Hard Lefschetz Conjecture and Hodge Standard Conjecture on blowing-up of Projective Spaces Pranav Haridas Advisor : Prof. (Dr.) Pascal Boyer 1 Contents 1 Blowing-up of Projective Spaces. 4 1.1 Preliminaries . 4 1.2 Blowing-up of Projective Spaces . 7 n 1.2.1 Construction of B ..................... 8 n 1.2.2 Divisors on B ....................... 8 d 1.2.3 The Drinfeld Upper Half spaces ΩbK . 13 2 Hard Lefschetz and Hodge Standard Conjecture 14 2.1 The Hard Leftschetz Conjecture . 14 2.2 Hodge Standard Conjecture . 15 3 Cohomology of Varieties obtained by Blowing-up 20 3.1 Some Fundamental Thoerems . 20 3.2 Proof of Results . 27 3.2.1 Cohomology of Blowing-ups . 27 3.2.2 The Main result . 32 2 Acknowledgements I wish to take this opportunity to express my sincere thanks to Prof. (Dr.) Pascal Boyer for giving me his valuable time to discuss the subject matter and suggesting the framework of this thesis. But for his encouragement and consideration, this work would not have been possible. I am thankful to Prof. (Dr.) Fontanne of the University of Paris Sud, who was instrumental in inculcating a sense of confidence in me to carry on my academic endeavor in the University. I am also thankful to the authorities, who preside over the Erasmus Mundus ALGANT program and the European Commission for affording me an oppor- tunity to study Mathematics in two celebrated Universities in Europe. I shall be failing in my duty if I do not place on record my sincere gratitude to all my Professors in the University of Padova, Italy and the University of Paris Sud, France, who have helped me to peep into the world of Mathematics. And finally, I wish to thank my father, my mother and my brother for their unflinching support throughout the time of preparation of this thesis and for making my life so beautiful and meaningful with their love and affection. 3 Chapter 1 Blowing-up of Projective Spaces. 1.1 Preliminaries Let X be a noetherian scheme and B a quasi-coherent sheaf of OX -modules L which has a structure of a sheaf of graded OX -algebras. i.e B = d≥0 Bd, where Bd is the homogenous part of degree d with B0 = OX , B1 a coherent OX -module and B generated by B1 as an OX -algebra. For an open affine subset U = Spec A of X, B(U) is a graded A-Algebra. πU : Proj B(U) −! U is a projective A-scheme. Then B quasi-coherent ∼ −1 implies that Proj B(Uf ) = πU (Uf ) . Hence for two affine open subests U −1 −1 and V , we have a natural isomorphism πU (U \ V ) = πV (U \ V ). Thus we can glue these schemes to obtain a scheme which we denote by Proj B. Also, there is a natural morphism π : Proj B −! X with the property that for every affine open subset U = Spec A, π−1(U) =∼ Proj B(U). Since a closed subscheme corresponds to a coherent sheaf of ideals, we can speak of blowing up along a coherent sheaf of ideals I on X . L d d Definition 1.1.1. Consider a sheaf of graded algebras B = d≥0 I , I being th 0 the d power of I with I = OX . We define Xe = Proj B to be the Blowing- up of X with respect to I. If Y is the closed subscheme corresponding to I, then we say Xe is the Blowing-up of X along Y . 4 1.1. PRELIMINARIES Definition 1.1.2. Let f : X −! Y be a morphism of schemes, I ⊆ OY be a sheaf of ideals on Y . Let f −1I be the inverse image of the sheaf I. f −1I is a −1 −1 sheaf of ideals in f OY . Using the natural map f OY −! OX , we define 0 the Inverse Image Ideal Sheaf of I ⊆ OX to be the sheaf of ideals generated by f −1I. n Let A be a ring. On PA we have the invertible sheaf O(1) and the homoge- n nous co-ordinates (x0; x1; :::::; xn) as global sections in Γ(PA; O(1)) generate n O(1). Let X be a scheme over A and φ : X −! PA be an A-morphism. Then L = φ∗(O(1)) is an invertible sheaf on X and the global sections ∗ si = φ (xi) 2 Γ(X; L) generate the sheaf L. Conversely Theorem 1.1.3. If L is an invertible sheaf on X and if (s0; s1; :::::; sn) 2 n Γ(X; L) generate L, the there exists a unique A-morphism φ : X −! PA such ∼ ∗ ∗ that L = φ (O(1)) and si = φ (xi). Proof. Let us define the open subset Xi = fP 2 Xj(si)P 2= mP LP g for i = 0; 1; ::::; n. If P2 = [iXi, then (si)P 2 mP LP 8i which implies that fsigi=0;:::;n do not generate LP , a contradiction. Hence, the fXigi cover X. We prove the theorem by providing a map from Xi to Ui where Ui is the open n n subset fxi 6= 0g of PA and then glueing them to obtain a map φ : X −! PA. n ∼ We have that Ui is isomorphic to AA = spec A[T0; ::::; Tn] with Ti = xj=xi where xi as above. Now consider the ring homomorphism A[T1; ::::; Tn] −! Γ(OX ;Xi) given by Tj −! sj=si. Now si 2= mP LP 8P 2 Xi and because L is locally free of rank 1, sj=si makes sense as an element of Γ(OX ;Xi). Also this map is well defined. Hence we have a map Xi −! Ui. By glueing n ∗ these maps, we get a A-morphism φ : X −! PA such that si = φ (xi). Any map φ satisfying the given properties is unique by construction above. Also, L =∼ φ∗(O(1)). Theorem 1.1.4 (Universal Property of blowing-up). Let X be a noetherian scheme, I a coherent sheaf of ideals and π : Xe −! X the blowing-up with 5 1.1. PRELIMINARIES −1 respect to I. If f : Z −! X is any morphism such that f I.OZ is an invertible sheaf of ideals on Z, then there exists a unique morphism g : Z −! Xe factoring through f. g ... Z ............. ............. ............. ............. .................... ... Xe ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... π ..... ..... f ..... ..... ..... ..... ..... ..... ...... ..... ........ ... X Proof. We may assume without loss of generality that X = spec A is affine with A noetherian. Further we have that the sheaf of ideals I is given by the ideal I whose generators are fa1; ::::ang. We have the natural morphism d A[x1; :::::; xn] −! S = ⊕d≥0I sending xi to ai with kernel fhjh(a1; :::::; an) = n−1 −1 0. This induces a closed immersion Xe −! PA . Let L = f I.OZ be the invertible sheaf. Then the inverse images of fa1; :::::; ang, as global sections, generate L. n−1 By the previous theorem there is a unique morphism g : Z −! PA such ∗ ∗ that L = g (O(1)) and si = g Xi. Then the map g factors through the closed n−1 subscheme Xe of PA . If h is such that h(a1; ::::an) = 0, then f(s1; ::::sn) = 0 in Γ(Z; Ld) where h is homogenous of deg d. Thus g is a map factoring f. This forces f −1I.O = g−1(π−1I.O ):O = g−1O .O . Thus there is a Z Xe Z Xe(1) Z surjective map g∗O (1) ∼ L. Xe = Since a surjective map of invertible sheaves on a locally ringed spaces is an isomorphism, we have g∗O ∼ L. Further we have that the sections s of Xe(1) = i L pull-backs sections xi of O(1). Now by Theorem 1.1.3 we have that g is unique. Corollary 1.1.5. Let f : X −! Y be a morphism of noethreian schemes and let I be a coherent sheaf of ideals on X. Let Xe be the blowing-up of I and let Ye be the blowing-up of Y with respect to the inverse image sheaf 6 1.2. BLOWING-UP OF PROJECTIVE SPACES −1 J = f I:OY . Then there exists a unique morphism fe : Ye −! Xe such that the diagram f e ... Y ............. ............. ............. ............. .................... X e. e. ....... ....... ..... ..... ... XY...................................................................................................................................... f is commutative. Also, if f is a closed immersion, then so is fe. Proof. The existence and uniqueness follows from the theorem above. The only thing that is to be checked is that fe is a closed immersion if f is. L d 0 0 L d Xe = ProjB where B = d≥0 I and Ye = Proj B where B = d≥0 J . Y a closed subscheme implies that B0 is a sheaf of graded algebras on X. Hence there is a natural surjectove homomorphism of graded rings B −! B0 which gives a closed immersion. Definition 1.1.6. Let Y be a closed subscheme. We call the closed subscheme Ye of Xe as above to be the strict transform of Y under the blowing-up π : Xe −! X. 1.2 Blowing-up of Projective Spaces n n n Definition 1.2.1. Let us denote by P for P . We define V ⊂ P to be Fq −1 n+1 a linear subvariety if φ (V [ f0g) ⊂ A is a linear subspace, where φ : n+1 n A nf0g −! P is the natural map. n Let Grd(P ) be the grassmann variety of linear subvarieties of dimension d n ` n n n and Gr∗(P ) = d Grd(P ). Let Grd(P )(Fq) be the linear subvarieties in P n of dimesion d defined over Fq.