hn−1 (A ∪ L) ∼ = Z2 . The case n = 2 is obvious. The same arguments with n replaced by any other dimension show that all groups H̃i (A ∪ L) with i 6= n − 1 are trivial. 2 The fundamental group of the space of generic hyperplanes ∨ ∨ ∗ In this section we calculate the fundamental group π1 (CPn \ (A ∪ A)), and in the next one we describe its action on Hn−1 (A ∪ L). ∨ ∨ ∗ Theorem 1. If n ≥ 3 then the group π1 (CPn \ (A ∪ A) is generated by three elements α, β, κ with relations κα = βκ, κ2 = 1. Remark 1. Obviously, this presentation of the group can be reduced to one with only two generators α, κ with the single relation κ2 = 1. However, the previous more symmetric presentation is more convenient for us. ∗ n−1 , which are tangent to Denote by P A the set of all hyperplanes in CP∞ the hypersurface ∂A ≡ A \ A of “infinitely distant” points of A. 3 ∨ ∗ Thus P A = (A \ A). Associating with any affine hyperplane in Cn its infinitely distant part, we obtain the down-left arrow in the commutative diagram of maps: ∨ ∨ ∗ ∨ n−1 CP∞ ∨ inclusion CPn \ (A ∪ A) C1 \ {2 points} ∗ ∗ CPn \ A (2) C1 \ PA ∗ Indeed, an affine hyperplane belongs to A if and only if its image under ∗ this map belongs to P A. On the other hand, the fiber of this map over ∨ n−1 any point of CP∞ consists of a pencil of affine hyperplanes parallel to one ∨ another, so it is a line bundle. Any such fiber C1 intersects the set A at exactly two points: indeed, for any non-asymptotic hyperplane there are exactly two hyperplanes parallel to it and tangent to A. Considering the fiber bundle represented by the left-hand part of the diagram (2), E ↓F B ∨ ∨ ∗ ∨ ∗ let F = C1 \ {2 points}, E = CPn \ (A ∪ A), B = CPn−1 \ P A. We have the exact sequence for the fiber bundle. ... → π2 (E) → π2 (B) → π1 (F ) → π1 (E) → π1 (B) → π0 (F )... (3) F is connected, so the rightmost arrow is trivial. Lemma 1. If n > 2 then π1 (B) = Z2 ; if n = 2 then π1 (B) = Z. Proof. The statement for n = 2 is obvious: in this case B is the complex projective line less two points. For n = 3 this statement follows by the Zariski 4 theorem (using the case n = 2 as the base), see e.g. [4], Chapter 6, §3. Finally, for n > 3 it follows from the case n = 3 by the strong Lefschetz theorem, see [2]. Lemma 2. Let C be a smooth quadratic hypersurface in CPn−1 . If n 6= 3 then π2 (CPn−1 \ C) is trivial. π2 (CP2 \ C) ∼ = Z. In particular, this is true for the base of our fiber bundle (3). Proof. Let [C] ⊂ Cn be the union of lines corresponding to the points of C. We have a fiber bundle Cn \ [C] ↓ C∗ CPn−1 \ C This fiber bundle is trivial because it is a restriction of the tautological bundle of CPn−1 on the complement of a non-trivial divisor, so its first Chern class is equal to 0. Therefore π2 (Cn \ [C]) = π2 (CPn−1 \ C) ⊕ π2 (C∗ ) = π2 (CPn−1 \ C). Let ϕ : Cn → C be the quadratic polynomial defining the sets [C] and C. It defines the Milnor fibration ϕ : Cn \ [C] → C∗ . Let E ′ = Cn \[C], B ′ = C∗ , F ′ = Vλ . In this notation, π2 (B) = π2 (CPn−1 \ C) = π2 (Cn \ [C]) = π2 (E ′ ). We have the exact sequence for the fiber bundle. ...π3 (B ′ ) → π2 (F ′ ) → π2 (E ′ ) → π2 (B ′ ) → π1 (F ′ ) → π1 (E ′ ) → π1 (B ′ )... The base B ′ is homotopy equivalent to S 1 , in particular the groups π3 (B ′ ) and π2 (B ′ ) are trivial. Also, according to the Milnor theorem, F ′ is homotopy equivalent to S n−1 . 5 ( 0 for n 6= 3; Thus π2 (B) = π2 (E ′ ) = π2 (F ′ ) = π2 (S n−1 ) = Z for n = 3. So for n 6= 3 the interesting fragment of the exact sequence (3) reduces to 1 → π1 (F ) → π1 (E) → π1 (B) → 1. (4) Lemma 3. In the case n = 3 the map π2 (E) → π2 (B) in (3) is epimorphic. Proof. By the construction of the generator of the group π2 (B) ∼ Z in this case, this generator can be realised by the sphere consisting of complexifications of all oriented planes through the origin in R3 . All these planes do ∨ ∗ not meet the set A ∪ A, and hence define a 2-spheroid in E. So, the map π2 (B) → π1 (F ) in (3) is trivial, and we can use the exact sequence (4) also in the case n = 3. π1 (F ) = Z ∗ Z, π1 (B) = Z2 Thus π1 (E) has three generators α, β, κ, where α and β are two free generators of π1 (F ), and κ is an element of the coset π1 (E) \ π1 (F ). We can realize these elements as follows. Choose the linear coordinates in Cn in which A is given by the equation z12 + · · · + zn2 = 1. Take for the ∨ ∨ ∗ base point in CPn \ (A ∪ A) the hyperplane {z1 = 0}. The fiber F containing this point consists of all complex hyperplanes {z1 = const} parallel to this one, they are characterized by the corresponding value of z1 . The exceptional ∨ points of intersection with A in this fiber correspond to the values 1 and −1. Then for α and β we take the classes of two simplest loops in C1 going along line intervals from 0 to the points 1 − ε (respectively, −1 + ε), ε > 0 very small, then turning counterclockwise around the point 1 (respectively, −1) along a circle of radius ε, and coming back to 0. For κ we take the 1-parameter family of planes given by the equation (cos τ )z1 + (sin τ )z2 = 0, τ ∈ [0, π]. 6 Lemma 4. The element κ thus defined does not belong to the image of π1 (F ) in π1 (E) under the second map in (4), i.e. its further map to π1 (B) defines a generator of the latter group. Indeed, it is easy to check this in the case n = 2, which provides (via the Zariski theorem) the generator of the latter group. The loop κ defines also a loop in the base of our fiber bundle. Moving the fibers over it and watching the corresponding movement of two exceptional points, we get that κ acts on π1 (F ) by permuting α and β. Theorem 1 is proved. 3 Monodromy representation We know that Hn (Cn , A ∪ L) = Z2 , (5) see Proposition 1. ∨ Proposition 2. For any n, the monodromy action of the group π1 (CPn \ ∨ ∗ (A ∪ A)) on Hn (Cn , A ∪ L) has a 1-dimensional invariant subspace. Proof. This subspace is the image of the group Hn (Cn , A) ∼ = Z under the n n obvious map Hn (C , A) → Hn (C , A ∪ L); it corresponds via the boundary isomorphism Hn (Cn , A ∪ L) → Hn−1 (A ∪ L) in (1) to the image of the map Hn−1 (A) → Hn−1 (A ∪ L). Indeed, this image does not depend on L. It is convenient to fix the generators of this group (5) as follows. Suppose again that A is given by the equation z12 + · · · + zn2 = 1, (6) and the basepoint L0 in the space of planes is given by z1 = 0. Then we have two relative cycles in Cn (and even in Rn ) modulo A ∪ L0 : they are given by the two half-balls bounded by the the surface (6) and (the real part of) the hyperplane L0 ; we supply these half-balls with the orientations induced from a fixed orientation of Rn . It follows immediately from the proof of Proposition 1 that these two chains indeed generate the group Hn (Cn , A ∪ L) = Z2 . 7 Denote these two generators by a and b. Namely, a (respectively, b) is the part placed in the half-space where z1 > 0 (respectively, z1 < 0). The invariant subspace of the monodromy action is then generated by the sum of these two elements: indeed, it is a relative cycle mod A only. Let we study the action of loops α, β, and κ on a and b. Proposition 3. For any n, κ(a) = b, κ(b) = a. Proof. This follows immediately from the construction of both cycles a and b and of the loop κ: when the hyperplane Lτ moves along this loop, the parts of the space Rn bounded by the sphere (6) and real parts of these hyperplanes move correspondingly and permute at the end of this movement. Proposition 4. If n is odd, then the action of both loops α and β is trivial. If n is even, then α(a) = −a, α(b) = 2a + b, β(b) = −b, β(a) = 2b + a. Proof. Both these statements follow immediately from the Picard–Lefschetz formula, see Chapter III in [5]. So, in the case of odd n the monodromy action reduces to that of the group Z2 . In the case of even n the monodromy group is infinite: for instance the orbit of any generating element a or b consists of all points of the integer lattice Z2 satisfying the conditions u − v = 1 or u − v = −1. 8 References [1] V.I.Arnold, A.N.Varchenko, S.M.Gusein-Zade. Singularities of Differentiable Maps. Vol. II Monodromy and Asymptotic Integrals. Birkhäuser 1988. [2] M. Goresky, R. MacPherson. Stratified Morse Theory. Springer, 1988. [3] F. Pham, Introduction a l’Etude Topologique des Singularites de Landau. 1967. [4] V.Prasolov, Elements of combinatorial and Differential Topology, AMS, 2006. [5] V.A.Vassiliev. Applied Picard-Lefschetz Theory. American Mathematical Society, 2002. Daodao Yang, Faculty of Mathematics, Higher School of Economics, Russia Address: 33/1 Studencheskaya, 121165, Moscow E-mail : dyang@edu.hse.ru 9