ICTP Day 3 Part I

This is the end of the preliminary background lectures. This afternoon, we start Hodge Theory itself, with Hodge Structures, Variations, and Hodge Theory of Maps as the three lecture series. Tonight, when I post the notes for the next three lectures here, I’ll also link to my website, where lecture notes for each series should be available in PDF format, and a bit nicer than the wordpress ones.

1. Tu 4 – Sheaves and the Cech-deRham Isomorphism

Version 6 of lecture notes on Summer School website.

Compute ${H^*}$ of ${S^n}$, ${\mathbb{R}^n\setminus\{0\}}$, ${\mathbb{C}\mathbb{P}^1}$, ${\mathbb{C}\mathbb{P}^2}$ for problem session today.

Problem Session: Does ${H^*(\mathcal{U},\mathscr{F})}$ depend on the order on the index set?

1.1. Sheaves

A sheaf on a topological space ${X}$ is a presheaf such that for any open set ${U}$ and any open cover ${U_i}$ of ${U}$, we have

1. Uniqueness: If ${s\in\mathscr{F}(U)}$ such that ${s|_{U_i}=0}$ for all ${i}$, then ${s=0}$.
2. Gluing: If ${s_i\in \mathscr{F}(U_i)}$ such that ${s_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j}}$ for all ${i,j}$, then there exists ${s\in\mathscr{F}(U)}$ such that ${s|_{U_i}=s_i}$.
1. ${\mathcal{A}^k}$ is a sheaf for any manifold.
2. ${Z^k}$, the closed ${C^\infty}$ ${k}$-forms on a manifold is a sheaf.
3. ${\mathcal{A}^{p,q}}$ is a sheaf for any complex manifold.
4. ${\underline{\mathbb{R}}}$, the locally constant functions to ${\mathbb{R}}$, is a sheaf.

However, the presheaf of constant functions is not a sheaf.

1.2. Cohomology in Degree 0

Let ${\mathscr{F}}$ be a sheaf on a topological space ${X}$. We want to know ${\check{H}^0(\mathcal{U},\mathscr{F})}$. Let ${\mathcal{U}=\{U_i\}}$ be an open cover of ${X}$. Then as ${\mathscr{F}}$ is a sheaf, we have the Cech sequence.

So then ${\check{H}^0(\mathcal{U},\mathscr{F})=\ker\delta}$ as ${C^{-1}=0}$ which is just ${\mathrm{im}(r)}$ where ${r}$ is the restriction map ${\mathscr{F}(X)\rightarrow \prod \mathscr{F}(U_i)}$. But this is precisely ${\mathscr{F}(X)}$.

Now, let ${\mathcal{V}}$ be a refinement of ${\mathcal{U}}$. Both Cech groups are isomorphic to ${\mathscr{F}(X)}$, and so they are isomorphic.

If ${\mathscr{F}}$ is a sheaf on a topological space ${X}$, then ${\check{H}^0(X,\mathscr{F})=\mathscr{F}(X)}$.

1.3. Further Computations

For all ${q}$ and all ${k>0}$, ${\check{H}^k(M,\mathcal{A}^q)=0}$. But how do we prove this? The standard method for proving that cohomology is zero is finding a map ${K:C^p\rightarrow C^{p-1}}$ such that ${1-0=\delta K+K\delta}$. Applying ${(\delta K+K\delta)}$ to a cocycle just gives ${\delta K}$. Therefore, this induces the zero map on cohomology, and so ${1^*=0}$, which can only happen when ${H^q=0}$. We call ${K}$ a chain homotopy. There is a theorem guaranteeing that such a ${K}$ exists when the cohomology is zero.

Let ${\mathcal{U}=\{U_\alpha\}}$ be an open cover of ${M}$, and ${\{\rho_\alpha\}}$ be a ${C^\infty}$ partition of unity subordinate to ${\mathcal{U}}$. For ${k\geq 1}$, then ${K:C^p(\mathcal{U},\mathscr{F})\rightarrow C^{p-1}(\mathcal{U},\mathscr{F})}$ by ${(K\omega)_{\alpha_0,\ldots,\alpha_{p-1}}=\sum_\alpha \rho_\alpha \omega_{\alpha,\alpha_0,\ldots,\alpha_{p-1}}}$.

We must check that ${\delta K+K\delta=1}$ for ${k\geq 1}$, and then we have the vanishing of the Cech cohomology, when we take ${\mathscr{F}=\mathcal{A}^q}$. We in fact have ${H^k(M,\mathcal{A}^{p,q})=0}$ for ${k\geq 1}$ on a complex manifold.

But ${H^k(M,\Omega^q)}$ is not necessarily zero!

1.4. Sheaf Morphism

A morphism of sheaves is a collection of maps, one for each open set, compatible with the restriction maps

We define the stalk of a presheaf ${\mathscr{F}_p}$ at ${p\in X}$ to be ${\varinjlim_{p\in U} \mathscr{F}(U)}$. So a presheaf homomorphism ${\phi_p:\mathscr{F}_p\rightarrow \mathscr{G}_p}$ by ${(U,s)\mapsto (U,\phi(s))}$.

1.5. Exact Sequences of Sheaves

A short exact sequence of sheaves ${0\rightarrow\mathscr{E}\rightarrow \mathscr{F}\rightarrow \mathscr{G}\rightarrow 0}$ induces a long exact sequence ${\ldots\rightarrow H^k(X,\mathscr{E})\rightarrow H^k(X,\mathscr{F})\rightarrow H^k(X,\mathscr{G})\rightarrow H^{k+1}(X,\mathscr{E})\rightarrow\ldots}$.

The sequence of sheaves ${0\rightarrow\mathbb{R}\rightarrow\mathcal{A}^0\rightarrow\ldots}$ is exact.

Proof: Look at ${0\rightarrow\mathbb{R}\rightarrow\mathcal{A}^0\rightarrow \mathrm{im}(d_0)\rightarrow 0}$, and similarly, we can make ${0\rightarrow Z^{k-1}\rightarrow \mathcal{A}^{k-1}\rightarrow Z^k\rightarrow 0}$ for all ${k}$.

So we have ${0\rightarrow H^{k-1}(M,Z^1)\rightarrow H^k(M,\underline{\mathbb{R}})\rightarrow H^k(M,\mathcal{A}^0)=0}$, and we can finish the proof by basic homological manipulations. $\Box$

2. Trang 4 – Sheaves of Modules

Let ${(X,\mathscr{O}_X)}$ be a ringed space and ${\mathscr{M}}$ a sheaf on ${X}$ such that for all ${U}$, ${\mathscr{M}(U)}$ is an ${\mathscr{O}_X(U)}$-module and these structures are compatible with the restriction maps.

2.1. Locally Free Modules

${\mathscr{M}}$ is free if for all ${U}$, ${\mathscr{M}(U)}$ is a free ${\mathscr{O}_X(U)}$-module.

On a scheme ${(X,\mathscr{O}_X)}$, say ${(\mathrm{Spec} A,\mathscr{O}_A)}$, then an ${\mathscr{O}_A}$-module on ${\mathrm{Spec} A}$ is given by an ${A}$-module ${M}$ in the following way. Define ${\tilde{M}(U)}$ by the sections over ${U}$ are ${s:U\rightarrow \prod_{P\in U} M_P}$ such that for all ${P\in U}$, there exists ${V,m,a}$ such that for all ${Q\in V}$, we have ${s(Q)=\frac{m}{a}}$. Then ${\tilde{M}_P=M_P}$. These are the prototypes of “good” sheaves.

A sheaf of ${\mathscr{O}_X}$-modules is quasi-coherent if there exists covering by open affines such that it’s restrictions are of the form ${\tilde{M}_i}$.

${\mathscr{F}}$ is coherent if the ${M_i}$ are finite ${A_i}$-modules.

2.2. Differential Forms

Let ${A}$ be a ring, ${B}$ an ${A}$-algebra and ${M}$ a ${B}$-module. Then an ${A}$-derivation of ${B}$ into ${M}$ is a map ${d:B\rightarrow M}$ such that ${d}$ is additive, for all ${b,b'\in B}$ we have ${d(bb')=bd(b')+b'd(b)}$ and for all ${a\in A}$, we have ${d(a1)=0}$.

There exists a universal object, a module ${\Omega_{B/A}}$ and derivation ${\delta:B\rightarrow \Omega_{B/A}}$ such that any other derivation factors through it. We can construct it by looking at ${\Delta:B\otimes_A B\rightarrow B}$ given by ${\Delta(b\otimes b')=bb'}$, setting ${I=\ker\Delta}$ and then ${\Omega_{B/A}=I/I^2}$, with ${\delta:B\rightarrow I/I^2}$ given by ${\delta(b)=1\otimes b-b\otimes 1}$ modulo ${I^2}$.

Now, letting ${X\rightarrow Y}$ be a morphism of schemes, we can take ${\Delta:X\rightarrow X\times_Y X}$ and and let ${\mathscr{I}}$ be the ideal sheaf of the image of ${\Delta}$. Then ${\Omega_{X/Y}=\mathscr{I}/\mathscr{I}^2}$.

Let ${A\stackrel{h}{\rightarrow}B\stackrel{k}{\rightarrow}C}$. Then we have exact sequences ${\Omega_{B/A}\otimes_B C\rightarrow \Omega_{C/A}\rightarrow \Omega_{B/A}\rightarrow 0}$ and if ${I}$ is an ideal of ${B}$ and ${C=B/I}$, then we have ${I/I^2\rightarrow \Omega_{B/A}\otimes_B C\rightarrow \Omega_{C/A}\rightarrow 0}$.

The sheaf is well-behaved with respect to base change: let ${f:X\rightarrow Y}$ a morphism, and base change along ${g:Y'\rightarrow Y}$, then ${\Omega_{X'/Y'}=(g')^*(\Omega_{X/Y})}$.

More generally, let ${f:X\rightarrow Y}$ and ${g:Y\rightarrow Z}$ morphisms of schemes. Then ${f^*\Omega_{Y/Z}\rightarrow \Omega_{X/Z}\rightarrow\Omega_{X/Y}\rightarrow 0}$ is exact, and, if ${Z\subset X}$ is a closed subscheme with ideal ${\mathscr{I}}$, we have ${\mathscr{I}/\mathscr{I}^2\rightarrow \Omega_{X/Y}\otimes_{\mathscr{O}_X}\mathscr{O}_Z\rightarrow \Omega_{X/Z}\rightarrow 0}$.

2.3. Nonsingular Varieties

Let ${Y\subset k^n}$ affine, with ${I(Y)=(f_1,\ldots,f_k)}$. Then we say that ${\dim Y}$ is the Krull dimension of ${A(Y)}$, and we’ll denote it by ${r}$. Then a point ${x}$ is a nonsingular point of ${Y}$ if and only if ${\mathrm{rank} \left(\frac{\partial f_i}{\partial x_j}\right)=n-r}$ at ${x}$. We say that ${Y}$ is nonsingular if it is nonsingular at every point.

It is a theorem that ${y\in Y}$ is nonsingular if and only if ${\mathscr{O}_{Y,y}}$ is a regular local ring, so we take this as the definition, on schemes.

What does it mean to be regular? For a Noetherian local ring of dimension ${r}$, the following are equivalent and taken to define regularity:

1. ${\mathfrak{m}}$ is generated by ${r}$ elements.
2. The associated graded ring with respect to ${\mathfrak{m}}$ is a polynomial ring in ${r}$ variables.
3. ${\dim(\mathfrak{m}/\mathfrak{m}^2)=r}$.

Now, let ${X}$ be a locally ringed space isomorphic to an analytic space. This is locally a local analytic space, and it is Hausdorff, and there exists an analytification functor which takes varieties to analytic spaces.

3. Cattani 5

We’ll be working on a compact Kähler manifold. We know that ${\Delta=2\Delta_{\bar{\partial}}}$.

If ${\alpha\in A^k(M)}$ is harmonic and ${\alpha=\sum \alpha_{p,q}}$, then ${\Delta\alpha_{p,q}=0}$.

Let ${H^{p,q}(M)}$ be the set of classes in ${H^{p+q}(M)}$ such that ${\alpha}$ has a representation in bidegree ${p,q}$.

${H^k_{dR}(M,\mathbb{C})\cong\oplus_{p+q=k}H^{p,q}(M)}$ and ${H^{p,q}(M)\cong \mathscr{H}^{p,q}(M)=H^{p,q}_{\bar{\partial}}(M)\cong H^q(M,\Omega^p)}$.

We in fact have ${H^{q,p}(M)=\overline{H^{p,q}(M)}}$.

Now, if ${k}$ is odd, we have ${H^k(M)=H^{k,0}\oplus\ldots\oplus H^{0,k}}$, and there is no middle term, so the dimension of ${H^k(M)}$ is even when ${k}$ is odd.

The cup product will actually respect the bigrading: ${H^{p,q}\cup H^{p',q'}\subset H^{p+p',q+q'}}$. Voisin has found examples where every condition is satisfied except this property.

Why should all of these nice things be true? Let ${\omega\in H^2(M,\mathbb{R})\cap H^{1,1}(M)}$. We have a Lefschetz map ${L_\omega:A^*\rightarrow A^*}$ by ${\alpha\mapsto \omega\wedge \alpha}$ increasing degree by ${(1,1)}$. We have ${L_\omega^{n+1}=0}$. Now, we can define ${Y:A^*\rightarrow A^*}$ by ${Y(\alpha)=(n-k)\alpha}$ for ${\alpha\in A^k}$, and we have ${[Y,L]=-2L}$. Now, define ${N_+=(-1)^k*L*}$ on ${A^*}$. This is the adjoint of ${L}$, and it satisfies ${[Y,N_+]=2N_+}$ and ${[N_+,L]=Y}$, so this actually gives us a representation of the Lie algebra ${\mathfrak{sl}_2}$!

In this representation, ${L}$ and ${N_+}$ are shifts, and the eigenvalues of ${Y}$ are the degrees of forms, ${L}$ increases degree and ${N_+}$ decreases. Then there’s the Lefschetz Theorem which says, first, that ${L^k_\omega:\bigwedge^{n-k}(T_p^*)\rightarrow\bigwedge^{n+k}(T_p^*)}$ is an isomorphism. Moreover, it tells us that there are two types of cohomology classes ${\bigwedge^{n-k}(T_p^*)=P^{n-k}\oplus L(\bigwedge^{n-k-2})}$. We call this the Lefschetz decomposition for forms.

We in fact have the following relations:

1. ${[\partial,L]=[\bar{\partial},L]=[\partial^*,N_+]=[\bar{\partial}^*,N_+]=0}$
2. ${[\bar{\partial}^*,L]=i\partial}$, ${[\partial^*,L]=-i\bar{\partial}}$, ${[\bar{\partial},N_+]=i\partial^*}$, ${[\partial,N_+]=-i\bar{\partial}^*}$.

These imply that ${[\Delta_\partial,L]=[\Delta_\partial,Y]=[\Delta_\partial,N_+]=0}$, by just writing it out and putting ${L}$ between the partials. This implies that the Lefschetz theorem holds if we replace forms by cohomology classes. We call this the Hard Lefschetz Theorem.

This gives another topological restriction: the betti numbers (even and odd separately) must be increasing to the middle degree.

Now, let us assume ${\dim_\mathbb{C} M=1}$, so we are on a Riemann surface. So then ${H^1=H^{1,0}\oplus H^{0,1}}$, and by the Dolbeault theorem, ${H^{1,0}\cong H^{1,0}_{\bar{\partial}}\cong H^0(M,\Omega^1)}$, the holomorphic 1-forms, that can be written ${f(z)dz}$ locally, for ${f}$ holomorphic. Then there is ${H^{0,0}}$ and ${H^{1,1}}$, so the Hodge numbers are easy: ${h^{0,0}=h^{1,1}=1}$ and ${h^{1,0}=h^{0,1}=g}$.

On a complex surface, things are slightly more interesting, and ${H^{1,1}}$ splits into ${H^{1,1}_0+\mathbb{C}\omega}$, where subscript of zero will indicate the primitive cohomology.