J. Huisman. Abstract. let bxc be the fundamental Z=2Z-homology class of X. We show that

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1 On the dual of a real analytic hypersurface J. Huisman Abstract Let f be an immersion of a compact connected smooth real analytic variety X of dimension n into real projective space P n+1 (R). We say that f is homologically trivial if the induced maps f? : H i (X;Z=2Z)?! H i (P n+1 (R);Z=2Z) are zero maps for all i > 0. Denote by f g the dual map of f and let bxc be the fundamental Z=2Z-homology class of X. We show that the dual map f g is homologically trivial in each of the following two cases: 1. f is homologically trivial and of odd degree when considered as a map from X into f(x). 2. f? bxc 6= 0 in H n (P n+1 (R);Z=2Z). As an application, we show that f(x) has an inection point if f(x) is not a real projective hyperplane and if f? bxc 6= 0. Roughly speaking, this implies that a minimal real analytic hypersurface in a real projective space realizing the nontrivial Z=2Z-homology class is a real projective hyperplane. MSC 1991: 14P15, 32C07 Keywords: point real analytic hypersurface, dual hypersurface, inection 1 Introduction In the literature [4, proof of Lemma 5], one comes across the following curious statement. Theorem 1. Let C be a compact connected smooth real analytic curve in the real projective plane. Then, the dual curve C g is homologically trivial in the dual real projective plane. 1

2 I nd this statement curious because it says that, whether or not C is homologically trivial in P 2 (R), the dual curve C g is homologically trivial in P 2 (R) g. It seems contradictory in view of duality: if one applies Theorem 1 to C g one would get that C = (C g ) g is homologically trivial in P 2 (R). This seems contradictory because C might have been homologically nontrivial in P 2 (R). Of course, the statement is not really contradictory. The point is that one cannot apply Theorem 1 to C g if C is homologically nontrivial, i.e., C g either is reduced to a point, or is a singular real analytic curve. In [4] this is used to obtain the following statement. Corollary 2. Let C be a compact connected smooth real analytic curve in the real projective plane. Suppose that C satises the following properties. 1. C is homologically nontrivial in P 2 (R), 2. C is not a real projective line in P 2 (R), and 3. C has no bitangent lines. Then, C contains an inection point. Roughly speaking, Corollary 2 implies that a minimal homologically nontrivial smooth compact connected real analytic curve in the real projective plane is a real projective line. In the present paper, we generalize Theorem 1 to a statement about immersions into real projective (n + 1)-space of smooth compact connected real analytic varieties of arbitrary dimension n (Theorem 10). The main ingredients of its proof are an adjunction formula for such immersions (Lemma 12), and a formula that relates the homologically triviality of such an immersion and the homologically triviality of its dual (Proposition 13). As an application, we generalize Corollary 2 to arbitrary dimension and, moreover, we get rid of the condition about bitangent lines (Theorem 16). Convention. In order to avoid the usual pathologies, a real analytic variety will be the germ at the real points of a complex analytic variety dened over the reals, a morphism between real analytic varieties will be a morphism of such germs, etc. Due to this convention we can apply results of [3] to real analytic varieties. We will do this without further mention. 2 Homological and cohomological triviality Let n be a natural integer and let X be a compact connected topological manifold of dimension n. Let f : X! P n+1 (R) be a continuous map. We 2

3 say that f is cohomologically trivial if the induced morphism f? : H (P n+1 (R);Z=2Z)?! H (X;Z=2Z) of graded Z=2Z-algebras factorizes through the trivially graded Z=2Z-algebra Z=2Z. In order to put it otherwise, let = n+1 be the unique nonzero element of the cohomology group H (P 1 n+1 (R);Z=2Z). Dene! f = f? () 2 H 1 (X;Z=2Z): Since generates the Z=2Z-algebra H (P n+1 (R);Z=2Z), the map f is cohomologically trivial if and only if! f = 0. By abuse of language, we will also say that X is cohomologically trivial in P n+1 (R) if f is so. If X P n+1 (R) is a submanifold then we will say that X is cohomologically trivial in P n+1 (R) if X is so with respect to the inclusion map of X into P n+1 (R). Examples 3. A projective hyperplane in P n+1 (R) is cohomologically nontrivial in P n+1 (R), unless n = 0. The n-sphere in P n+1 (R), dened by the ane equation x x2 n+1 = 1, is cohomologically trivial in Pn+1 (R). We say that f is homologically trivial if for all i > 0 the map f? : H i (X;Z=2Z)?! H i (P n+1 (R);Z=2Z) is the zero map. By abuse of language, we will also say that X is homologically trivial in P n+1 (R) if f is so. If X P n+1 (R) is a submanifold then we will say that X is homologically trivial in P n+1 (R) if X is so with respect to the inclusion map of X into P n+1 (R). By Kronecker Duality [5], f is homologically trivial if and only if f is cohomologically trivial. Therefore we may and will interchangeably use the notions of homological and cohomological triviality. There is also a weaker notion of homological triviality. Let bxc be the fundamental class of X, i.e., bxc is the unique nonzero element of the homology group H n (X;Z=2Z) [5]. We say that f is homologically trivial in its highest degree if f? bxc = 0 in H n (P n+1 (R);Z=2Z). Of course, f is homologically trivial in its highest degree whenever f is homologically trivial, provided that n > 0. The converse does not necessarily hold, as is shown in the following example. Example 4. Let X P 3 (R) be the hyperboloid dened by the ane equation z 2 = x 2 + y 2? 1. Then, X is a compact connected submanifold of P 3 (R) of dimension 2. Let f be the inclusion of X into P 3 (R). We show that the map f is homologically trivial in its highest degree but is not homologically trivial. 3

4 O Let L X be the projective line dened by the ane equations x = 1 and z = y. Consider the fundamental class blc of L as an element of the homology group H 1 (X;Z=2Z). Of course, f? blc is nonzero. Hence, f is not homologically trivial. However, the complement P 3 (R)n X is not connected. Therefore, f? bxc = 0, i.e., f is homologically trivial in its highest degree. Here is a useful criterion for a map to be homologically trivial when one knows it to be homologically trivial in its highest degree. Proposition 5. Let n be a natural integer and let X be a compact connected topological manifold of dimension n. Let f : X! P n+1 (R) be a continuous map. Suppose that f? : H n?1 (X;Z=2Z)?! H n?1 (P n+1 (R);Z=2Z) is injective. Then f is homologically trivial if f is homologically trivial in its highest degree. Proof. Consider the commutative diagram [5] H 1 (X;Z=2Z) f? H 1 (P n+1 (R);Z=2Z) \bxc \f?bxc / H n?1 (X;Z=2Z) f? / H n?1 (P n+1 (R);Z=2Z) By hypothesis, f? bxc is zero. Hence, the lower horizontal arrow in the diagram is the zero map. Also by hypothesis, the vertical arrow to the right is an injective map. Since the upper horizontal arrow is an isomorphism by Poincare Duality [5], the vertical arrow to the left is the zero map. In particular,! f = f? () = 0. Hence, f is homologically trivial. 3 The dual of a real analytic hypersurface Let us recall the classical construction of the dual of a real analytic hypersurface of real projective space. Let n be a natural integer. Let X be a smooth compact connected real analytic variety of dimension n. Suppose that f : X! P n+1 (R) is a real analytic immersion of X into the (n + 1)-dimensional real projective space P n+1 (R). Let x 2 X. Since f is an immersion, the image of the tangent map T x f of f at x is an n-dimensional real vector subspace of the tangent space T f (x) P n+1 (R) of P n+1 (R) at f(x). Let : R n+2 n f0g! P n+1 (R) be the projection map. We identify the tangent space to R n+2 n f0g at a point y 4

5 with R n+2. For y 2 R n+2 n f0g such that (y) = f(x), the inverse image V x of im(t x f) by the map T y is a (n + 1)-dimensional real vector subspace of R n+2. It is clear that V x does not depend on y, but only on x. The n-dimensional real projective subspace T x = (V x n f0g) of P n+1 (R) is the embedded projective tangent space to f(x) at f(x). The subspace T x is an element of the dual real projective space P n+1 (R) g. We denote by f g the map from X into P n+1 (R) g dened by x 7?! T x. That map is the dual map of f. It is a real analytic map. If X P n+1 (R) is a smooth real analytic subvariety then the image X g = f g (X) is the dual variety of X, where f is the inclusion map of X into P n+1 (R). This is a slight abuse of language since X g is not necessarily a real analytic subset of P n+1 (R), as the following example shows. Example 6. Let X be the smooth compact connected real analytic subvariety of P 3 (R) dened by the ane equation x 3 + y 3 + z 3 = 1. We show that X g is not a real analytic subset of P 3 (R) g. Indeed, let L be the real projective line in P 3 (R) dened by the ane equations x = 1 and z =?y. Then, L is contained in X. The set of all hyperplanes in P 3 (R) containing L is a real projective line L g in P 3 (R) g. We show that the intersection L g \X g is a proper closed interval in L g. Let H be a real projective plane in P 3 (R) which is tangent to X and contains L. Then, it can be readily checked that H is tangent to X at a point of L. It follows that L g \ X g = f g (L), where f is the inclusion map from X into P 3 (R). Let ': P 1 (R)! L be the bijective parametrization of L anely dened by '(t) = (1; t;?t). It can be readily checked that the embedded projective tangent space to X at '(t) has ane coordinates (?1;?t 2 ;?t 2 ). It follows that f g (L) is a closed interval of L g, properly contained in L g. Therefore, X g is not a real analytic subset of P 3 (R) g. Proposition 7. Let n be a natural integer. Let X be a smooth compact connected real analytic variety of dimension n and let f : X! P n+1 (R) be a real analytic immersion. Then f(x) is a closed real analytic subset of P n+1 (R). Proof. Let x 2 X. Since f is an immersion, there are an open neighborhood U of x in X and an open neighborhood V of f(x) in P n+1 (R) such that f(u) is a smooth closed real analytic subvariety of V and fj U is an isomorphism from U onto f(u). Therefore, in order to show that f(x) is a real analytic subset of P n+1 (R), it suces to show that, for any x 2 X there are only nitely many y 2 X such that f(y) = f(x). But this follows from the compactness of X and the fact that f is an immersion. 5

6 Let X be a smooth compact connected real analytic variety of dimension n. Denote by bxc the fundamental Z=2Z-homology class of X [5]. Since under the hypotheses of Proposition 7 f(x) is a real analytic subset of P n+1 (R), it has a fundamental Z=2Z-homology class [2] bf(x)c 2 H n (f(x);z=2z); i.e., bf(x)c is the unique nonzero element of the above homology group. Dene the degree mod 2 of f to be the unique integer d mod 2 such that f? bxc = dbf(x)c in H n (f(x);z=2z). Since we are interested in real analytic hypersurfaces and not in immersions into real projective spaces, we will concentrate on immersions of odd degree. The following statement shows that this is not very restrictive, as far as real analytic hypersurfaces are concerned. Proposition 8. Let n be a natural integer. Let X be a smooth compact connected real analytic variety of dimension n and let f : X! P n+1 (R) be a real analytic immersion. Then, there are a smooth compact connected real analytic variety X of dimension n, an immersion f of X into P n+1 (R), and a real analytic map : X! X such that the diagram X H f HHHHHHHH$ X wwwwwwww; f w P n+1 (R) commutes and such that (X; f) is universal in having these properties. The map f is injective on the complement of a proper closed real analytic subset of X. In particular, f(x) = f(x) and f is of odd degree. Proof. Let R be the equivalence relation on X dened by (x; y) 2 R if and only if there are open neighborhoods U and V of x and y, respectively, such that f(u) = f(v ). Then, R is a smooth closed real analytic subvariety of X X. Moreover, the restriction to R of the projection on any factor of X X is a local isomorphism. It follows that the quotient X = X=R is a smooth compact connected real analytic variety of dimension n. Obviously, one has induced real analytic maps : X! X and f : X! P n+1 (R) such 6

7 that f = f. Moreover, is a local isomorphism. Hence, f is an immersion. It is clear that (X; f) is universal. By construction, if x; y 2 X, x 6= y, such that f(x) = f(y), then f(u) 6= f(v ) for all open neighborhoods U and V in X of x and y, respectively. It follows that f is injective on the complement of a proper closed real analytic subset of X. Corollary 9. Let n be a natural integer. Let X be a smooth compact connected real analytic variety of dimension n and let f : X! P n+1 (R) be a real analytic immersion. Suppose that f g is also an immersion. Then, f g = (f) g. In particular, the topological degrees mod 2 of f and f g coincide. Proof. Let (X; f) be universal for f : X! P n+1 (R) and let (X 0 ; f g ) be universal for f g : X! P n+1 (R) g. By universality, X 0 = X and f g = f g. Let : X! X and 0 : X! X 0 be the quotient maps. Then, = 0. Since f g and f g are of odd degree, deg(f) = deg() = deg( 0 ) = deg(f g ) (mod 2): Here is the principal result of the paper concerning the dual of a real hypersurface. Theorem 10. Let n be a nonzero natural integer. Let X be a smooth compact connected real analytic variety of dimension n and let f : X! P n+1 (R) be a real analytic immersion. Let f g : X! P n+1 (R) g be the dual morphism. Suppose that 1. f is homologically trivial and of odd degree, or 2. f is homologically nontrivial in its highest degree. Then f g is homologically trivial. Before attacking the proof of Theorem 10, we need some preparation. First of all, we will often consider a closed real analytic subspace Z of codimension 1 in a smooth compact connected real analytic variety of dimension m. Let i: Z! Y be the inclusion map. We will denote by dze 2 H 1 (Y;Z=2Z) the Poincare Dual of the homology class i? bzc, where bzc is the fundamental Z=2Z-homology class of Z [2]. Since Z is of codimension 1 in Y, one 7

8 has an invertible sheaf O Y (Z) of O Y -modules and a global section s such that div(s) = Z as real analytic spaces. The cohomology class dze is nothing but the rst Stiefel-Whitney class w 1 (O Y (Z)) of O Y (Z) (compare [1, x12.4]). From now on, we x, throughout the rest of this section, a nonzero natural integer n, a smooth compact connected real analytic variety X of dimension n and an immersion f : X! P n+1 (R). Let N g f be the conormal sheaf of X over P n+1 (R). Lemma 11. w 1 (N g f ) = f? df(x)e. Proof. By Proposition 7, f(x) is a closed real analytic subset of P n+1 (R) of codimension 1. Let I be the sheaf of O-ideals dening f(x). The sheaf f? I is isomorphic to N g f. Since f(x) is of codimension 1 in Pn+1 (R), the sheaf I is isomorphic to the dual O(f(X)) g of the invertible sheaf O(f(X)). It follows that w 1 (N g f ) = w 1 (f? I) = = f? w 1 (I) = = f? w 1 (O(f(X)) g ) = = f? w 1 (O(f(X))) = = f? dxe: If Y is a smooth compact connected real analytic variety of dimension m then we denote by w 1 (Y ) the rst Stiefel-Whitney class of the invertible sheaf m Y of real analytic dierential m-forms on Y. Lemma 12 (Adjunction formula). w 1 (X) = f? (w 1 (P n+1 (R)))+f? df(x)e. Proof. Since f is an immersion, one has an isomorphism f? n+1 P n+1 (R) = n X OX N g f : Taking rst Stiefel-Whitney classes, one gets by Lemma 11. f? (w 1 (P n+1 (R))) = w 1 (f? n+1 P n+1 (R) ) = = w 1 ( n X) + w 1 (N g f ) = = w 1 (X) + f? df(x)e; 8

9 Recall from Section 2 that! f is the pull-back by f? of the nontrivial cohomology class n+1 of H 1 (P n+1 (R);Z=2Z). Proposition 13.! f g =! f + f? df(x)e. Proof. Let P 2 P n+1 (R) be a point such that P 62 f(x). Denote by P linear projection of P n+1 (R)nfP g onto P n (R). More precisely, identify P n (R) with the set of projective lines of P n+1 (R) that pass through P. Then, P (Q) is dened to be the projective line of P n+1 (R) that passes through P and Q, for Q 2 P n+1 (R) n fp g. Consider the map g : X?! P n (R) dened by g = P f. Clearly, g is real analytic. Let us show that g is a nite map. Let L be a projective line in P n+1 (R) passing through P. By Proposition 7, f(x) is a closed real analytic subset of P n+1 (R). Hence, the intersection f(x) \ L is a closed real analytic subset of L. Since P 62 f(x), it is also a proper subset of L. Therefore, f(x) \ L is nite. It follows that g is nite. The morphism g induces a pull-back morphism g? : g? n P n (R)?! n X : Note that both sheaves g? n P n (R) and n X are invertible sheaves of O X -modules. Since g is nite, the morphism g? is nonzero. Let I be the annihilator of the sheaf coker(g? ). Then, I is a sheaf of ideals of O X. Let Z be the real analytic subspace of X dened by I. Then, Z is equal to the real analytic subspace (f g )?1 (P g ) of X, where P g denotes the projective hyperplane in P n+1 (R) g dual to P. In particular, dze = (f g )? dp g e =! f g: The morphism g? corresponds to a global section s of the invertible sheaf (g? n P n (R))?1 OX n X : Moreover, s is not the zero section since g is nite. Since the divisor div(s) of s is equal to Z, in H 1 (X;Z=2Z). By Lemma 12,! f g = ddiv(s)e = g? (w 1 (P n (R))) + w 1 (X)! f g = g? (w 1 (P n (R))) + f? (w 1 (P n+1 (R))) + f? df(x)e (1) the 9

10 If n is odd, P n (R) is orientable, i.e., w 1 (P n (R)) = 0, and P n+1 (R) is not orientable, i.e., w 1 (P n+1 (R)) = n+1. Hence, if n is odd, equation (1) gives! f g =! f + f? df(x)e: (2) If n is even, w 1 (P n (R)) = n and w 1 (P n+1 (R)) = 0. But g? n = f? n+1 =! f. Substituting in equation (1) shows that equation (2) holds as well if n is even. The conclusion is that the equation (2) holds for all n. Remark 14. In the proof above, it is particularly important to consider following our convention a real analytic space as the germ of the real points of a complex analytic space dened over R. For example, above, Z is a codimension 1 real analytic subspace of X. However, forgetting about the germ and taking into account only its real points, Z may well be of higher codimension in X! In fact, Z may even be empty! Proof of Theorem 10. We treat separately the two cases 1 and 2. In case 1, f is homologically trivial and of odd degree. In particular,! f = 0 and f? bxc = 0. Let i be the inclusion map of f(x) into P n+1 (R). Since f is of odd degree, i? bf(x)c = f? bxc = 0. Hence, df(x)e = 0. By Proposition 13,! f g = 0, i.e., f g is homologically trivial. In case 2, f is homologically nontrivial in its highest degree, i.e., f? bxc 6= 0. Then, necessarily, i? bf(x)c = f? bxc 6= 0. Hence, df(x)e 6= 0, i.e., df(x)e =. Then, f? df(x)e =! f and! f g = 0 by Proposition 13. As we have seen in the above proof, if f is homologically nontrivial in it highest degree then f is necessarily of odd degree. Hence, Theorem 10 only concerns immersions of odd degree that are either homologically trivial, or homologically nontrivial in their highest degree. As shows the following example, the statement of Theorem 10 cannot be weakened as to include all real analytic immersions f : X! P n+1 (R) of odd degree. Example 15. Let again X P 3 (R) be the hyperboloid of Example 4, let f be the inclusion of X into P 3 (R) and let L X be the projective line dened by the ane equations x = 1 and z = y. The dual L g of L is the projective line in P 3 (R) g consisting of all projective planes in P 3 (R) that contain L. It is clear that L g is contained in X g. Therefore, X g is not homologically trivial. 4 Inflection points on real analytic hypersurfaces Let n be a natural integer. Let X be a smooth compact connected real analytic variety of dimension n. Let f : X! P n+1 (R) be a real analytic 10

11 immersion. The map f is said to be nondegenerate if the image f(x) is not contained in a projective hyperplane. Let f g : X! P n+1 (R) g be the dual map of f. We say that x 2 X is an inection point of f if f is nondegenerate and the map f g is not an immersion at x. By abuse of language, we will also say that f(x) is an inection point of f(x) if x is an inection point of f. Theorem 16. Let n be a natural integer. Let X be a smooth compact connected real analytic variety of dimension n. Let f : X! P n+1 (R) be a real analytic immersion. Suppose that 1. f has no inection points, and 2. f is homologically nontrivial in its highest degree. Then f(x) is a projective hyperplane in P n+1 (R) and f is an isomorphism from X onto f(x). In particular, X is isomorphic to P n (R). Proof. The statement obviously holds if n = 0. We may assume, therefore, that n > 0. By Theorem 10, the dual map f g of X into P n+1 (R) g is homologically trivial. We show that f g is not an immersion. Suppose, to the contrary, that f g is an immersion. Since f is homologically nontrivial in its highest degree, the topological degree of f is odd. Applying Corollary 9 to f g, the topological degrees mod 2 of f g and f are equal. Hence, f g is of odd topological degree. Since f g is homologically trivial, Theorem 10 implies that f = (f g ) g is homologically trivial. This contradicts the hypothesis that f is not homologically trivial in its highest degree. Therefore, f g is not an immersion. Since f g is not an immersion, f has an inection point or f is degenerate. By hypothesis, f does not have an inection point. Hence, f is degenerate, i.e., f(x) is contained in a projective hyperplane H of P n+1 (R). Then, f considered as a map from X into H is a topological covering. Since the fundamental group 1 (P n (R)) of P n (R) is isomorphic to Z=2Z, the map f is of degree 1 or 2. But f is not of degree 2 since f is not homologically trivial in its highest degree. Therefore, the degree of f is equal to 1 and f is an isomorphism from X onto H. Theorem 16 has some nice equivalent formulations: Corollary 17. Let n be a nonzero natural integer. Let X be a smooth compact connected real analytic variety of dimension n. Let f : X! P n+1 (R) be a nondegenerate real analytic immersion. If f has no inection points then f is homologically trivial in its highest degree, i.e., f? bxc = 0 in the homology group H n (P n+1 (R);Z=2Z). In particular, f(x) is the boundary of some closed region in P n+1 (R). 11

12 Corollary 18. Let n be a nonzero natural integer. Let X be a smooth compact connected real analytic variety of dimension n. Let f : X! P n+1 (R) be a nondegenerate real analytic immersion. If f is homologically nontrivial in its highest degree then f has an inection point. Remark 19. The statement of Corollary 18 admits an easier proof if, moreover, n is odd and f(x) is a branch of a real algebraic variety V in P n+1 (R). Indeed, in that case, V is of odd degree d. The Hessian H of V is of degree (n + 2)(d? 2). In particular, the degree of H is odd. Then, there is a real analytic branch B of H such that bbc 6= 0 in H n (P n+1 (R);Z=2Z). Since bf(x)c 6= 0 in H n (P n+1 (R);Z=2Z) as well, B \ f(x) 6= ;, i.e., f has an inection point. References [1] Bochnak, J., Coste, M., Roy M-F.: Geometrie algebrique reelle. Ergeb. Math. 3.Folge, Bd. 12, Springer Verlag, 1987 [2] Borel, A., Haeiger, A.: La classe d'homologie fondamentale d'un espace analytique. Bull. Soc. Math. France 89 (1961), 461{513 [3] Grauert, H., Remmert, R.: Coherent analytic sheaves. Grundlehren Math. Wiss. 265, Springer Verlag, 1984 [4] Huisman, J.: Inection points on real plane curves having many pseudolines. Beitrage Algebra Geom. (to appear) [5] Munkres, J. R.: Elements of algebraic topology. Addison-Wesley, 1984 Institut de Recherche Mathematique de Rennes Universite de Rennes 1 Campus de Beaulieu Rennes Cedex France huisman@univ-rennes1.fr Home page: Typeset by AMS-L A TEX and XY-pic 12