## ICTP Day 2

More of the background stuff. Tomorrow I’ll be making two posts, one around lunchtime here, with the last background material and links to complete notes without having to deal with the vagaries of wordpress formatting (I’ve noticed that theorem, example, proposition etc environments are all gone…just doing a quick latex2wp and then making sure everything compiles…the final notes will be marginally nicer), and one in the evening when we get to actual Hodge theory.

1. Tu 2 – Computation of deRham Cohomology

1.1. Pullback of Forms

If ${f:M\rightarrow N}$ is a ${C^\infty}$ map, then there is a pullback map ${f^*:\mathcal{A}^k(N)\rightarrow\mathcal{A}^k(M)}$.

For ${k=0}$, then ${f^*(h)=h\circ k}$. In generall,y locally ${\omega\in \mathcal{A}^k(N)}$ can be written ${\omega=\sum a_Idx^I=\sum a_Idx^{i_1}\wedge\ldots \wedge dx^{i_k}}$ and we define ${f^*\omega=\sum (f^*a_I)d(f^*x^{i_1})\wedge\ldots\wedge d(f^*x^{i_k})}$.

\mbox{}

1. ${1_M^*=1_{\mathcal{A}^k(M)}}$
2. ${(f\circ g)^*=g^*\circ f^*}$
3. ${f^*d=df^*}$

By 3, we know that ${f^*}$ of a closed form is closed. If ${d\omega=0}$, then ${d(f^*\omega)=f^*d\omega=0}$. Now, ${f^*(d\tau)=d(f^*\tau)}$, and so exact forms pullback to exact forms. Thus, ${f^*}$ induces a map (also denoted by ${f^*}$ on cohomology ${H^k(N)\rightarrow H^k(M)}$.

If ${f:M\rightarrow N}$ is a diffeomorphism, then there exists ${g:N\rightarrow M}$ such that ${f\circ g=1_N}$ and ${g\circ f=1_M}$, then ${g^*\circ f^*=1_{H^*(N)}}$ and ${f^*\circ g^*=1_{H^*(M)}}$.

If ${f:M\rightarrow N}$ is a diffeomorphism, then ${f^*:H^*(N)\rightarrow H^*(M)}$ is an isomorphism.

1.2. Homological Algebra

A cochain complex ${\mathcal{C}}$ is ${0\rightarrow C^0\stackrel{d}{\rightarrow}C^1\rightarrow\ldots}$ such that ${d^2=0}$. So ${H^k(\mathcal{C})=\ker d_k/\mathrm{im} d_{k-1}}$ is defined.

A sequence of vector spaces ${A\stackrel{i}{\rightarrow} B\stackrel{j}{\rightarrow} C}$ is exact at ${B}$ if ${\mathrm{im}(i)=\ker j}$.

A linear map ${f:(\mathrm{A},d)\rightarrow (\mathrm{B},d)}$ of cochain complexes is a cochain map if ${f\circ d=d\circ f}$ (this is the data of a map in each degree).

A cochain map induces ${f^*:H^k(\mathcal{A})\rightarrow H^k(\mathcal{B})}$.

A short exact sequence of cochain complexes ${0\rightarrow\mathcal{A}\stackrel{i}{\rightarrow}\mathcal{B}\stackrel{j}{\rightarrow}\mathcal{C}\rightarrow 0}$ induces a long exact sequence on cohomology ${\ldots\rightarrow H^k(\mathcal{A})\rightarrow H^k(\mathcal{B})\rightarrow H^k(\mathcal{C})\stackrel{d^*}{\rightarrow}H^{k+1}(\mathcal{A})\rightarrow\ldots}$.

1.3. Mayer-Vietoris Sequence

Suppose that ${M}$ is covered by open sets ${U,V}$ so that ${M=U\cup V}$. Then we have ${\mathcal{A}^k(M)\stackrel{i}{\rightarrow}\mathcal{A}^k(U)\oplus \mathcal{A}^k(V)\stackrel{j}{\rightarrow}\mathcal{A}^k(U\cap V)}$ by the restriction maps followed by the difference of restrictions.

The sequence above is exact.

1.4. ${H^*(S^1)}$

We cover ${S^1}$ by the union of two copies of ${\mathbb{R}}$. Then ${H^*(U)=H^*(V)=H^*(\mathbb{R})}$ and ${H^*(U\cap V)}$ is two copies of ${\mathbb{R}}$, and so ${H^*(U\cap V)=H^*(\mathbb{R})\oplus H^*(\mathbb{R})}$.

Then the Mayer-Vietoris sequence becomes ${0\rightarrow\mathbb{R}\rightarrow\mathbb{R}^2\rightarrow\mathbb{R}^2\rightarrow H^1(S^1)\rightarrow 0\rightarrow0\rightarrow\ldots}$ So by exactness, ${H^1(S^1)=\mathrm{im}(d^*)\cong\mathbb{R}^2/\ker d^*=\mathbb{R}^2/\mathrm{im}(j^*)\cong\mathbb{R}}$, because the image of ${j^*}$ is the diagonal.

1.5. Smooth Homotopy

Two ${C^\infty}$ maps ${f_0,f_1:M\rightarrow N}$ are smoothly homotopic if there existsa ${C^\infty}$ map ${F:M\times [0,1]\rightarrow N}$ such that ${F(x,0)=f_0(x)}$ and ${f(x,1)=f_1(x)}$ (where we say that ${F}$ is ${C^\infty}$ if it can be extended to a ${C^\infty}$ function in a neighborhood of ${M\times[0,1]}$ in ${M\times\mathbb{R}}$). We write ${f_0\sim f_1}$.

${f:M\rightarrow N}$ has a homotopy inverse if there exists ${g:N\rightarrow M}$ such that ${f\circ g\sim 1_N}$ and ${g\circ f\sim 1_M}$

Then we say that ${M}$ and ${N}$ have the same homotopy type and ${f}$ is a homotopy equivalence.

\underline{Homotopy Axiom}:Homotopic maps ${f,g:M\rightarrow N}$ induce the same map on cohomology ${f^*=g^*:H^*(N)\rightarrow H^*(M)}$.

${\mathbb{R}^n}$ has the homotopy type of a point ${\{0\}}$. We have ${i}$ the inclusion of the origin and ${\pi}$ the unique map to the point. Then ${\pi\circ i}$ is the identity map on the point, and ${i\circ \pi:\mathbb{R}^n\rightarrow\mathbb{R}^n}$ sends every point to ${0}$. We claim this is homotopic to the identity. Define ${F:\mathbb{R}^n\times[0,1]\rightarrow \mathbb{R}^n}$ by ${F(x,t)=(1-t)x}$ to be the homotopy.

If ${f:M\rightarrow N}$ is a homotopy equivalence, then ${f^*:H^*(N)\rightarrow H^*(M)}$ is an isomorphism.

${H^*(\mathbb{R}^n)=H^*(pt)}$ is ${\mathbb{R}}$ in degree 0 and ${0}$ else.

2. Trang 2 – Sheaves

Let ${X}$ be a topological space, ${\mathscr{F},\mathscr{G}}$ sheaves. A morphism of sheaves ${\mathscr{F}\rightarrow\mathscr{G}}$ is a map for each open set compatible with the restriction maps.

We define the stalk at ${x\in X}$ of ${\mathscr{F}}$ to be ${\varinjlim_{x\in U}\mathscr{F}(U)}$ as abelian groups.

We define the associated sheaf to a given presheaf ${\mathscr{F}}$ to be the sheaf such that every map from the presheaf to any sheaf must factor through, and denote it ${\tilde{\mathscr{F}}}$. This is unique up to unique isomorphism.

The associated sheaf has the property that ${\mathscr{F}_x\cong \tilde{\mathscr{F}}_x}$ for all ${x\in X}$.

Now, let ${f:X\rightarrow Y}$ be a continuous map and ${\mathscr{F}}$ a sheaf on ${X}$. We define ${f_*\mathscr{F}}$ by ${f_*\mathscr{F}(U)=\mathscr{F}(f^{-1}(U))}$.

For ${Y\subset X}$, and ${\mathscr{F}}$ on ${X}$, we define ${f^{-1}(\mathscr{F})}$ to be ${\mathscr{F}|_Y}$.

2.1. Ringed Spaces

A pair ${(X,\mathscr{O}_X)}$ where ${\mathscr{O}_X}$ is a sheaf of rings on ${X}$ is a ringed space. A morphism of ringed spaces ${(X,\mathscr{O}_X)\rightarrow (Y,\mathscr{O}_Y)}$ is a continuous map ${f:X\rightarrow Y}$ and a map ${\mathscr{O}_Y\rightarrow f_*\mathscr{O}_X}$.

A locally ringed space is a ringed space such that the stalks of the sheaf ${\mathscr{O}_X}$ are all local rings, and a morphism of locally ringed spaces is required to induce on the stalks maps ${f^{-1}(\mathfrak{m}_{X,x})=\mathfrak{m}_{Y,y}}$.

Take ${(E,\mathscr{O}_E)}$ where ${E}$ is an algebraic set and ${\mathscr{O}_E}$ is the sheaf of regular functions. This example is the fundamental one in algebraic geometry.

The locally ringed spcae ${(\mathbb{C}^n,\mathscr{O}_{\mathbb{C}^n})}$ with the sheaf being the sheaf of local holomorphic functions is the fundamental example in analytic geometry.

2.2. Local Analytic Spaces

For ${U\subset\mathbb{C}^n}$, ${f_i:U\rightarrow \mathbb{C}}$ holomorphic for ${i\in I}$. Then we know ${(\mathbb{C}^n,\mathscr{O}_{\mathbb{C}^n})}$ is a locally ringed space, look at ${(U,\mathscr{O}_{\mathbb{C}^n}|_U)}$. Set ${X=\{f_i=0,i\in I\}}$ and then ${(f_i)\mathscr{O}_U=\mathscr{I}}$ is a sheaf of ideals.

So, we can now distinguish between ${(0,\mathbb{C})}$ and ${(0,\mathbb{C}\{x\}/x^2)}$, the first is just a point, the second is a double point, and can be viewed as the intersection of a parabola and a line tangent to its vertex.

2.3. Affine Schemes

Let ${A}$ be a ring (commutative with identity). Then ${\mathrm{Spec} A}$ is the set of prime ideals of ${A}$, and the closed sets are given by tkaing an ideal ${I}$ in ${A}$ and setting ${V(I)}$ to be the set of prime ideals containing ${I}$. The open sets are their complements.

We define on ${\mathrm{Spec} A}$ the sheaf ${\mathscr{O}_A}$, whose stalks at ${P}$ is ${A_P}$, and for any open set ${U}$, we define ${\mathscr{O}_A(U)}$ by ${s}$ is a section if ${s:U\rightarrow \coprod_{P\in U}A_P}$ with for all ${P}$ there exists an open neighborhood ${V}$ and ${a,b\in A}$ such that for all ${Q\in V}$ we have ${b\notin Q}$ and ${s(Q)=a/b}$.

The set of morphisms ${(\mathrm{Spec} A,\mathscr{O}_A)\rightarrow (\mathrm{Spec} B,\mathscr{O}_B)}$ is the same as the set of homomorphisms ${B\rightarrow A}$.

2.4. Schemes

A locally ringed space ${(X,\mathscr{O}_X)}$ is a scheme if for every ${x}$ there exists a ${U}$ such that ${(U,\mathscr{O}_X|_U)}$ is isomorphic to an affine scheme.

If ${R}$ is a graded ring ${\oplus_{g\geq 0} R_d}$, then we define a scheme ${(\mathrm{Proj} R,\mathscr{O}_R)}$ by ${\mathrm{Proj} R}$ is the set of homogeneous prime ideals in ${R}$. We want to set up ${\mathscr{O}_{R,P}}$ to be ${R_{(P)}}$, which is the set of elements of degree zero in ${T^{-1}R}$, where ${T}$ is the set of homogeneous elements which are not in ${P}$. We set ${s\in \mathscr{O}_R(U)}$ if ${s:U\rightarrow \coprod_{P\in U} R_{(P)}}$ such that for all ${P\in U}$ there exists ${V}$ and ${a,b}$ homogeneous of the same degree such that for all ${Q\in V}$, ${b\notin Q}$ and ${s(Q)=a/b}$.

3. Cattani 3

Let ${V}$ be a real vector space along with an operator ${J^2=-I}$. This makes it a complex vector space. We can also say ${V_\mathbb{C}=V\oplus iV=V'\oplus V''}$ where ${V'}$ is the ${i}$-eigenspace and ${V''}$ the ${-i}$-eigenspace. We write ${V^*_\mathbb{C}=V^{1,0}\oplus V^{0,1}}$ along with ${J^*}$, which are the ${i}$ and ${-i}$ eigenspaces. Then ${\bigwedge^rV^*_\mathbb{C}=\bigoplus \bigwedge^{p,q}}$ with ${p+q=r}$. So ${\bigwedge^2V^*_\mathbb{C}=\bigwedge^{2,0}\oplus \bigwedge^{1,1}\oplus\bigwedge^{0,2}}$, which are ${-1,1,-1}$ eigenspaces respectively.

So now, we have ${d:A^{p,q}(M)\rightarrow A^{p+1,q}\oplus A^{p,q+1}}$ and for each ${p}$ we get a complex ${0\rightarrow \Omega^p(U)\rightarrow A^{p,0}(U)\rightarrow A^{p,1}\rightarrow\ldots}$ which is exact for a small enough ${U}$ (more precisely, exact as a complex of sheaves.

For ${p=n=\dim M}$, then ${\Omega^n(U)=fdz_1\wedge\ldots\wedge dz_n}$ with ${f}$ holomorphic.

The Following are equivalent

1. A symmetric bilinear form ${B:V\times V\rightarrow \mathbb{R}}$ such that ${B(Ju,Jv)=B(u,v)}$
2. An alternating form ${\omega:V\times V\rightarrow \mathbb{R} }$such that ${\omega(Ju,Jv)=\omega(u,v)}$
3. A hermitian form ${H:(V,J)\times (V,J)\rightarrow \mathbb{C}}$ with ${H(Jv,Ju)=H(v,u)}$.

Now, we move to manifolds. Every complex manifold has a positive definite Hermitian structure on the holomorphic tangent bundle, which is equivalent to every complex manifold has a Riemannian metric compatible with ${J}$.

By this, we mean that on ${(U,z_1,\ldots,z_n)}$, we have that ${h_{jk}=H(\frac{\partial}{\partial x_j},\frac{\partial}{\partial x_k})}$ and ${\omega=\frac{1}{2}\sum_{j,k} h_{jk} dz_j\wedge d\bar{z}_k}$.

We define a hermitian structure on ${M}$ to be Kahler if ${d\omega=0}$. This implies that ${h_{jk}(z)=\delta_{jk}+O(|z|^2)}$.

3.1. Symplectic and Kähler Manifolds

A symplectic manifold is a pair ${(M,\omega)}$ where ${d\omega=0}$ and ${\omega^n\neq 0}$, with ${\omega}$ a 2-form. We’ll assume that ${M}$ is compact. Then ${M}$ being Kähler implies that ${M}$ is symplectic, because ${\int_M \omega^n\neq 0}$, and so ${0\neq [\omega]^n\in H^{2n}(M)}$, so each ${[\omega^k]}$ is nonzero.

Calabi and Eckmann proved that for ${n,m\geq 1}$, there was a complex structure on ${S^{2n+1}\times S^{2m+1}}$, and these can never be Kähler.

In fact, any compact symplectic manifold has an almost complex structure.

${\mathbb{C}^n}$ is Kähler with metric ${\frac{i}{2}\sum_{i,j}\delta_{ij}dz_i\wedge d\bar{z}_j}$.

${\mathbb{P}^n}$ has a Kähler structure, by taking on each ${U_j}$ the sunftion ${\rho_j([z])=\frac{\sum |z_i|^2}{|z_j|^2}>0}$. On ${U_j\cap U_k}$, we have ${|z_j|^2\rho_j=|z_k^2|\rho_k}$, we then take logs and apply ${\partial\bar{\partial}}$, and we find that ${\partial\bar{\partial}\log(\rho_j)=\partial\bar{\partial}\log(\rho_k)}$. So we set ${\omega_j=-\frac{1}{2\pi i}\partial\bar{\partial}\log(\rho_j(z))}$, and the metric we construct is the Fubini-Study metric.

Let ${N\subset M}$. Then for all ${p\in N}$, there exist coordinates ${(U,z_1,\ldots,z_n)}$ on ${M}$ around ${p}$ Such that ${N\cap U}$ is described by ${z_1=\ldots=z_k=0}$. If ${M}$ is Kähler and ${N\subset M}$ is a submanifold, then ${N}$ is Kähler.

Thus, if ${M}$ is a submanifold of ${\mathbb{P}^n}$, then ${M}$ is Kähler. Thus, ${[\omega]\in H^2(M)\cap H^2(M,\mathbb{Z})}$ is necessary.

4. Tu 3 – Presheaves and Cech Cohomology

A presheaf on a topological space ${X}$ is a function that assigns to each open ${U\subset X}$ an abelian group ${\mathscr{F}(U)}$ and to every inclusion ${i_U^V:U\rightarrow V}$ a group homomorphism ${\mathscr{F}(i^V_U)=\rho^U_V:\mathscr{F}(U)\rightarrow \mathscr{F}(V)}$ such that ${\rho^U_U=1_{\mathscr{F}(U)}}$, and ${\rho^U_W=\rho^V_W\circ\rho^U_V}$.

${\mathcal{A}^k(U)}$ is the ${C^\infty}$ ${k}$-forms on ${U}$. This is a presheaf on a manifold ${M}$.

If ${G}$ is an abelian group, for every open ${U\subset X}$, define ${\underline{G}(U)}$ to be the locally constant functions ${f:U\rightarrow G}$. Then ${\underline{G}}$ is a presheaf.

4.1. Cech Cohomology of an Open Cover

Let ${\{U_\alpha\}}$ be an open cover of a topological space indexed by a totally ordered set. We’ll denote intersections by putting the subscripts together.

When ${\alpha=0,1}$ gives the cover, we have Mayer-Vietoris, which says ${0\rightarrow \mathcal{A}^k(M)\rightarrow \prod \mathcal{A}^k(U_i)\rightarrow \mathcal{A}^k(U_{01})\rightarrow 0}$.

Now, let ${\mathscr{F}}$ be a presheaf on a topological space ${X}$. We then have a sequence

$\displaystyle 0\rightarrow\mathscr{F}(X)\rightarrow\prod_\alpha \mathscr{F}(U_\alpha)\rightarrow \prod_{\alpha<\beta}\mathscr{F}(U_{\alpha\beta})\rightarrow\ldots$

Define ${C^p(\mathcal{U},\mathscr{F})}$ to be the term involving ${p}$ open sets. Then we define ${\delta:C^p(\mathcal{U},\mathscr{F})\rightarrow C^{p+1}(\mathcal{U},\mathscr{F})}$ by ${(\delta\omega)_{\alpha_0,\ldots,\alpha_{p+1}}=\sum_{i=0}^{p+1}(-1)^i\omega_{\alpha_0,\ldots,\hat{\alpha}_i,\ldots,\alpha_{p+1}}}$.

It turns out that ${\delta^2=0}$, and so we define the cohomology of this complex to be the Cech cohomology ${\check{H}^k(\mathcal{U},\mathscr{F})}$.

4.2. Direct Limits

A directed set is a set ${I}$ with a binary relation ${<}$ that is reflexive, transitive and such that any two elements have a common upper bound.

Fix ${p\in X}$. Let ${I}$ be the set of neighborhoods of ${p}$ in ${X}$ and say that ${U iff ${U\subset V}$.

Fix a topological space ${X}$. Then ${I}$ be the set of all open covers of ${X}$. An open cover ${\mathcal{V}}$ refines ${\mathcal{U}}$ if every ${V\in\mathcal{V}}$ is contained in some ${U\in\mathcal{U}}$. Refinement gives a directed set structure to the set of covers. A refinement ${\mathcal{V}}$ of ${\mathcal{U}}$ can be given by a refinement map on the index sets stating which ${U}$ each ${V}$ is contained in.

A directed system of groups is a collection ${\{G_i\}_{i\in I}}$ of groups indexed by a directed set ${U}$ such that for all ${a we have a homomorphism ${f_{ab}:G_a\rightarrow G_b}$ satisfying that ${f_{aa}=1}$ and ${f_{ac}=f_{bc}\circ f_{ab}}$.

Let ${I}$ be the neighborhoods of ${p\in X}$. Then set ${G_U=C^\infty(U)}$ and say that ${(U,f)}$ and ${(V,g)}$ are equivalent iff there exists ${W\subset U\cap V}$ such that ${f|_W=g|_W}$. We call these the germs of functions at ${p}$.

In ${\coprod_{i\in I} G_i}$, let ${g_a\in G_a}$ and ${g_b\in G_b}$. Then we say ${g_a\sim g_b}$ if there exists ${c>a,b}$ such that ${f_{ac}(g_a)=f_{bc}(g_b)}$ and define ${\varinjlim_{i\in I} G_i=\coprod G_i/\sim}$.

4.3. Cech Cohomology of a Topological Space

For each open cover, we have ${\check{H}^k(\mathcal{U},\mathscr{F})}$, and we have restrictions making it into a directed system of abelian groups. So we define ${\check{H}^k(X,\mathscr{F})}$ to be the limit of this system.

4.4. ${C^\infty}$ partitions of unity

A ${C^\infty}$ partition of unity on a manifold ${M}$ is a collection of ${C^\infty}$ functions ${\rho_\alpha}$ with ${\sum\rho_\alpha=1}$ and ${\rho_\alpha:M\rightarrow [0,1]}$.

We define the support of a function to be the set where it is nonzero.

A collection of subsets ${\{A_\alpha\}}$ in ${X}$ is locally finite if every ${p\in X}$ has a neighborhood that meets only finitely many of the ${A_\alpha}$.

Given any open cover of a manifold, there exists a ${C^\infty}$ partition of unity with each element’s support contained in one of the open sets of the cover.

5. Trang 3 – Projective Schemes

Let ${R}$ be a graded ring, look at ${(\mathrm{Proj} R,\mathscr{O}_R)}$. Let ${f\in \oplus_{d>0}R_d=R_+}$, then we define ${D_+(f)}$ to be the homogeneous primes not containing ${f}$. ${(D_+(f),\mathscr{O}_R|_{D_+(f)})}$ is isomorphic to ${(\mathrm{Spec} R_{(f)},\mathscr{O}_{R_{(f)}})}$.

For any ring ${R=A[x_0,\ldots,x_n]}$, we have ${(\mathrm{Proj} R,\mathscr{O}_R)}$ and we’ll call it ${\mathbb{P}^n_A}$.

5.1. Gluing Schemes

Let ${(X_1,\mathscr{O}_1)}$ and ${(X_2,\mathscr{O}_2)}$ be two schemes wuch that ${(U_1,\mathscr{O}_1|_{U_1})\rightarrow (U_2,\mathscr{O}_2|_{U_2})}$ is an isomorphism. Then we can construct a new scheme by identifying them along this map.

5.2. Schemes over a scheme

Let ${\phi:(X,\mathscr{O}_X)\rightarrow (S,\mathscr{O}_S)}$. We call this an ${S}$-scheme, and often abuse notation by calling ${X}$ an ${S}$-scheme.

In particular, we will look at schemes over ${(\mathrm{Spec} k,k)}$.

5.3. Varieties and Schemes

Any variety is covered by a finite number of affine algebraic varieties.

This means that we can take any variety ${V}$ over ${k}$, and make a scheme over ${\mathrm{Spec} k}$ out of it, by just taking affine varieties to ${\mathrm{Spec} A(E)}$.

Now, we say that a scene is connected if ${X}$ is, irreducible if ${X}$ is, it is reduced if the rings are all reduced (have no nilpotents) and integral similarly.

A scheme is integral if and only if it is reduced and irreducible.

A scheme is locally noetherian if it has a covering by spectra of noetherian rings.

Now, let ${f:(X,\mathscr{O}_X)\rightarrow (Y,\mathscr{O}_Y)}$ be a morphisms of schemes.

We say ${f}$ is locally of finite type if ${(Y,\mathscr{O}_Y)}$ is covered by ${(\mathrm{Spec} A_i,\mathscr{O}_{A_i})}$ such that ${f^{-1}(\mathrm{Spec} A_i)=\mathrm{Spec} B_{ij}}$ with the ${B_{ij}}$ being ${A_i}$-algebras of finite type, that is, are finitely generated as algebras. We say that it is of finite type if the ${B_{ij}}$ are finite as modules.

An example is the map from a parabola to a line, which induces ${k[x]\rightarrow k[x,y]/(y-x^2)}$.

We can construct fiber products, take ${\phi:X\rightarrow S}$ and ${\psi:Y\rightarrow S}$, we can get ${X\times_S Y}$, it’s the unique scheme such that for all maps ${Z\rightarrow X}$ and ${Z\rightarrow Y}$ that are equal after composing with the maps to ${S}$, we get a unique map ${Z\rightarrow X\times_S Y}$.

This allows us to define base change: If we have a map ${X\rightarrow Y}$ and another ${Y'\rightarrow Y}$, we can define ${X'=X\times_Y Y'}$ and we have ${X'\rightarrow Y'}$, the base change of the morphism.

For any ${X\rightarrow Y}$, we have a map ${X\rightarrow X\times_Y X}$ by using the same map to create the fiber product. If this morphism is closed, then we say that ${f}$ is separated.

We call a morphism proper if it is separated, of finite type and universally closed, and we say that a scheme which is proper over ${\mathrm{Spec} k}$ is complete.

5.4. Projective Morphisms

Let ${(Y,\mathscr{O}_Y)}$ be a scheme. We define ${\mathbb{P}^n_Y\rightarrow Y}$ to be projective space over ${Y}$, where ${\mathbb{P}^n_Y=Y\times_{\mathrm{Spec} \mathbb{Z}}\mathbb{P}^n_\mathbb{Z}}$.

We say that a morphism is projective if it factors through ${\mathbb{P}^n_Y\rightarrow Y}$ and the map ${X\rightarrow \mathbb{P}^n_Y}$ is a closed immersion.

Projective morphisms of Noetherian schemes are proper, and quasi-projective morphisms are separated and of finite type.

So a variety over ${k}$ turns out to just be a scheme over ${k}$ which is integral and of finite type.

6. Cattani 4

Let’s look at the real, smooth case. Let ${M}$ be a compact oriented Riemannian manifold.

If ${V}$ is a real vector space which is oriented with an inner product, then ${\bigwedge^k(V^*)}$ has an inner product as well.

Show that ${\langle \alpha_1\wedge\ldots\wedge\alpha_r,\beta_1\wedge\ldots\wedge\beta_r\rangle=\det(\langle\alpha_i,\beta_j\rangle)}$.

Volume element ${\Omega\in \bigwedge^n(V^*)}$ given by ${\xi_1\wedge\ldots\wedge \xi_n}$, where the ${\xi_i}$ form an orthonormal basis.

We have a map ${*:\bigwedge^k(V^*)\rightarrow \bigwedge^{n-k}(V^*)}$, and ${*(\xi_{i_1}\wedge\ldots\wedge\xi_{i_r})=\mathrm{sign}(I,J)\xi_{j_1}\wedge\ldots\wedge \xi_{j_{n-r}}}$. So then ${\alpha\wedge *\beta=\langle \alpha,\beta\rangle\Omega}$.

Now, ${*}$ is an isomorphism, and it satisfies ${*^2=(-1)^{r(n-r)}}$.

Back to the manifold ${M}$. We can define an inner product on forms ${\alpha,\beta}$ by ${\int_M \alpha\wedge *\beta}$. This is a positive definite bilinear form.

Now, we define ${\delta=(-1)^{nr+1}*d*}$, and it takes ${r}$-forms to ${r-1}$-forms, using this ${*}$ operator. We claim that ${d}$ and ${\delta}$ are adjoints, that is, ${(d\alpha,\beta)=(\alpha,\delta\beta)}$.

${\int_M d\alpha\wedge*\beta=\int_M d(\alpha\wedge *\beta)-(-1)^r\alpha\wedge d*\beta}$, but this will just be ${-(-1)^r\int_M \alpha\wedge d*\beta}$, up to sign, we can just insert a ${*^2}$ in front of ${d}$, and the signs work out.

Now, if ${d\alpha=0}$, then ${*\alpha}$ may not be closed, but it is if and only if ${\delta\alpha=0}$.

${d\alpha=\delta\alpha=0}$ if and only if ${(d\delta+\delta d)\alpha=0}$.

One direction is simple, for the other, we have ${0=\langle (d\delta+\delta d)\alpha,\alpha\rangle}$, which using the adjoint property proves the result.

We call this operator ${\Delta}$, the Laplace-Beltrami operator, or the Laplacian, and we call any form ${\alpha}$ with ${\Delta\alpha=0}$ a harmonic form.

${*\Delta=\Delta*}$.

Why should we hope that every cohomology class has a harmonic form in it?

Heuristically, start with ${d\alpha=0}$. Then ${[\alpha]}$ is the set of forms of the form ${\alpha+d\beta}$. Then ${\|\alpha+td\beta\|^2=\langle\alpha,\alpha\rangle+2t\langle \alpha,d\beta\rangle+t^2\|d\beta\|^2}$. Now, suppose ${\|\alpha\|^2}$ is a maximum. Then for all ${\beta}$ we’ll have that ${\langle \alpha,d\beta\rangle=0}$ and ${\langle \delta\alpha,\beta\rangle=0}$, so we can assume that ${\delta\alpha=0}$.

1. ${\mathscr{H}^k(M)}$, the harmonic forms, is finite dimensional
2. ${\mathcal{A}^r(M)=\mathscr{H}^k\oplus\Delta(\mathcal{A}^r(M))}$.

In particular, every form is a harmonic form, plus ${d}$ of something plus ${\delta}$ of something. So then any form is ${\alpha+d\beta+\delta\gamma}$. But if it’s closed, then ${d\delta\gamma=0}$, which implies that ${\delta\gamma=0}$, so for any closed form, it is of the form ${\alpha=\eta+d\beta}$.

Thus, ${H^r_{dR}(M,\mathbb{R})\cong \mathscr{H}^r(M)}$.

Take a submanifold ${Z^{n-k}\subset M^n}$, then for any form in ${H^{n-k}(M)}$, we can restrict it to ${Z}$ and integrate to get a map to ${\mathbb{R}}$. By Poincaré duality, this gives us a class ${\eta_Z\in H^k(M)}$.

Now, we define everything in a “hermitian way.” So we take ${\int_M \alpha\wedge*\bar{\beta}}$, and note that ${*}$ takes ${(p,q)}$ to ${(n-q,n-p)}$.

We define ${\bar{\partial}^*=-*\partial *}$ and ${\partial ^*=-*\bar{\partial }*}$, and these are of type ${(0,-1)}$ and ${(-1,0)}$ and match with ${\partial }$ and ${\bar{\partial }}$. So then we have ${\Delta_{\bar{\partial }}=\bar{\partial }\bar{\partial }^*+\bar{\partial }^*\bar{\partial }}$, and we can write any form as a sum ${\alpha=\alpha^{k,0}+\alpha^{k-1,1}+\ldots}$.

And, we leave off with the fact that, on a Kähler manifold ${\Delta_d=2\Delta_{\bar{\partial }}}$.