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Complex Algebraic Geometry Jean Gallier∗ and Stephen S. Shatz∗∗ ∗Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] ∗∗Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] February 25, 2011 2 Contents 1 Complex Algebraic Varieties; Elementary Theory 7 1.1 What is Geometry & What is Complex Algebraic Geometry? . .......... 7 1.2 LocalStructureofComplexVarieties. ............ 14 1.3 LocalStructureofComplexVarieties,II . ............. 28 1.4 Elementary Global Theory of Varieties . ........... 42 2 Cohomologyof(Mostly)ConstantSheavesandHodgeTheory 73 2.1 RealandComplex .................................... ...... 73 2.2 Cohomology,deRham,Dolbeault. ......... 78 2.3 Hodge I, Analytic Preliminaries . ........ 89 2.4 Hodge II, Globalization & Proof of Hodge’s Theorem . ............ 107 2.5 HodgeIII,TheK¨ahlerCase . .......... 131 2.6 Hodge IV: Lefschetz Decomposition & the Hard Lefschetz Theorem............... 147 2.7 ExtensionsofResultstoVectorBundles . ............ 162 3 The Hirzebruch-Riemann-Roch Theorem 165 3.1 Line Bundles, Vector Bundles, Divisors . ........... 165 3.2 ChernClassesandSegreClasses . .......... 179 3.3 The L-GenusandtheToddGenus .............................. 215 3.4 CobordismandtheSignatureTheorem. ........... 227 3.5 The Hirzebruch–Riemann–Roch Theorem (HRR) . ............ 232 3 4 CONTENTS Preface This manuscript is based on lectures given by Steve Shatz for the course Math 622/623–Complex Algebraic Geometry, during Fall 2003 and Spring 2004. The process for producing this manuscript was the following: I (Jean Gallier) took notes and transcribed them in LATEX at the end of every week. A week later or so, Steve reviewed these notes and made changes and corrections. After the course was over, Steve wrote up additional material that I transcribed into LATEX. The following manuscript is thus unfinished and should be considered as work in progress. Nevertherless, given that Principles of Algebraic Geometry by Griffith and Harris is a formidable source, we feel that the material presented in this manuscript will be of some value. We apologize for the typos and mistakes that surely occur in the manuscript (as well as unfinished sections and even unfinished proofs!). Still, our hope is that by its “freshness,” this work will be of value to algebraic geometry lovers. Please, report typos, mistakes, etc. (to Jean). We intend to improve and perhaps even complete this manuscript. Philadelphia, February 2011 Jean Gallier Acknowledgement. My friend Jean Gallier had the idea of attending my lectures in the graduate course in Complex Algebraic Geometry during the academic year 2003-04. Based on his notes of the lectures, he is producing these LATEX notes. I have reviewed a first version of each LATEX script and corrected only the most obvious errors which were either in my original lectures or might have crept in otherwise. Matters of style and presentation have been left to Jean Gallier. I owe him my thanks for all the work these LATEXed notes represent. Philadelphia, September 2003 SSS 5 6 CONTENTS Chapter 1 Complex Algebraic Varieties; Elementary Local And Global Theory 1.1 What is Geometry & What is Complex Algebraic Geometry? The presumption is that we study systems of polynomial equations f1(X1,...,Xq) = 0 . ( ) . † fp(X1,...,Xq) = 0 where the fj are polynomials in C[X1,...,X q]. Fact: Solving a system of equations of arbitrary degrees reduces to solving a system of quadratic equations (no restriction on the number of variables) (DX). What is geometry? Experience shows that we need (1) A topological space, X. (2) There exist (at least locally defined) functions on X. (3) More experience shows that the “correct bookkeeping scheme” for encompassing (2) is a “sheaf” of functions on X; notation . OX Aside on Presheaves and Sheaves. (1) A presheaf , , on X is determined by the following data: P (i) For every open U X, a set (or group, or ring, or space), (U), is given. ⊆ P V (ii) If V U (where U, V are open in X) then there is a map ρU : (U) (V ) (restriction) such that U ⊆ P → P ρU = idU and ρW = ρW ρV , for all open subsets U,V,W with W V U. U V ◦ U ⊆ ⊆ (2) A sheaf , , on X is just a presheaf satisfying the following (patching) conditions: F 7 8 CHAPTER 1. COMPLEX ALGEBRAIC VARIETIES; ELEMENTARY THEORY (i) For every open U X and for every open cover U of U (which means that U = U , notation ⊆ { α}α α α Uα U ), if f,g (U) so that f ↾ Uα = g ↾ Uα, for all α, then f = g. { → } ∈F (ii) For all α, if we are given f (U ) and if for all α, β we have α ∈F α ρUα∩Uβ (f )= ρUα∩Uβ (f ), Uα α Uβ β (the f agree on overlaps), then there exists f (U) so that ρUα (f)= f , all α. α ∈F U α Our X is a sheaf of rings, i.e, X (U) is a commutative ring, for all U. We have (X, X ), a topological space andO a sheaf of rings. O O Moreover, our functions are always (at least) continuous. Pick some x X and look at all opens, U X, ∈ ⊆ where x U. If a small U x is given and f,g X (U), we say that f and g are equivalent, denoted f g, iff∈ there is some open∋V U with x V so∈ that O f ↾ V = g ↾ V . This is an equivalence relation and [f]∼ = the equivalence class of f ⊆is the germ of∈ f at x. Check (DX) that lim X (U) = collection of germs at x. O −→U∋x The left hand side is called the stalk of X at x, denoted X,x. By continuity, X,x is a local ring with maximal ideal m = germs vanishing at xO. In this case, Ois called a sheaf of localO rings. x OX In summary, a geometric object yields a pair (X, X ), where X is a sheaf of local rings. Such a pair, (X, ), is called a local ringed space (LRS). O O OX LRS’s would be useless without a notion of morphism from one LRS to another, Φ: (X, ) (Y, ). OX → OY (A) We need a continuous map ϕ: X Y and whatever a morphism does on X , Y , taking a clue from the case where and are sets of→ functions, we need something “ O .”O OX OY OY −→ OX Given a map ϕ: X Y with X on X, we can make ϕ∗ X (= direct image of X ), a sheaf on Y , as follows: For any open U→ Y , considerO the open ϕ−1(U) X,O and set O ⊆ ⊆ (ϕ )(U)= (ϕ−1(U)). ∗OX OX This is a sheaf on Y (DX). Alternatively, we have on Y (and the map ϕ: X Y ) and we can try making a sheaf on X: Pick OY → x X and make the stalk of “something” at x. Given x, we make ϕ(x) Y , we make Y,ϕ(x) and define ϕ∗∈ so that ∈ O OY (ϕ∗( )) = . OY x OY,ϕ(x) More precisely, we define the presheaf ϕ on X by P OY ϕ (U)= lim (V ), P OY OY V ⊇−→ϕ(U) where V ranges over open subsets of Y containing ϕ(U). Unfortunately, this is not always a sheaf and we ∗ need to “sheafify” it to get ϕ Y , the inverse image of Y . For details, consult the Appendix on sheaves and ringed spaces. We now haveO everything we need to defineO morphisms of LRS’s. (B) A map of sheaves, ϕ: ϕ , on Y , is also given. OY → ∗OX ∗ It turns out that this is equivalent to giving a map of sheaves, ϕ: ϕ Y X , on X (This is because ∗ O → O ϕ∗ and ϕ are adjoint functors, again, see the Appendix on sheaves.) 1.1. WHAT IS GEOMETRY & WHAT IS COMPLEX ALGEBRAIC GEOMETRY? 9 In conclusion, a morphism (X, ) (Y, ) is a pair (ϕ, ϕ) (or a pair (ϕ, ϕ)), as above. OX −→ OY When we look at the “trivial case”’ (of functions) we see that we want ϕ to satisfy ϕ(m ) m , for all x X. ϕ(x) ⊆ x ∈ This condition says that ϕ is a local morphism. We get a category . LRS After all these generalities, we show how most geometric objects of interest arise are special kinds of LRS’s. The key idea is to introduce “standard” models and to define a corresponding geometric objects, X, to be an LRS that is “locally isomorphic” to a standard model. First, observe that given any open subset U X, we can form the restriction of the sheaf X to U, denoted X ↾ U or ( U ) and we get an LRS (U, ⊆ ↾ U). Now, if we also have a collection ofO LRS’s (the standardO models),O we consider LRS’s, OX (X, X ), such that (X, X ) is locally isomorphic to a standard model. This means that we can cover X by opensO and that for everyO open U X in this cover, there is a standard model (W, ) and an isomorphism ⊆ OW (U, ↾ U) = (W, ), as LRS’s. OX ∼ OW Some Standard Models. (1) Let U be an open ball in Rn or Cn, and let be the sheaf of germs of continuous functions on U OU (this means, the sheaf such that for every open V U, U (V ) = the restrictions to V of the continuous functions on U). If (X, ) is locally isomorphic to a⊆ standard,O we get a topological manifold. O k (2) Let U be an open as in (1) and let U be the sheaf of germs of C -functions on U, with 1 k . If (X, ) is locally isomorphic to a standard,O we get a Ck-manifold (when k = , call these smooth≤manifolds).≤ ∞ O ∞ n ω (3) Let U be an open ball in R and let U be the sheaf of germs of real-valued C -functions on U (i.e., real analytic functions).
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  • §4 Grassmannians 73

    §4 Grassmannians 73

    §4 Grassmannians 4.1 Plücker quadric and Gr(2,4). Let 푉 be 4-dimensional vector space. e set of all 2-di- mensional vector subspaces 푈 ⊂ 푉 is called the grassmannian Gr(2, 4) = Gr(2, 푉). More geomet- rically, the grassmannian Gr(2, 푉) is the set of all lines ℓ ⊂ ℙ = ℙ(푉). Sending 2-dimensional vector subspace 푈 ⊂ 푉 to 1-dimensional subspace 훬푈 ⊂ 훬푉 or, equivalently, sending a line (푎푏) ⊂ ℙ(푉) to 한 ⋅ 푎 ∧ 푏 ⊂ 훬푉 , we get the Plücker embedding 픲 ∶ Gr(2, 4) ↪ ℙ = ℙ(훬 푉). (4-1) Its image consists of all decomposable¹ grassmannian quadratic forms 휔 = 푎∧푏 , 푎, 푏 ∈ 푉. Clearly, any such a form has zero square: 휔 ∧ 휔 = 푎 ∧ 푏 ∧ 푎 ∧ 푏 = 0. Since an arbitrary form 휉 ∈ 훬푉 can be wrien¹ in appropriate basis of 푉 either as 푒 ∧ 푒 or as 푒 ∧ 푒 + 푒 ∧ 푒 and in the laer case 휉 is not decomposable, because of 휉 ∧ 휉 = 2 푒 ∧ 푒 ∧ 푒 ∧ 푒 ≠ 0, we conclude that 휔 ∈ 훬 푉 is decomposable if an only if 휔 ∧ 휔 = 0. us, the image of (4-1) is the Plücker quadric 푃 ≝ { 휔 ∈ 훬푉 | 휔 ∧ 휔 = 0 } (4-2) If we choose a basis 푒, 푒, 푒, 푒 ∈ 푉, the monomial basis 푒 = 푒 ∧ 푒 in 훬 푉, and write 푥 for the homogeneous coordinates along 푒, then straightforward computation 푥 ⋅ 푒 ∧ 푒 ∧ 푥 ⋅ 푒 ∧ 푒 = 2 푥 푥 − 푥 푥 + 푥 푥 ⋅ 푒 ∧ 푒 ∧ 푒 ∧ 푒 < < implies that 푃 is given by the non-degenerated quadratic equation 푥푥 = 푥푥 + 푥푥 .