MATH 250B: COMPLEX ALGEBRAIC GEOMETRY

Contents Introduction/Syllabus 1 1. Complex manifolds vs almost complex manifolds 2 1.1. Definition 2 1.2.Complexlinearstructure 2 1.3.Almostcomplexmanifolds 4 1.4. Distributions and Frobenius Theorem 5 1.5.Newlander-NirenbergTheorem 8 2. Complex di↵erential forms 10 2.1. Definitions 10 2.2. Coordinate invariant definitions 12 2.3. Local exactness of the @ complex 14 2.4. One complex variable 15 2.5. Back to di↵erentialforms 16 2.6. Dolbeault complex 18 3. K¨ahler manifolds 19 3.1. K¨ahlerforms 19 3.2. K¨ahlermetrics 20 3.3. K¨ahlermanifolds 22 3.4. Connections 22 3.5. Examples of K¨ahlermanifolds 28 4. Harmonic forms 31 4.1. L2 formsandtheHodgestaroperator 31 4.2.Adjointoperators 35 4.3. Laplacian 37 4.4. Di↵erential operators 37 4.5. Symbols and elliptic operators 40 4.6. Fundamental Theorem 43 4.7. Serre duality 46 5. Lefschetz operator and applications 49 5.1.Hodgedecomposition 49 5.2. Lefschetz decomposition 52 5.3. Lefschetz decomposition in cohomology 56 5.4. Representations of sl2(C)57 6. Polarization 153 6.1. Outline of Kodaira’s theorem 153 6.2. Construction of L 156 7. Hypercohomologyandspectralsequences 160 0 MATH 250B: COMPLEX ALGEBRAIC GEOMETRY 1 7.1. Definition of hypercohomology 160 References 162 11. Chap 1: Basic complex analysis 201 101. Homework 1 301 102. Homework 2 301 MATH 250B: COMPLEX ALGEBRAIC GEOMETRY 1 Introduction/Syllabus Lectures in Room 226 Goldsmith, 3:30-4:50 MW Shifting to online lectures on March 16, 2020. Instructor: Kiyoshi Igusa, 305 Goldsmith Math 250b is called “Complex Algebraic Geometry” since it is the study of smooth projective varieties over C from a di↵erentiable point of view. Smooth complex projective varieties are compact complex manifolds with additional structure, namely (by Kodaira’s Embedding Theorem), they are K¨ahler manifolds with integral K¨ahler class. In this course we will build up this structure one layer at a time. The emphasis will be on concepts, definitions, statements. We need to skip technical details of many proofs of some very important theorems. I will do this by formulating technical lemmas and showing that the theorems follow from these lemmas. Since we already have notes for the course, my ambitious project is to append some of these technical details at the end of the notes. We start in Chapter 2 of [Voisin]: Almost Complex Manifolds. (We will come back to Chap 1 when we need it: See section 2.4.) -Almost Complex Manifolds are even dimensional real manifolds X with complex structures on their tangent bundles. The Newlander-Nirenberg Theorem will tell us when X has the structure of a complex manifold. -Next we look at Complex Di↵erential Forms. We will review basic complex anal- ysis (Cauchy’s Theorem which implies: Complex di↵erentiable implies complex ana- lytic.) The main point is that the vector space of C-values di↵erential forms is bigraded: k p,q p,q ⌦ (X, C)= p+q=k ⌦ (X). Elements of ⌦ (X)arecalledforms of type (p, q). -A K¨ahler manifold is a complex manifold with a K¨ahler form. This is a di↵erential form of typeL (1, 1) which is closed. Being closed implies that we obtain a cohomology operation called the Lefschetz operator.ThisleadstotheHodgeDecompositionTheorem on compact Kahler manifolds, the Lefschetz Decomposition Theorem and the Kodaira Embedding Theorem. At this point we will be forced to “review” the cohomology of sheaves. This follows my philosophy that we want motivation first before going into any technicalities. By the end of the course, we should have an appreciation of the amazing, elegant side of the analytic and cohomological structure of smooth projective varieties over the complex numbers and a good idea of what are the lemmas and theorems whose proofs we are skipping. pages 1-100: New notes pages 101-200: Old notes pages 201-300: Planned technical details. pages 301-400: Homework 2MATH250B:COMPLEXALGEBRAICGEOMETRY 1. Complex manifolds vs almost complex manifolds What is the di↵erence between a complex manifold and a real even dimensional manifold with a complex structure on its tangent bundle? 1.1. Definition. Definition 1.1.1. A(topological)n-manifold is a Hausdor↵space X together with a collection of pairs called charts (U,') where U are open subsets of X which cover X and ' : U Rn is a homeomorphism of U onto an open subset V of Rn. ! The structure of a manifold is given by the transition mappings. These are 1 ' = ' ' : ' (U U ) V ij j i i i \ j ! j where (Ui,'i : Ui ⇠= Vi)and(Uj,'j : Uj ⇠= Vj)aretwochartswhichintersect.Ifthese transition maps are Ck (k times continuously di↵erentiable) then M is a Ck-manifold. If they are analytic (given locally by converging power series) then M is an analytic manifold. Given a chart (U,')wegetlocal coordinates which are function x1, ,xn : U R given by composition: ··· ! n xi = pi ' : U R R. ! ! Definition 1.1.2. A complex n-manifold is a real 2n manifold X with charts (U,' : U, Cn)sothatthetransitionmapsareholomorphic(complexdi↵erentiableandthus complex! analytic: Theorem 11.0.2). This formal definition does not help understand what we are talking about. We will later go through some of the technical details when we need it (sec 2.4). 1.2. Complex linear structure. The first point to understand is that complex struc- ture is locally defined. A complex manifold X locally looks like Cn.Foracoordinate chart (U,')wegetn complex coordinate functions zi : U C given by 2n real coordi- ! nate functions xi,yi : U R given by zi = xi + iyi. ! The tangent plane at one point TX,x is a complex vector space of dimension n: n TX,x ⇠= C . Recall the definition of the tangent plane TX,x.

Definition 1.2.1. A tangent vector to X at x0 is an R-linear function: : Ck(X) R ! satisfying the Leibnitz rule: (fg)=g(x0)(f)+f(x0)(g). Tangent vectors are “point derivations” on Ck(X), the ring of Ck functions X R. ! Since X is a real 2n-manifold, the tangent vectors at any x0 X form a 2n- dimensional real vector space spanned by the 2n tangent vectors 2 @ @ @ @ x0 , x0 , x0 , , x0 . @x1 | ···@xn | @y1 | ··· @yn | Call this vector space T 2n. X,x0,R ⇠= R MATH 250B: COMPLEX ALGEBRAIC GEOMETRY 3 When X is a complex manifold, this vector space has a complex structure given by an operator I which we think of as multiplication by i: @ @ @ @ I x0 = x0 ,I x0 = x0 . @xj | @yj | @yj | @xj | From now on, we suppress the notation . |x0 Definition 1.2.2. Take the complexified tangent space 2n TX,x0, C = C . R ⌦R ⇠ This has a complex basis of the 2n vectors @ 1 @ @ := i @z 2 @x @y j ✓ j j ◆ @ 1 @ @ := + i . @z 2 @x @y j ✓ j j ◆ When X is a complex manifold, we also have the operator I which will never be the same as multiplication by i. Why not? By definition of I we would have: @ 1 @ @ 1 @ @ := + I = =0. @z 2 @x @y 2 @x @x j ✓ j j ◆ ✓ j j ◆ Definition 1.2.3. For U Cn open, a (real-) di↵erentiable (C1) function f : U C is holomorphic if ⇢ ! @f =0 @zj for j =1, ,n at each point in U.Thisisequivalenttosayingthatthederivativeof f is a complex··· linear map: df : TU C ! at each point in the domain of f.Wealsowritedf C for the C-linear extension: C df : TU C C. ⌦ ! Example: n =1,U = C and f(z)=az2 = a(x + iy)2 = ax2 ay2 +2aixy.Then,by the chain rule, @ @a(x + iy)2 df = =2ax +2aiy =2az @x @x ✓ ◆ @ @a(x + iy)2 df = = 2ay +2aix =2iaz. @y @y ✓ ◆ So, 1 @ @ @f df C + i = az az =0= . 2 @x @y @z ✓ ◆ Thus, f is holomorphic. In general, f is holomorphic if df I = idf since, in that case @f 1 @ @ 1 @ @ (1.1) = df C + i = df + iI =0. @z 2 @x @y 2 @x @x j ✓ j j ◆ ✓ j j ◆ 4MATH250B:COMPLEXALGEBRAICGEOMETRY 1.3. Almost complex manifolds. Definition 1.3.1. An almost complex manifold is a Ck real 2n-manifold together with a Ck endomorphism I of its tangent bundle I : T T x X,x ! X,x so that I2 = id. This is a local structure. A neighborhood of each point is (equivalent to) an open subset U R2n with a complex structure on the tangent plane (also R2n)ateachpoint. An almost✓ complex structure is equivalent to an honest complex manifold structure on X if, for each such nbh U,thereisadi↵erentiableembedding ' : U, Cn ! n so that, at each point x U,thederivative' : TU,x C is complex linear. I.e., 2 ⇤ ! (1.2) ' (Iv)=i' (v) ⇤ ⇤ (and a linear isomorphism). We will rephrase this condition to resemble (1.1). n Since U is a real manifold, the derivative ' : TU,x C is apriori only linear ⇤ ! over R. Tensoring this R linear map with C (and composing with the C-linear map Cn C Cn)willalwaysgiveaC-linear map: ⌦R ! C n ' : TU,x R C C . ⇤ ⌦ ! Condition (1.2) is equivalent to: (1.3) 'C (v + iIv)=0. ⇤ Proof: Since 'C is C-linear, ⇤ 'C (v + iIv)='C(v)+i'C(Iv)='C(v)+i2'C(v)=0. ⇤ ⇤ ⇤ ⇤ ⇤ The condition that ' : R2n Cn is a linear isomorphism (over R)translatesinto ⇤ ! 'C : C2n Cn being an epimorphism. ⇤ ! Definition 1.3.2. We will have two complex structures on the same vector space. To avoid confusion, we will refer to I,operatingonthefirstfactorofTU,x C,asthe geometric complex structure map or rotation (since it “rotates” tangent⌦ vectors). The scalar i acting on the second tensor factor will be called the coecient or scalar. We will rephrase Equation (1.3) in a fancy language and derive necessary and sucient conditions for an almost complex manifold to be equivalent to a complex manifold. Remark 1.3.3. Some linear algebra: For any C-vector space V and any C-linear map J : V V with J 2 = id , V will decompose (as complex vector space) ! V V = V 1,0 V 0,1 where V 1,0 is the i-eigenspace of J and V 0,1 is the i-eigenspace of J: V 1,0 := v V Jv = iv { 2 | } V 0,1 := v V Jv = iv . { 2 | } MATH 250B: COMPLEX ALGEBRAIC GEOMETRY 5 For each v V ,thecomponentsare: 2 1 1 v1,0 = (v iJv) ,v0,1 = (v + iJv) . 2 2 For example, take V = C C = C2 with complex structure given by multiplication ⌦R ⇠ on the second factor. Over R, V = R4 has basis: 1 = 1 1,i=1 i, I = i 1, iI = i i. ⇠ 1 1 ⌦ ⌦ ⌦ ⌦ V 1,0 = C = R2 has real basis 11,0 = (1 iI),i1,0 = (i + I)=i(11,0)andV 0,1 = C = R2 ⇠ ⇠ 2 2 ⇠ ⇠ has real basis 10,1 = 1 (1 + iI),i0,1 = 1 (i I)=i(10,1). 2 2 Definition 1.3.4. Let X be an almost complex manifold (this includes complex mani- 1,0 0,1 folds). Define TX ,TX to be the subbundles of the complexified tangent bundle

TX, := TX C C ⌦R C 1,0 which are the i and i eigenspaces of the C-linear endomorphism I .I.e.,TX,x is the 0,1 set of all elements of TX,x C on which I = i and T is the set of all elements of ⌦R X,x TX,x C on which I = i. ⌦R 1,0 0,1 When are TX ,TX isomorphic as complex vector bundles over X? n When ' : TX,x C is a linear isomorphism so that ' I = i' ,Equation(1.3)says ⇤ ⇤ ⇤ C ! n 0,1 that ' : TX,x C C is a complex linear epimorphism with kernel equal to TX.x.We need conditions⇤ ⌦ under! which such a map exists.

1.4. Distributions and Frobenius Theorem. An almost complex manifold has the 0,1 structure of a complex manifold if and only if the “distribution” TX is “integrable”. We will go over what this means. For simplicity we assume all structures are C1.

Definition 1.4.1. A k-distribution on a real n-manifold X is defined to be a k-dimensional subbundle F TX of the tangent bundle of X.Thedistributioniscalledintegrable if, in a nbh U of✓ each point x,thereisasubmersion

n k ' : U V R . ! ✓ So that the kernel of ' U is F U. ⇤| | Theorem 1.4.2 (Frobenius). A distribution F on X is integrable if and only if

[F, F] F, ✓ i.e., for any two C1 vector fields ,⌧ on X with values in F , [,⌧] is in F .

Hiding a ton of technicalities under the rug this will give:

Theorem 1.4.3 (Newlander-Nirenberg). An almost complex manifold X is equivalent 0,1 to a complex manifold if and only if the distribution TX TX C is integrable. I.e., 0,1 0,1 0,1 ✓ ⌦ i↵ [T ,T ] T . X X ✓ X 6MATH250B:COMPLEXALGEBRAICGEOMETRY 1.4.1. Review of bracket [,⌧]. A Ck-vector field is defined to be a derivation : C`(X) Ck(X) ! where `>k.Inotherwords, (fg)=(f)g + f(g). It is a standard fact that all Ck-derivations are given locally by @ = i @xi where i is a Ck function on the coordinateX chart U.Conversely,anysuch gives a derivation Ck+1(X) Ck(X). Thus, even if is only defined on C`(X) Ck+1(X), it will extend uniquely! to the larger domain. ⇢ k 1 The bracket [,⌧]isdefinedtobetheC derivation given by k+1 k 1 [,⌧ ]=⌧ ⌧ : C (X) C (X). ! Proposition 1.4.4. If = i @ ⌧ = ⌧ j @ then @xi @xj P P@⌧j @j @ [,⌧]= i ⌧ i . @xi @xi @xj X ✓ ◆ Proof. GIven f Ck+1(X), ⌧(f)= ⌧ j @f .BytheLeibnitzrule, 2 @xj @f @f ⌧(f)= P(⌧ j) + ⌧ j @xj @xj X X @⌧j @f @ @f = i + ⌧ ji . @xi @xj @xi @xj Switch ,⌧: X X @j @f @ @f ⌧(f)= ⌧ i + ⌧ ji . @xi @xj @xj @xi Subtract to get the Proposition.X (We need f CX2 for the second terms to cancel.) 2 ⇤ Example 1.4.5. If is the “vertical” vector field = @ then n @xn @ @⌧j @ [ ,⌧]= ,⌧ = n @x @x @x n j n j  X since the coecients of n are constant. This is the “vertical component” of the deriva- tive of the section ⌧ of TX in the direction of the vector n in X. The following lemma, the key step in the proof of Frobenius’ Theorem, follows from this example and the proof of the Constant Rank Theorem in [Lee], Thms 7.13, 7.8. I believe C1 is sucient for the proof. Lemma 1.4.6. Suppose that F is a k distribution on an open nbh U of 0 in Rn so @ n 1 that F includes the vertical vector field 0 = at each point (y, t) U R R. @xn 2 ⇢ ⇥ Suppose that [F, F] F . Then the plane F(y,t) is independent of t. I.e., F(y,t) = Ey R ⇢ n 1 ⇥ for all (y, t) U for some k 1 distribution E on a nbh of 0 in R . 2 MATH 250B: COMPLEX ALGEBRAIC GEOMETRY 7

Proof. Observation from [Lee]: Any k plane close to Rk 0inRn is the graph of a unique k n k ⇥ n k linear map R R since any such plane does not meet 0 R (except at 0). It is enough! to prove the theorem for y =0.Bychangingcoordinates,wemayassume⇥ k 1 @ @ that F(0,0) = R 0 R is spanned by unit vectors n = and for i