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ABUNDANCE OF REAL LINES ON REAL PROJECTIVE HYPERSURFACES 2 S. Finashin, V. Kharlamov 1 0 2 There is geometry in the humming of the strings. n There is music in the spacing of the spheres. a PYTHAGORAS (6th century BC) J 3 1 1. Revelations Our aim is to show that in the case of a generic real hypersurface X of degree ] G 2n−1inaprojectivespaceofdimensionn+1thenumberNRofreallinesonX isnot A less than approximately the square root of the number NC of complex lines. More h. precisely, NR ≥(2n−1)!!, while due to Don Zagier[GM] NC ∼q2π7(2n−1)2n−23, t so that a 1 m logNR ≥nlog2n+O(n)= logNC. 2 [ Note that NR, unlike NC, depends not only on n but also on the choice of X. The key point of our estimate is an appropriate signed count of the real lines that 1 v makes the sum invariant. This sum, which we denote by NRe, is nothing but the 7 Euler number of a suitable vector bundle (see 3.1). Its evaluation gives finally the 9 conclusion: 8 2 (1.0.1) NR ≥NRe =(2n−1)!!. . 1 We came to it in 2010–2011,when we started workingon a survey on real cubic 0 hypersurfaces. Realizing that Segre’s division of real lines on cubic surfaces in two 2 species,ellipticandhyperbolic,leadstoaremarkablerelationh−e=3betweenthe 1 number of hyperbolic lines, h, and number e of elliptic, we looked for conceptual : v explanations and generalizations. i X Later, we recollected that the non-vanishing of the Stiefel-Whitney number of the same bundle (which implies existence of real lines) was proved much earlier r a by O. Debarre and L. Manivel in [DM], and that the Euler number was used by J. Solomon in his thesis [So] to perform a signed count of real lines on quintic 3- folds. InSeptember2011wewereinformedbyC.OkonekandA.Telemanthatthey obtained the bound (1.0.1) and that they followed a similar approach. It is when anaccountoftheirproofwaspostedonarXivthatwedecidedtocomposethisnote and to present our proof (in our opinion, a bit simpler and more geometric) and a few related observations, like disclosing precise relation to Welschinger’s indices, see 5.2, and revealing the “secret” of the “magic formula” h−e=3, see 4.1.1. 2000Mathematics Subject Classification. 14N15,14P25,14J80. The second author was partially funded by the ANR-09-BLAN-0039-01 grant of Agence Na- tionale de la Recherche, and is a member of FRG Collaborative Research ”Mirror Symmetry & TropicalGeometry” (AwardNo. 0854989). 1 2 S. FINASHIN, V. KHARLAMOV 2. Origin of Symmetry 2.1. The polar correspondence. The Grassmannian G (Rn+2) of oriented 2- 2 planes can be canonically identified with the purely imagienary quadric Qn(C) defined in Pn+1(C) by equation x2 +x2 +···+x2 = 0. Namely, a real oriented 0 1 n+1 plane p ⊂ Rn+2 defines a real oriented line ℓR ⊂ Pn+1(R), which decomposes its complexificationℓC ⊂Pn+1(C) into apair ofhalvespermutedby the complex conjugation. Each half intersects Qn(C) at a single point. We give privilege to that half, hl, whose complex orientation bounds the given orientation of ℓR. The map p7→q =Qn(C)∩h yields a diffeomorphismκ: G (Rn+2)→Qn(C), whichwe call l 2 the polar correspondence. e In the reverse direction, the polar correspondence can be described as follows. A point q ∈ Qn(C) ⊂ Pn+1(C) is represented by a complex line L ⊂ Cn+2, the q complex line L projects to Rn+2 into a real plane p by averaging v 7→ 1(v+v¯), q 2 and the complex orientation of L is pushed forwardto orient the plane p. q This yields the following interpretation for the tautological bundles of the oriented Grassmannian, G (Rn+2), and of the non-orientable one, G (Rn+2). 2 2 e 2.1.1. Proposition. The polar correspondence G (Rn+2)→Qn(C) identifies the 2 tautological (oriented 2-plane) bundle τ : E (eRn+2) → G (Rn+2) with the re- 2,n+2 2 2 striction, τn : En(C) → Qn(C), to Qne(C) of tehe tautologicael (complex line) bundle on Pn+1(C). This yields identification of the tautological bundle τ over 2,n+2 G (Rn+2) with the quotient of τ by the complex conjugation. (cid:3) 2 n 2.2. Characteristic classes. Thepolarcorrespondenceleadstoasimpleexplicit formula for the Euler class, e(Sym2m−1(F)), of symmetric powers of the dual tautological bundles, F =τ∗ . Namely, the polar correspondence identifies F with 2,n+2 the real 2-bundle underelying complex line bundle L=κ∗(τn∗), so, F ⊗C =L⊕L¯, and Symk(F)⊗C = Symk(F ⊗C) = Lk ⊕Lk−1L¯···⊕L¯k, for any k ≥ 1. The complex conjugation interchanges LaL¯b with LbL¯a, thus, for any odd k =2m−1, the real vector bundle Sym2m−1(F) is the real part of L2m−1 ⊕L2m−2L¯ ⊕···⊕ L¯2m−1. We define an isomorphismbetween L2m−1⊕L2m−2L¯⊕···⊕LmL¯m−1 and Sym2m−1(F) by projecting v 7→ 1(v+v¯). It transports the complex orientationof 2 L2m−1⊕L2m−2L¯ ⊕···⊕LmL¯m−1 to an orientation of Sym2m−1(F), so that the Euler class of the latter becomes well defined and equal to the Chern class of the former one. 2.2.1. Proposition. Under the above orientation convention, e(Sym2m−1(F))=c (L2m−1⊕L2m−2L¯⊕···⊕LmL¯m−1)=(2m−1)!!c (L)m. (cid:3) m 1 3. Absolution 3.1. Euler’s numbers. By Proposition2.1.1,the polarcorrespondenceidentifies thetotalspaceE ofthebundleSym2n−1(τ∗ )withthequotientbycomplexcon- 2,n+2 jugation of the total space of a complex n-bundle (τ∗)2n−1⊕(τ∗)2n−2(τ¯∗)⊕···⊕ n n n (τ∗)n(τ¯∗)n−1 overa complex n-manifold, Qn(C). Since any anti-holomorphicinvo- n n lution ona complex manifold ofevencomplex dimension preservesthe orientation, the manifold E inherits an orientation. Let N e denote the Euler number of Sym2n−1(τ∗ ) with respect to this ori- R 2,n+2 entation. Given a section s: G (Rn+2) → E and an isolated zero x of s, let Ie(s) 2 x ABUNDANCE OF REAL LINES ON REAL PROJECTIVE HYPERSURFACES 3 denote its index, that is the local Euler number of s at x. If s has only isolated zeros, then NRe =PxIxe(s), and we let NR(s)=Px|Ixe(s)|. 3.1.1. Theorem. (1) N e =(2n−1)!!. R (2) For any section s having only isolated zeros, NR(s)≥(2n−1)!!. Proof. Bydefinition,twiceN eequalstheEulernumberofSym2n−1(τ∗ ),which R 2,n+2 in accordance with Proposition 2.2.1 is equal to (2n−1)!!c1(τn∗)n[Qen(C)], while c (τ∗)n[Qn(C)]=degQn =2. Thisgives(1),wherefrom(2)isstraightforward. (cid:3) 1 n Consider a real homogeneous polynomial f ∈ H0(Pn+1,O(d)) defining a real hypersurface X of degree d = 2n−1 in Pn+1. Restricting f to planes p ⊂ Rn+2, we obtain a section s of Sym2n−1(τ∗ ). Its zeros represent real lines lying on f 2,n+2 X. We call a real line ℓ ⊂ X isolated if it represents an isolated zero of s , and f attribute to it the real multiplicity equal, by definition, to |Ie(s )|. Thus, Theorem ℓ f 3.1.1 implies the following result. 3.1.2. Corollary. Assume that all the real lines ℓ ⊂ X of a real hypersurface X ⊂Pn+1 of degree 2n−1 are isolated. Then |Ie(s )|≥(2n−1)!!. (cid:3) Pℓ⊂X ℓ f As is known (see, for example, [BVV]), for a generic f (that is for a generic X) the section s is transverse to the zero section, and so, all the lines ℓ ⊂ X are f isolated and all have indices Ie(s ) = ±1. Let N + and N − denote the number ℓ f R R of reallines of indices 1 and −1, respectively. Thus, for a generic f, the number of real lines is N ++N − ≥N +−N − =N e =(2n−1)!! R R R R R 3.1.3. Corollary. For a generic hypersurface X ⊂ Pn+1 of degree (2n−1), the number of real lines ℓ⊂X is finite and bounded from below by (2n−1)!!. (cid:3) Remark. Our choice of an orientation of E and of the multiplicities of ℓ ⊂ X is a bit different from that in [OT]. The orientation is not essential for us, since |N e| R and|Ie(s)| remaininvariant,while the multiplicities makea difference. Infact,the x complex multiplicities used in [OT] are bounded from below by our indices |Ie(s)|. x 4. Showbiz 4.1. Cubic surfaces: 3 versus 27. Recall, that any nonsingular cubic surface contains exactly 27 = c (Sym3(τ∗ (C)))[G (C4)] complex lines (here τ (C) de- 4 2,4 2 2,4 notes the tautological bundle over the complex Grassmannian G (C4)), and it 2 implies, in particular, that for a nonsingular cubic surface, a section s , which f can be defined for Sym3(τ∗ (C)) as well as for Sym3(τ∗ ), is in the both cases 2,4 2,4 transversal to the zero section. As a result, Theorem 3.1.1 reads as identity NRe =NR+−NR− =3 and estimate NR ≥3. The estimate NR ≥ 3 was known already to Schläfli [Sc], who proved that the number of real lines on a non-singular real cubic surface takes only the values 27, 15, 7, and 3. Segre [Se] was probably the first to observe that the real lines on a nonsingular real cubic surface can be divided in two species, named by him hyperbolic and elliptic, and that numerically the distribution between the species, NR =h+e, is as follows: 27=15+12,15=9+6,7=5+2,and 3=3+0. It implies the following numerical coincidence. 4 S. FINASHIN, V. KHARLAMOV 4.1.1. Proposition. For any non-singular real cubic surface, h=N +, e=N −. (cid:3) R R 4.2. A direct proof of Proposition 4.1.1. Considerarealline ℓonarealnon- singular projective cubic surface X. Choose projective coordinates x,y,u,v in P3 so that the line ℓ is given by equations x = y = 0. Then the defining polynomial of X has the form f = u2L +2uvL + v2L +uQ +vQ + C, where L , 11 12 22 1 2 ij Q , and C are homogeneous polynomials in x and y of degree, respectively, one, i two, and three. According to Segre’s definitions, the line ℓ is elliptic (respectively, hyperbolic) if the quadratic form L2 −L L is definite (respectively, indefinite). 12 11 22 4.2.1. Lemma. A real line contributes 1 into N + if the line is hyperbolic, and to R N − if elliptic. R Proof. A line ℓ′ in a neighborhood of ℓ ∈ G (R4) hits the coordinate projective 2 planes v = 0 and u = 0 at some points [x : y :1:0] and [x : y :0:1], and thus, 1 1 2 2 ℓ′ can be givenin a parametricform as (u,v)7→u(x ,y ,1,0)+v(x ,y ,0,1). The 1 1 2 2 value of section s (ℓ′) is defined by the restriction of f to ℓ′, i.e, by substitution f x = ux +vx and y = uy +vy in the polynomial f. Letting L = l x+m y 1 2 1 2 ij ij ij for all i,j, we obtain s (ℓ′) = u2[l (ux +vx )+m (uy +vy )]+.... It is f 11 1 2 11 1 2 straightforward to check, following the polar correspondence, that our orientation of the total space of Sym3(τ∗ ) agrees with the order x ,x ,y ,y of the local 2,4 1 2 1 2 coordinatesinG (R4)andtheframeu3,u2v,uv2,v3 thatgivesalocaltrivialization 2 of Sym3(τ∗ ). The Jacobi matrix of s in these coordinates is as follows 2,4 f l 0 m 0 11 11 2l l 2m m  12 11 12 11 l 2l m 2m  22 12 22 12  0 l 0 m  22 22 Now, it remains to notice that the determinant of this matrix, whose sign is the index Ie(s ), is opposite to the discriminant of the binary quadratic form L2 − ℓ′ f 12 L L . (cid:3) 11 22 5. Black Holes 5.1. Congruences. As was disclosedby Gru¨nberg and Moree [GM], the residues of NC modulo 2q form a 2q-periodic sequence for all q ≥ 1. For the modulo 2q residues of N e = (2n−1)!! the 2q-periodicity is evident. Up to q = 2 the two R sequences of residues coincide, NC =NRe mod 4. The relation between 8-residues is less trivial; the period is 1,1,3,3,5,5,7,7 for NC and 1,1,3,7,1,1,3,7 for NRe. What can be general rules and conceptual explanations? In the case ofcubic surfaces another congruenceholds, NR =NC mod 4. How- ever, for any fixed dimension n ≥ 3, the range of NR includes all the odd integers in a certain interval from some NRmin ≥ NRe to some NRmax ≤ NC. In a sense, this phenomenon reflects the difference in the Galois groups of the corresponding enumerativeproblems (starting from3-folds the Galois groupis the full symmetric group, see [H]). ABUNDANCE OF REAL LINES ON REAL PROJECTIVE HYPERSURFACES 5 5.2. Welschinger invariants. There is a method, introduced by J.Y. Welschin- ger [We], of counting real rational curves in a given homology class on arbitrary real rational surfaces and certain real 3-folds. As was observed by J.Solomon [So], in the case of quintic 3-folds the number N e =15 can be seen as the Welschinger R invariant of the generator in the second homology group of the complexification of the quintic. To get N e = 3 in the case of cubic surfaces, one should take the R sum of the Welschinger invariants over all homology classes of projective degree 1. Though, the invariants should be taken not with the signs traditionally applied in the case of rational surfaces, but with signs copied from that used in the case of 3-folds (just consider the Pin−-structure q : H1(XR;Z/2) → Z/4 inherited by the reallocusXR ofanon-singularcubicsurfacefromtheambientrealprojectivespace and define the sign of a line l to be equal to iq(l)−1). As an additional similarity, let us point that Welschinger invariants tend to have asymptotic and arithmetic properties analogous to that of the Euler numbers of real lines treated in this note (see, for example, [IKS]). Is there a general theory of real enumerative invariants that bring all these results together? 5.3. Hypersurfaces of other degrees. IfX isarealhypersurfaceofodddegree d<2n−1inPn+1(R),thedimensionofthevarietyFR(X)ofreallinescontainedin X is≥2n−1−d,andisequalto2n−1−dforgenericX;inparticular,itisalways nonempty (see [DM]). What is about abundance? Is there a lower bound for the total Betti number, b∗(FR(X)) comparable in logarithmic scale with b∗(FC(X))? References [BVV] W.Barth,A.VandeVen,Fanovarietiesoflinesonhypersurfaces,Arch.Math.(Basel) 31(1978/79), 96–104. [DM] O. Debarre, L. Manivel, Sur les intersections complètes réelles, CRAS 331 (Série I) (2000), 887–992. [GM] D.B.Gru¨nberg,P.Moree,Sequencesofenumerativegeometry: congruencesandasymp- totics(with Appendix by Don Zagier),Experiment.Math.17(2008), 409–426. [H] J.Harris,Galois groups of enumerative problems, DukeMath.J.46(1979), 685–724. [IKS] I.Itenberg,V.Kharlamov,E.Shustin,WelschingerinvariantsofrealDelPezzosurfaces of degree ≥3,arXiv:1108.3369,32pages. [OT] C. Okonek, A. Teleman, Intrinsic signs and lower bounds in real algebraic geometry, arXiv:1112.3851,17pages. [Sc] L.Schläfli, An attempt to determine the twenty-seven lines upon a surface of the third order, and to divide such surfaces into species in reference to the reality of the lines upon the surface,Quart.J.PureAppl.Math.2(1858), 110–120. [Se] B.Segre,TheNon-SingularCubicSurfaces.AnewmethodofInvestigationwithSpecial Reference toQuestions of Reality,OxfordUniv.Press,London,1942. [So] J.P. Solomon, Intersection theory on the moduli space of holomorphic curves with La- grangian boundary conditions,arXiv:math/0606429, 79pages. [We] J.-Y. Welschinger, Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry,C.R.Acad.Sci.Paris,Sér.I336(2003), 341–344. Middle East Technical University, Department of Mathematics Ankara 06531Turkey Université de Strasbourg et IRMA (CNRS) 7 rue René Descartes 67084Strasbourg Cedex, France