# APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY

Size: px
Start display at page:

Transcription

1 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY In this appendix we begin with a brief review of some basic facts about singular homology and cohomology. For details and proofs, we refer to [Mun84]. We then discuss the Leray-Hirsch theorem and the Thom isomorphism, we review some special features of the cohomology of algebraic varieties, and finally, we carry out some simple computations that we need: the cohomology of a projective space and that of a smooth blow-up. 1. Basic facts about singular homology and cohomology For every Abelian group A and every non-negative integer p, we have a covariant functor H p, A and a contravariant functor H p, A from the category of topological spaces to the category of Abelian groups. This is the p th singular homology group respectively, the p th singular cohomology group with coefficients in A. More generally, we have a covariant functor H p,, A and a contravariant functor H p,, A from the category of pairs of topological spaces 1 to the category of Abelian groups. This is the p th relative homology group respectively, the p th relative cohomology group with coefficients in A. If f : X Y is a continuous map, the morphism induced in homology is denoted by f and that induced in cohomology is denoted by f. These functors satisfy the following properties: i If X consists of one point, then H 0 X, A A and H 0 X, A A, and H p X, A = 0 = H p X; A for p > 0. ii If X = i I X i, then H p X, A i I H p X i, A for all p 0, and H p X, A i I H p X i, A for all p 0. iii If f, g : X Y are homotopic continuous maps, then they induce the same maps in homology H p X, A H p Y, A and in cohomology H p Y, A H p X, A. iv For every pair X, Y, we have the following functorial long exact sequences: and... H p Y, A H p X, A H p X, Y, A H p 1 Y, A H p X, Y, A H p X; A H p Y, A H p+1 X, Y, A A pair X, Y consists of a topological space X and a subspace Y ; a morphism in this categoy from X, Y to X, Y is given by compatible continuous maps X X and Y Y. 1

2 2 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY v Excision If X, Y is a pair and U is a subset of X such that U is contained in the interior of Y, then the inclusion induces isomorphisms H p X U, Y U, A H p X, Y, A and H p X, Y, A H p X U, Y U, A for all p 0. In what follows we will mostly be interested in the case when the cohomology has coefficients in Z, Q, R, or C. An immediate consequence of the definition of singular homology and cohomology groups is that if π 0 X is the set of path-components of X, then 1 H 0 X, A A π 0X and H 0 X, A Hom Z A, Z π 0X. In particular, we have a canonical morphism of Abelian groups deg: H 0 X, A A, which is an isomorphism if X is path-connected. A useful tool for computing cohomology is provided by the Mayer-Vietoris sequence. Suppose that U and V are open subsets of X such that X = U V. If the following maps are given by inclusions: i U : U X, j U : U V U, i V : V X, j V : U V V, then for every Abelian group A, we have a long exact sequence:... H p X, A where α and β are given by α H p U, A H p β V, A H p U V, A H p+1 X, A..., αy = i Uy, i V y and βy 1, y 2 = j Uy 1 j V y 2. A similar sequence holds for homology. The following is the Universal Coefficient theorem, which describes the groups H p X, A and H p X, A in terms of the groups H p X, Z see [Mun84, Theorems 55.1, 56.1]. Theorem 1.1. For every Abelian group A and for every p 0, we have canonical short exact sequences 2 : and 0 Ext 1 ZH p 1 X, Z, A H p X, A Hom Z H p X, Z, A 0 0 H p X, Z Z A H p X, A Tor Z 1 H p 1 X, Z, A 0. Both sequences are split, but the splitting is non-canonical. Remark 1.2. An immediate consequence of the theorem is that if K is a field containing Q, then H p X, K H p X, Z Z K and H p X, K H p X, Z Z K H p X, K. 2 We make here the convention that H 1 X, Z = 0.

3 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY 3 Remark 1.3. Since H 0 X, Z is a free Abelian group, it is an immediate consequence of Theorem 1.1 that H 1 X, Z Hom Z H1 X, Z, Z. In particular, we see that H 1 X, Z has no torsion. Remark 1.4. Note that if M is a finitely generated Abelian group, then Ext 1 ZM, Z is torsion and Hom Z M, Z has no torsion. We thus deduce from Theorem 1.1 that if all homology groups H i X, Z are finitely generated, then the torsion subgroup H p X, Z tors of H p X, Z is isomorphic to Ext 1 ZH p 1 X, Z, Z, and H p X, Z/H p X, Z tors Hom Z H p X, Z, Z. If R is a ring, then H X, R := p 0 H p X, R has the structure of a graded-commutative graded ring with respect to the cup-product We also have a cap product map H p X, R H q X, R a, b a b H p+q X, R. H p X, R H q X, R H q p X, R. This makes H X, R := p H px, R a left module over H X, R. These operations satisfy the projection formula: if f : X Y is a continuous map, then 2 f f α β = α f β for every α H Y, R, β H X, R. One can define the cup-product more generally for relative cohomology: if R is a ring and Y 1, Y 2 X, then we have a cup-product map H p X, Y 1, R H q X, Y 2, R H p+q X, Y 1 Y 2, R that extends the one in the absolute case and which satisfies similar properties. Given two topological spaces X and Y, the Künneth theorem computes the cohomology of the product X Y in terms of the cohomologies of X and Y. The simplest form concerns homology and says that for every m 0, there is a natural short exact sequence 0 H p X, Z Z H q Y, Z H m X Y, Z Tor Z 1 Hp X, Z, H q Y, Z 0, p+q=m p+q=m 1 which is split, but non-canonically. If all homology groups H p X, Z are finitely generated, then for every m 0 we have a natural short exact sequence for cohomology: 0 H p X, Z Z H q ϑ Y, Z H m X Y, Z Tor Z 1 H p X, Z, H q Y, Z 0, p+q=m p+q=m+1 where ϑα β = pr 1α pr 2β. Moreover, this sequence splits non-canonically if also all H q Y, Z are finitely generated. For these results, see [Mun84, 59, 60]. Suppose now that M is a compact, orientable, n-dimensional C -manifold. The orientation on M defines on M a fundamental class µ M H n M, Z. The following result is one of the incarnations of Poincaré duality.

4 4 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY Theorem 1.5. With the above notation, the cap product map 3 H p M, Z H n p M, Z, α α µ M is an isomorphism. One can show that for a compact manifold M, the groups H p M, Z hence also the groups H p M, Z are finitely generated. We can now rephrase Poincaré duality as follows. Consider the following bilinear pairing 4 H p X, Z H n p X, Z Z, α, β deg α β µ M. Via the isomorphism 3, this gets identified with the morphism H p X, Z H p X, Z Z, α, β degα β. Using the Universal Coefficient theorem, we see that the map induced after killing the torsion is a perfect pairing. We thus conclude that 4 induces after killing the torsion a perfect pairing: 5 H p X, Z/H p X, Z tors H n p X, Z/H n p X, Z tors Z. By tensoring with Q, we obtain a perfect pairing H p X, Q H n p X, Q Q, α, β deg α β µ M. 2. The Leray-Hirsch theorem and the Thom isomorphism We now discuss two results that we will need later, in order to compute the cohomology of projective bundle and that of smooth blow-ups. We begin with a result that allows the description of the cohomology for the total space of a locally trivial map. Recall that a continuous map f : Y X is locally trivial, with fiber F, if X can be covered by open subsets U such that f 1 U is homeomorphic over U with U F. In the cases of interest to us, X will be covered by finitely many such subsets, and in this case the results below are easier to prove. Theorem 2.1. Let f : Y X be a continuous map, which is locally trivial, with fiber F. Suppose that all cohomology groups H p F, Z are finitely generated, free Abelian groups. If α 1,..., α r H Y, Z are such that for every x X, the inclusion i x : f 1 x Y has the property that i xα 1,..., i xα r give a basis of H f 1 x, Z, then we have an isomorphism of Abelian groups r Z r Z H X, Z H Y, Z, m 1,..., m r β = m i α i f β. Proof. If Y = X F, then the assertion is an easy consequence of the Künneth theorem. When X has a finite cover by open subsets U i such that f 1 U i is isomorphic over U i with U i F, one argues by induction on the number of open subsets, using the Mayer-Vietoris sequence. We do not give the proof in the general case, as we will not need it. i=1

5 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY 5 We now turn to the definition of the Thom class and to the statement of the Thom isomorphism theorem. Suppose that π : E X is an oriented, real vector bundle of rank r 1, on a topological space X. In particular, for every x X, the fiber E x = π 1 x is an r-dimensional, oriented real vector space. We will consider X embedded in E via the 0-section. Recall that H r R r, R r {0}, Z Z, and choosing a generator is equivalent to the choice of an orientation on R r. Indeed, it follows from the long exact sequence in cohomology for R r, R r {0} and from the fact that R r is contractible, while R r {0} is homotopically equivalent to the sphere S r 1, that 3 H r R r, R r {0}, Z H r 1 R r {0}, Z H r 1 S r 1, Z Z. Theorem 2.2. Let π : E X be an oriented, real vector bundle of rank r 1, on a topological space X. i There is a unique η E H r E, E X, Z the Thom class of E such that for every x X, the restriction of η E to H r π 1 x, π 1 x {0}, Z corresponds to the given orientation on the fiber. ii For every closed subset W of X and every p 0, we have and the map H p E, E W, Z = 0 for p < r H p r X, X W, Z H p E, E W, Z, α π α η E is an isomorphism for every p r the Thom isomorphism. Proof. We only sketch the argument under the assumption that there are finitely many open subsets of X that cover X and such that E is trivial over each of them this is enough for our purpose. For both i and ii, arguing by induction on the number of open subsets and using the Mayer-Vietoris sequence, we reduce to the case when E is trivial. In this case the assertions follow easily from the Künneth theorem. Remark 2.3. We will apply the Thom isomorphism via the following consequence. Suppose that X is an oriented, n-dimensional C -manifold, and Y is a closed, oriented submanifold, of codimension r. The orientations on the tangent bundles T X and T Y induce an orientation on the normal bundle N = N Y/X, since we have a canonical isomorphism dett X dett Y detn. Recall that by the tubular neighborhood theorem, there is an open neighborhood U of Y in X, a retract r : U Y, and a homeomorphism h: U N, such that we have a commutative diagram Y j U r Y 1 Y Y h 1 Y i N π Y, 3 In order for this to also hold for r = 1, we need to replace the 0 th cohomology by the reduced version, the cokernel of the canonical map Z H 0, Z.

6 6 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY where j is the inclusion, i is the embedding as the 0-section, and π is the vector bundle map. If W a a closed subset of Y, we then obtain isomorphisms H p r Y, Y W, Z H p N, N W, Z H p U, U W, Z H p X, X, W, Z, for every p 0, where the first isomorphism is provided by Theorem 2.2, the second one is induced by h, and the third one is given by excision we make the convention that the first group is 0 for p < r. In particular, we have isomorphisms 6 H p r Y, Z H p X, X Y, Z for all p 0. Remark 2.4. If X and Y are as in the previous remark and ι: Y X is the inclusion, then the Gysin map ι : H p Y, Z H p+r X, Z is defined as the composition H p Y, Z H p+r X, X Y, Z H p+r X, Z, where the first map is the isomorphism in 6 and the second one is the canonical map induced by X, X, X Y. Therefore the long exact sequence in cohomology for the pair X, X Y becomes 7... H p r Y, Z ι H p X, Z H p X Y, Z H p r+1 Y, Z.... One can show that Gysin maps are functorial for closed embeddings of oriented C - manifolds. Remark 2.5. If f : Y X is any differentiable map between compact, oriented C - manifolds, with dimx = n and dimy = m, we get a morphism f : H p Y, Z H p+r X, Z, where r = n m. Indeed, Poincaré duality gives isomorphisms H p Y, Z H m p Y, Z and H p+r X, Z H m p X, Z and via these isomorphisms, f corresponds to the morphism in homology associated to f. It is clear that this is functorial. Moreover, one can show that if f embeds Y as a submanifold of X, then this definition coincides with the one in Remark Cohomology of complex algebraic varieties In this section we review some basic facts about the topology of complex algebraic varieties. For proofs and details, we refer to [Voi07]. We make use of the singular cohomology for complex algebraic varieties. Given such a variety X not necessarily irreducible, we denote by X an the same variety, endowed with the classical topology. Recall how this is defined: if X is affine, then we have a closed immersion X C n, and we endow X with the subspace topology, where we take on C n the Euclidean topology. It is straightforward to see that the topology we get is independent of the embedding. If X is not-necessarilyaffine, then we obtain the topology on X an by gluing the topology that we get on the subsets in an affine open cover. In this way, we get a functor from complex algebraic varieties to topological spaces. Whenever we consider the singular cohomology of an algebraic

7 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY 7 variety over C, we always consider it endowed with the classical topology. The following are some basic facts that we will use: i If X is irreducible or, more generally, connected, then X an is connected. ii Since X is assumed to be separated, then X an is Hausdorff. iii If X is a complete variety, then X an is compact. iv If X is smooth of dimension n, then X an is a complex manifold of dimension n, and therefore also a real C -manifold of dimension 2n. Like every complex manifold, it carries a canonical orientation. We note that for every complex algebraic variety X and every Abelian group A, we have canonical isomorphisms H p X an, A H p X an, A for all p 0, where on the left-hand side we have sheaf cohomology with values in the constant sheaf A, and on the right-hand side we have the singular cohomology groups with coefficients in A. In fact, such isomorphisms hold for all nice topological spaces see [God73]. From now on, we always identify these two groups. In order to simplify the notation, we write H i X, Z and H i X, Z instead of H i X an, Z and H i X an, Z, and similarly for the other singular cohomology groups. If Y is a complete complex algebraic variety of dimension n, then Y an admits a finite triangulation, with simplices of dimension 2n. More generally, if Z is a closed subvariety of Y, then Z an and Y an admit compatible finite triangulations, with simplices of dimension 2n. Given an arbitrary complex algebraic variety X, of dimension n, it can be embedded as an open subvariety of a complete variety Y by Nagata s theorem. Using the above-mentioned fact about triangulations, one can show that the groups H i X, Z and H i X, Z are finitely generated, and they vanish for i > 2n. The Betti numbers of X are the ranks of these cohomology groups: b i X := rk H i X, Z = rk H i X, Z. Let X be a complex algebraic variety. The topological space X an carries a sheaf of holomorphic functions O X an. In order to define this sheaf, suppose first that X is affine and consider an embedding X C n as a Zariski closed subset. By definition, holomorphic functions on open subsets of X an are the functions that can be locally extended to holomorphic functions on open subsets of C n. The definition is independent of embedding and in the case of an arbitrary variety X, the sheaf O X an is obtained by glueing the corresponding sheaves on the open subsets in an affine open cover. Note that we have a morphism of locally ringed spaces ι: X an, O X an X, O X whose corresponding morphism of topological spaces X an X is the identity map. If F is a coherent sheaf on X, then one gets a sheaf F an := ι F on X an and in this way we get a functor from coherent sheaves on X to coherent sheaves of O X an-modules. Serre s GAGA theorem says that if X is projective, then this functor is an equivalence of categories, preserving the locally free sheaves in each category. Moreover, for every coherent sheaf on X, we have a canonical isomorphism H p X, F H p X an, F an.

8 8 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY We note that as in the algebraic category, the group PicX an of rank 1 locally free O X an-modules is isomorphic to H 1 X an, OX an. For all these results, we refer to [Ser56]. A very useful tool in this analytic setting is provided by the exponential sequence. If X is any complex algebraic variety, then on X an we have a short exact sequence 0 Z X O X an exp2πi OX an 0, where the first map is the inclusion and the second map takes a holomorphisc function ϕ to exp2πiϕ. Exactness is an immediate consequence of the local existence of the logarithm function. By taking the long exact sequence in cohomology associated to the exponential sequence, we obtain an exact sequence H 1 X, Z H 1 X an, O X an H 1 X an, O X an We will also denote by c 1 the composition c1 H 2 X, Z. PicX PicX an = H 1 X an, O X an H2 X, Z. This is the first Chern class map for line bundles. Chern classes of line bundles behave functorially. Indeed, if f : Y X is a morphism, then we have a commutative diagram with exact rows 0 Z f an 1 O X an f an 1 O X an 0 0 Z O Y an O Y an 0, and by comparing the corresponding long exact sequences in cohomology, we see that for every L PicX, we have c 1 f L = f c 1 L. If X is any complete complex algebraic variety of dimension n, then X carries a fundamental class µ X H 2n X, Z, defined as follows. If X is smooth, then X an is a compact oriented C -manifold of dimension 2n and µ X is the corresponding fundamental class. Suppose now that X is an arbitrary complete complex n-dimensional variety. By Hironaka s theorem, we have a proper morphism f : Y X such that Y is smooth, and we put µ X = f µ Y H 2n X, Z. One can show that this is independent of the choice of f. Moreover, if g : W X is an arbitrary surjective, generically finite morphism of complete complex algebraic varieties, then 8 f µ W = degfµ X. On the other hand, if g is such that dim gw < dimw, then g µ W = 0 this is due to the fact that g factors through H gw, Z and Hi gw, Z = 0 for i > 2 dim gw. Recall that if f : W X is a surjective morphism of smooth, complete complex algebraic varieties, with dimw = dimx = n, then for every p we have a morphism f : H p W, Z H p X, Z

9 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY 9 induced via Poincaré duality by the push-forward in homology see Remark 2.5. It is easy to deduce from 8 and the projection formula 2 that f f = degf Id. In particular, f is injective if f is birational. Suppose now that X is a complete complex algebraic variety, of dimension n. We have a symmetric, multilinear map H 2 X, Z n Z, α 1,..., α n α 1... α n := deg α 1... α n µ X. This is compatible with intersection numbers of line bundles via the Chern class map: if L 1,..., L n PicX, then L 1... L n = c 1 L 1... c 1 L n. Example 3.1. If X is a smooth, n-dimensional, complete complex variety, Y is a smooth, Cartier divisor on X, and i: Y X is the inclusion, then i µ Y = c 1 O X Y µ X in H 2n 2 X, Z. Via the projection formula, this implies that for every α H p X, Z, we have i i α µ Y = α c 1 O X Y H p+2 X, Z, where we identify cohomology and homology via Poincaré duality. More generally, suppose that Y 1,..., Y r are smooth, effective divisors on X that intersect transversely, so that Y = Y 1... Y r is smooth, of codimension r in X. An easy induction on n implies that if i: Y X is the inclusion, then i µ Y = c 1 O X Y 1... c 1 O X Y r µ X. For example, if X = P n, then for every d n, we have c 1 O P n1 d µp n = i µ L, where L P n is any linear subspace of codimension d and i: L P n is the inclusion. We now recall the Hodge decomposition. Suppose that X is a smooth, projective complex algebraic variety. In this case, we have for every m 0 a decomposition H m X, Z Z C = H m X, C = H p,q, where H p,q H q X, Ω p X. p+q=m Moreover, the complex conjugation induces a real linear map on H m X; C that maps H p,q to H q,p. In particular, we have h q X, Ω p X = hp X, Ω q X for all p, q 0. Remark 3.2. If X is a smooth, projective complex algebraic variety, then Poincaré duality is compatible with the Hodge decomposition, inducing an isomorphism between H p,q and the dual of H n p,n q, where n = dimx. Suppose now that f : X Y is a morphism of such varieties. The pull-back f : H Y, C H X, C preserves the Hodge decomposition: for every p and q, we have f H p,q Y H p,q X,

10 10 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY with the map being induced by the morphism f Ω p Y Ωp X. Using the definition of f : H X, C H Y, C and the above compatibility between Poincaré duality and the Hodge decomposition, we deduce that where n = dimx and m = dimy. f H p,q X H m n+p,m n+q Y, Finally, we need Noether s formula for smooth complex projective surfaces. This says that if X is such a surface, then where χx, O X = 1 12 K 2 X + χ top X, χ top X = b 0 X b 1 X + b 2 X b 3 X + b 4 X = 2 2qX + b 2 X is the topological Euler-Poincaré characteristic. This formula is a consequence of the Hirzebruch-Riemann-Roch theorem for surfaces see [Ful98, Example ]. 4. Cohomology of projective bundles and smooth blow-ups We begin with the computation of the cohomology of the complex projective space. Proposition 4.1. For every n 0, we have a ring isomorphism H P n, Z Z[x]/x n+1, mapping c 1 O P n1 to the class of x. In particular, we have H i P n, Z = 0 if i is odd or i > 2n, and H i P n, Z Z, otherwise. Proof. We proceed by induction of n, following the argument in [Voi07, Theorem 7.14]. The assertion is trivial for n = 0. In order to prove the induction step, note that we have a closed subvariety Y of P n such that Y P n 1 and P n Y C n. The long exact sequence 7 becomes... H i 2 P n 1, Z H i P n, Z H i C n, Z H i 1 P n 1, Z.... Since C n is contractible, we have H i C n, Z = 0 for all i > 0, and we conclude from the exact sequence that H i P n, Z H i 2 P n 1, Z for all i 1 and H 0 P n, Z H 0 C n, Z Z. The last assertion in the proposition thus follows from the inductive assumption. In order to complete the proof of the induction step, we need to show that if h = c 1 O P n1, then h k is a generator of H 2k P n, Z for 0 k n. The key point is that h n = O P n1 n = 1. In particular, we have h k 0 for 0 k n. For every element α H 2k P n, Z, we can thus write α = qh k, for some q Q. On the other hand, since q = qh n = deg α h n k η P n Z, we conclude that q Z. This implies that H 2k P n, Z is generated by α k, completing the proof.

11 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY 11 Corollary 4.2. If X is a complex variety and π : PE X is a projective bundle on X, with rke = n + 1, then we have a ring isomorphism H X, Z[x]/x n+1 H PE, Z n n, α i x i π α i c 1 O PE 1 i. Proof. It is clear that the map is a ring homomorphism. In order to see that it is an isomorphism, it is enough to apply the Leray-Hirsch theorem for the elements c 1 O P n1 i H 2i PE, Z, for 0 i n; the fact that they satisfy the hypothesis in the theorem follows from Proposition 4.1. We can now compute the cohomology of a smooth blow-up. Let us first fix some notation. We consider a smooth complex variety X and a smooth closed subvariety Y of X, of codimension r. Let f : X X be the blow-up of X along Y, with exceptional divisor E. Let j : E X be the inclusion and g : E Y the map induced by f. Proposition 4.3. With the above notation, for every i 0, we have an isomorphism i=0 i=0 r 1 H p X, Z H p 2q Y, Z H p X, Z q=1 given by r 1 β, α 1,..., α r 1 f β + j c 1 g α q O E 1 q, q=1 with the convention that H p 2q Y, Z = 0 if p 2q < 0. Proof. Since E is a projective bundle over Y, with fibers isomorphic to P r 1, it follows from Corollary 4.2 that we have an isomorphism r r H p 2q Y, Z H p 2 E, Z, α 1,..., α r g α q c 1 O E 1 q 1. q=1 By considering the long exact cohomology sequences for the two pairs X, X Y and X, X E, we obtain the following commutative diagram with exact rows: H p X, X Y, Z f H p X, Z f q=1 H p X Y, Z H p+1 X, X Y, Z H p X, X E, Z H p X, Z H p X E, Z H p+1 X, X E, Z, in which the vertical maps are induced by the pull-back in cohomology corresponding to f. Note that the third vertical map is an isomorphism. Moreover, by using Thom isomorphisms as in Remark 2.4, we obtain a commutative diagram H p X, X Y, Z H p 2r Y, Z f f H p X, X E, Z ϕ p H p 2 E, Z,

12 12 APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY in which the horizontal maps are isomorphisms. Moreover, one can check that ϕ p α = g α c 1 O E 1 r 1 for al α H p 2r Y, Z. Using also the fact that f : H p X, Z H p X, Z is injective 4, the assertion in the proposition follows from the above exact sequences, using a straightforward diagram chase. References [Ful98] W. Fulton, Intersection theory, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2, Springer-Verlag, Berlin, [God73] R. Godement, Topologie algébrique et théorie des faisceaux, Troisième édition revue et corrigée, Publications de l Institut de Mathématique de l Université de Strasbourg, XIII, Actualités Scientifiques et Industrielles, No. 1252, Hermann, Paris, [Mun84] J. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, , 2, 3 [Ser56] J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble , [Voi07] C. Voisin, Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, , 10 4 When X and Y are complete, we have seen that this follows from the fact that f f = Id, and this is the only case that we need. The assertion in the general case follows similarly using Borel-Moore homology and Poincaré duality for non-compact manifolds.

### APPENDIX 3: AN OVERVIEW OF CHOW GROUPS

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss

### LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion

### Math 797W Homework 4

Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

### DELIGNE S THEOREMS ON DEGENERATION OF SPECTRAL SEQUENCES

DELIGNE S THEOREMS ON DEGENERATION OF SPECTRAL SEQUENCES YIFEI ZHAO Abstract. We present the proofs of Deligne s theorems on degeneration of the Leray spectral sequence, and the algebraic Hodge-de Rham

### Oral exam practice problems: Algebraic Geometry

Oral exam practice problems: Algebraic Geometry Alberto García Raboso TP1. Let Q 1 and Q 2 be the quadric hypersurfaces in P n given by the equations f 1 x 2 0 + + x 2 n = 0 f 2 a 0 x 2 0 + + a n x 2 n

### Chern classes à la Grothendieck

Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

### INTERSECTION THEORY CLASS 12

INTERSECTION THEORY CLASS 12 RAVI VAKIL CONTENTS 1. Rational equivalence on bundles 1 1.1. Intersecting with the zero-section of a vector bundle 2 2. Cones and Segre classes of subvarieties 3 2.1. Introduction

### Special cubic fourfolds

Special cubic fourfolds 1 Hodge diamonds Let X be a cubic fourfold, h H 2 (X, Z) be the (Poincaré dual to the) hyperplane class. We have h 4 = deg(x) = 3. By the Lefschetz hyperplane theorem, one knows

### Lecture on Equivariant Cohomology

Lecture on Equivariant Cohomology Sébastien Racanière February 20, 2004 I wrote these notes for a hours lecture at Imperial College during January and February. Of course, I tried to track down and remove

### CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents

CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally

### Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism

Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism UWO January 25, 2005 Marc Levine Prelude: From homotopy theory to A 1 -homotopy theory A basic object in homotopy theory is a generalized

### Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

### The Riemann-Roch Theorem

The Riemann-Roch Theorem TIFR Mumbai, India Paul Baum Penn State 7 August, 2015 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond ellipticity 5. The Riemann-Roch

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

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

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

### LECTURE 6: THE ARTIN-MUMFORD EXAMPLE

LECTURE 6: THE ARTIN-MUMFORD EXAMPLE In this chapter we discuss the example of Artin and Mumford [AM72] of a complex unirational 3-fold which is not rational in fact, it is not even stably rational). As

### TitleOn manifolds with trivial logarithm. Citation Osaka Journal of Mathematics. 41(2)

TitleOn manifolds with trivial logarithm Author(s) Winkelmann, Jorg Citation Osaka Journal of Mathematics. 41(2) Issue 2004-06 Date Text Version publisher URL http://hdl.handle.net/11094/7844 DOI Rights

### STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY

STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY ALEXANDER MERKURJEV 1. Introduction Let p be a prime integer. For a pair of topological spaces A X we write H i (X, A; Z/pZ) for the i-th singular cohomology group

### Curves on P 1 P 1. Peter Bruin 16 November 2005

Curves on P 1 P 1 Peter Bruin 16 November 2005 1. Introduction One of the exercises in last semester s Algebraic Geometry course went as follows: Exercise. Let be a field and Z = P 1 P 1. Show that the

### Algebraic geometry versus Kähler geometry

Algebraic geometry versus Kähler geometry Claire Voisin CNRS, Institut de mathématiques de Jussieu Contents 0 Introduction 1 1 Hodge theory 2 1.1 The Hodge decomposition............................. 2

### Non-uniruledness results for spaces of rational curves in hypersurfaces

Non-uniruledness results for spaces of rational curves in hypersurfaces Roya Beheshti Abstract We prove that the sweeping components of the space of smooth rational curves in a smooth hypersurface of degree

### An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations

An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations Mircea Mustaţă University of Michigan Mainz July 9, 2018 Mircea Mustaţă () An overview of D-modules Mainz July 9, 2018 1 The

### Hochschild homology and Grothendieck Duality

Hochschild homology and Grothendieck Duality Leovigildo Alonso Tarrío Universidade de Santiago de Compostela Purdue University July, 1, 2009 Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality

### DERIVED CATEGORIES OF COHERENT SHEAVES

DERIVED CATEGORIES OF COHERENT SHEAVES OLIVER E. ANDERSON Abstract. We give an overview of derived categories of coherent sheaves. [Huy06]. Our main reference is 1. For the participants without bacground

### AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES

AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are C-linear. 1.

### V. SRINIVAS. h p,q (X)u p v q

THE HODGE CHARACTERISTIC V. SRINIVAS 1. Introduction The goal of this lecture is to discuss the proof of the following result, used in Kontsevich s proof of the theorem that the Hodge numbers of two birationally

### The Riemann-Roch Theorem

The Riemann-Roch Theorem Paul Baum Penn State Texas A&M University College Station, Texas, USA April 4, 2014 Minicourse of five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology?

### EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY APPENDIX A: ALGEBRAIC TOPOLOGY

EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY APPENDIX A: ALGEBRAIC TOPOLOGY WILLIAM FULTON NOTES BY DAVE ANDERSON In this appendix, we collect some basic facts from algebraic topology pertaining to the

### ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

### Characteristic classes in the Chow ring

arxiv:alg-geom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic

### Math 6510 Homework 10

2.2 Problems 9 Problem. Compute the homology group of the following 2-complexes X: a) The quotient of S 2 obtained by identifying north and south poles to a point b) S 1 (S 1 S 1 ) c) The space obtained

### Introduction to surgery theory

Introduction to surgery theory Wolfgang Lück Bonn Germany email wolfgang.lueck@him.uni-bonn.de http://131.220.77.52/lueck/ Bonn, 17. & 19. April 2018 Wolfgang Lück (MI, Bonn) Introduction to surgery theory

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE

LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds

### Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

### Math Homotopy Theory Hurewicz theorem

Math 527 - Homotopy Theory Hurewicz theorem Martin Frankland March 25, 2013 1 Background material Proposition 1.1. For all n 1, we have π n (S n ) = Z, generated by the class of the identity map id: S

### arxiv: v1 [math.ag] 13 Mar 2019

THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show

### 1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

### Cup product and intersection

Cup product and intersection Michael Hutchings March 28, 2005 Abstract This is a handout for my algebraic topology course. The goal is to explain a geometric interpretation of the cup product. Namely,

### ON THE PICARD GROUP FOR NON-COMPLETE ALGEBRAIC VARIETIES. Helmut A. Hamm & Lê Dũng Tráng

Séminaires & Congrès 10, 2005, p. 71 86 ON THE PICARD GROUP FOR NON-COMPLETE ALGEBRAIC VARIETIES by Helmut A. Hamm & Lê Dũng Tráng Abstract. In this paper we show some relations between the topology of

### 3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).

3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)

### arxiv: v1 [math.ag] 28 Sep 2016

LEFSCHETZ CLASSES ON PROJECTIVE VARIETIES JUNE HUH AND BOTONG WANG arxiv:1609.08808v1 [math.ag] 28 Sep 2016 ABSTRACT. The Lefschetz algebra L X of a smooth complex projective variety X is the subalgebra

### A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY. We also have an isomorphism of holomorphic vector bundles

A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY LIVIU I. NICOLAESCU ABSTRACT. These are notes for a talk at a topology seminar at ND.. GENERAL FACTS In the sequel, for simplicity we denote the complex

### MATH 233B, FLATNESS AND SMOOTHNESS.

MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)

### ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1

ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP HONG GYUN KIM Abstract. I studied the construction of an algebraically trivial, but topologically non-trivial map by Hopf map p : S 3 S 2 and a

### Abelian Varieties and the Fourier Mukai transformations (Foschungsseminar 2005)

Abelian Varieties and the Fourier Mukai transformations (Foschungsseminar 2005) U. Bunke April 27, 2005 Contents 1 Abelian varieties 2 1.1 Basic definitions................................. 2 1.2 Examples

### Eilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )

II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups

### Diagonal Subschemes and Vector Bundles

Pure and Applied Mathematics Quarterly Volume 4, Number 4 (Special Issue: In honor of Jean-Pierre Serre, Part 1 of 2 ) 1233 1278, 2008 Diagonal Subschemes and Vector Bundles Piotr Pragacz, Vasudevan Srinivas

### ALGEBRAIC GEOMETRY I, FALL 2016.

ALGEBRAIC GEOMETRY I, FALL 2016. DIVISORS. 1. Weil and Cartier divisors Let X be an algebraic variety. Define a Weil divisor on X as a formal (finite) linear combination of irreducible subvarieties of

### Characteristic Classes, Chern Classes and Applications to Intersection Theory

Characteristic Classes, Chern Classes and Applications to Intersection Theory HUANG, Yifeng Aug. 19, 2014 Contents 1 Introduction 2 2 Cohomology 2 2.1 Preliminaries................................... 2

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### Notes on p-divisible Groups

Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete

### SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN

SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X -bimodule of rank

### Wild ramification and the characteristic cycle of an l-adic sheaf

Wild ramification and the characteristic cycle of an l-adic sheaf Takeshi Saito March 14 (Chicago), 23 (Toronto), 2007 Abstract The graded quotients of the logarithmic higher ramification groups of a local

### Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

### GK-SEMINAR SS2015: SHEAF COHOMOLOGY

GK-SEMINAR SS2015: SHEAF COHOMOLOGY FLORIAN BECK, JENS EBERHARDT, NATALIE PETERNELL Contents 1. Introduction 1 2. Talks 1 2.1. Introduction: Jordan curve theorem 1 2.2. Derived categories 2 2.3. Derived

### A TALE OF TWO FUNCTORS. Marc Culler. 1. Hom and Tensor

A TALE OF TWO FUNCTORS Marc Culler 1. Hom and Tensor It was the best of times, it was the worst of times, it was the age of covariance, it was the age of contravariance, it was the epoch of homology, it

### An Outline of Homology Theory

An Outline of Homology Theory Stephen A. Mitchell June 1997, revised October 2001 Note: These notes contain few examples and even fewer proofs. They are intended only as an outline, to be supplemented

### p,q H (X), H (Y ) ), where the index p has the same meaning as the

There are two Eilenberg-Moore spectral sequences that we shall consider, one for homology and the other for cohomology. In contrast with the situation for the Serre spectral sequence, for the Eilenberg-Moore

### Geometry 9: Serre-Swan theorem

Geometry 9: Serre-Swan theorem Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have

### PERVERSE SHEAVES ON A TRIANGULATED SPACE

PERVERSE SHEAVES ON A TRIANGULATED SPACE A. POLISHCHUK The goal of this note is to prove that the category of perverse sheaves constructible with respect to a triangulation is Koszul (i.e. equivalent to

### Manifolds and Poincaré duality

226 CHAPTER 11 Manifolds and Poincaré duality 1. Manifolds The homology H (M) of a manifold M often exhibits an interesting symmetry. Here are some examples. M = S 1 S 1 S 1 : M = S 2 S 3 : H 0 = Z, H

### Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

### THE p-smooth LOCUS OF SCHUBERT VARIETIES. Let k be a ring and X be an n-dimensional variety over C equipped with the classical topology.

THE p-smooth LOCUS OF SCHUBERT VARIETIES GEORDIE WILLIAMSON ABSTRACT. These are notes from talks given at Jussieu (seminaire Chevalley), Newcastle and Aberdeen (ARTIN meeting). They are intended as a gentle

### MOTIVES OF SOME ACYCLIC VARIETIES

Homology, Homotopy and Applications, vol.??(?),??, pp.1 6 Introduction MOTIVES OF SOME ACYCLIC VARIETIES ARAVIND ASOK (communicated by Charles Weibel) Abstract We prove that the Voevodsky motive with Z-coefficients

### On embedding all n-manifolds into a single (n + 1)-manifold

On embedding all n-manifolds into a single (n + 1)-manifold Jiangang Yao, UC Berkeley The Forth East Asian School of Knots and Related Topics Joint work with Fan Ding & Shicheng Wang 2008-01-23 Problem

### Cohomology and Vector Bundles

Cohomology and Vector Bundles Corrin Clarkson REU 2008 September 28, 2008 Abstract Vector bundles are a generalization of the cross product of a topological space with a vector space. Characteristic classes

### The diagonal property for abelian varieties

The diagonal property for abelian varieties Olivier Debarre Dedicated to Roy Smith on his 65th birthday. Abstract. We study complex abelian varieties of dimension g that have a vector bundle of rank g

### Dedicated to Mel Hochster on his 65th birthday. 1. Introduction

GLOBAL DIVISION OF COHOMOLOGY CLASSES VIA INJECTIVITY LAWRENCE EIN AND MIHNEA POPA Dedicated to Mel Hochster on his 65th birthday 1. Introduction The aim of this note is to remark that the injectivity

### Hodge Theory of Maps

Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar

### ALGEBRAIC CYCLES ON THE FANO VARIETY OF LINES OF A CUBIC FOURFOLD arxiv: v2 [math.ag] 18 Apr INTRODUCTION

ALGEBRAIC CYCLES ON THE FANO VARIETY OF LINES OF A CUBIC FOURFOLD arxiv:1609.05627v2 [math.ag] 18 Apr 2018 KALYAN BANERJEE ABSTRACT. In this text we prove that if a smooth cubic in P 5 has its Fano variety

### 18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Kiehl s finiteness theorems

18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Kiehl s finiteness theorems References: [FvdP, Chapter 4]. Again, Kiehl s original papers (in German) are: Der

### INTERSECTION THEORY CLASS 6

INTERSECTION THEORY CLASS 6 RAVI VAKIL CONTENTS 1. Divisors 2 1.1. Crash course in Cartier divisors and invertible sheaves (aka line bundles) 3 1.2. Pseudo-divisors 3 2. Intersecting with divisors 4 2.1.

### THE GROTHENDIECK-RIEMANN-ROCH THEOREM FOR VARIETIES

THE GROTHENDIECK-RIEMANN-ROCH THEOREM FOR VARIETIES PETER XU ABSTRACT. We give an exposition of the Grothendieck-Riemann-Roch theorem for algebraic varieties. Our proof follows Borel and Serre [3] and

### The Riemann-Roch Theorem

The Riemann-Roch Theorem Paul Baum Penn State TIFR Mumbai, India 20 February, 2013 THE RIEMANN-ROCH THEOREM Topics in this talk : 1. Classical Riemann-Roch 2. Hirzebruch-Riemann-Roch (HRR) 3. Grothendieck-Riemann-Roch

### Chow Groups. Murre. June 28, 2010

Chow Groups Murre June 28, 2010 1 Murre 1 - Chow Groups Conventions: k is an algebraically closed field, X, Y,... are varieties over k, which are projetive (at worst, quasi-projective), irreducible and

### 121B: ALGEBRAIC TOPOLOGY. Contents. 6. Poincaré Duality

121B: ALGEBRAIC TOPOLOGY Contents 6. Poincaré Duality 1 6.1. Manifolds 2 6.2. Orientation 3 6.3. Orientation sheaf 9 6.4. Cap product 11 6.5. Proof for good coverings 15 6.6. Direct limit 18 6.7. Proof

### EXOTIC SMOOTH STRUCTURES ON TOPOLOGICAL FIBRE BUNDLES B

EXOTIC SMOOTH STRUCTURES ON TOPOLOGICAL FIBRE BUNDLES B SEBASTIAN GOETTE, KIYOSHI IGUSA, AND BRUCE WILLIAMS Abstract. When two smooth manifold bundles over the same base are fiberwise tangentially homeomorphic,

### Nonabelian Poincare Duality (Lecture 8)

Nonabelian Poincare Duality (Lecture 8) February 19, 2014 Let M be a compact oriented manifold of dimension n. Then Poincare duality asserts the existence of an isomorphism H (M; A) H n (M; A) for any

### Algebraic Topology Final

Instituto Superior Técnico Departamento de Matemática Secção de Álgebra e Análise Algebraic Topology Final Solutions 1. Let M be a simply connected manifold with the property that any map f : M M has a

### SEPARABLE RATIONAL CONNECTEDNESS AND STABILITY

SEPARABLE RATIONAL CONNECTEDNESS AND STABILIT ZHIU TIAN Abstract. In this short note we prove that in many cases the failure of a variety to be separably rationally connected is caused by the instability

### 30 Surfaces and nondegenerate symmetric bilinear forms

80 CHAPTER 3. COHOMOLOGY AND DUALITY This calculation is useful! Corollary 29.4. Let p, q > 0. Any map S p+q S p S q induces the zero map in H p+q ( ). Proof. Let f : S p+q S p S q be such a map. It induces

### the complete linear series of D. Notice that D = PH 0 (X; O X (D)). Given any subvectorspace V H 0 (X; O X (D)) there is a rational map given by V : X

2. Preliminaries 2.1. Divisors and line bundles. Let X be an irreducible complex variety of dimension n. The group of k-cycles on X is Z k (X) = fz linear combinations of subvarieties of dimension kg:

### Kähler manifolds and variations of Hodge structures

Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic

### AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES

AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES MATTHEW H. BAKER AND JÁNOS A. CSIRIK This paper was written in conjunction with R. Hartshorne s Spring 1996 Algebraic Geometry course at

### Globalization and compactness of McCrory Parusiński conditions. Riccardo Ghiloni 1

Globalization and compactness of McCrory Parusiński conditions Riccardo Ghiloni 1 Department of Mathematics, University of Trento, 38050 Povo, Italy ghiloni@science.unitn.it Abstract Let X R n be a closed

### FAKE PROJECTIVE SPACES AND FAKE TORI

FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.

### MA 206 notes: introduction to resolution of singularities

MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k be

### 8 Perverse Sheaves. 8.1 Theory of perverse sheaves

8 Perverse Sheaves In this chapter we will give a self-contained account of the theory of perverse sheaves and intersection cohomology groups assuming the basic notions concerning constructible sheaves

### De Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1)

II. De Rham Cohomology There is an obvious similarity between the condition d o q 1 d q = 0 for the differentials in a singular chain complex and the condition d[q] o d[q 1] = 0 which is satisfied by the

### Vanishing theorems and singularities in birational geometry. Tommaso de Fernex Lawrence Ein

Vanishing theorems and singularities in birational geometry Tommaso de Fernex Lawrence Ein Mircea Mustaţă Contents Foreword vii Notation and conventions 1 Chapter 1. Ample, nef, and big line bundles 1

### A CRITERION FOR A DEGREE-ONE HOLOMORPHIC MAP TO BE A BIHOLOMORPHISM. 1. Introduction

A CRITERION FOR A DEGREE-ONE HOLOMORPHIC MAP TO BE A BIHOLOMORPHISM GAUTAM BHARALI, INDRANIL BISWAS, AND GEORG SCHUMACHER Abstract. Let X and Y be compact connected complex manifolds of the same dimension

### Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY

LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in

### Divisor class groups of affine complete intersections

Divisor class groups of affine complete intersections Lambrecht, 2015 Introduction Let X be a normal complex algebraic variety. Then we can look at the group of divisor classes, more precisely Weil divisor

### Topics in Algebraic Geometry

Topics in Algebraic Geometry Nikitas Nikandros, 3928675, Utrecht University n.nikandros@students.uu.nl March 2, 2016 1 Introduction and motivation In this talk i will give an incomplete and at sometimes

### Hungry, Hungry Homology

September 27, 2017 Motiving Problem: Algebra Problem (Preliminary Version) Given two groups A, C, does there exist a group E so that A E and E /A = C? If such an group exists, we call E an extension of