## ICTP Day 3 Part II

Here’s the second set of notes, the beginning of the crash course in Hodge theory before we get to advanced topics next week. My notes from the first day and a half are available here, please let me know if there are any errors or any bad code that should be fixed. Without further ado, Hodge Theory, day 0.5:

1. ElZein 1 – Hodge Structures and Mixed Hodge Structures

Work of Deligne, then Griffiths

Hodge decomposition is a geometric invariant: this means that if ${f:X\rightarrow Y}$ is a morphism of compact Kähler manifolds, then we have a map ${f^*:H^i(Y,\mathbb{Z})\rightarrow H^i(X,\mathbb{Z})}$, we have ${H^i(Y,\mathbb{C})=\oplus_{p+q=i}H^{p,q}}$ and ${f^*(H^{p,q}(Y))\subset H^{p,q}(X)}$. If ${f}$ is analytic, then we have ${f^*:H^{p,q}(Y)\rightarrow H^{p,q}(X)}$.

The Hodge decomposition is a linear structure on the cohomology.

The cohomology groups of algebraic varieties carry mixed Hodge structures.

A hodge structure of weight ${m}$ is defined by a finitely generated group ${H_\mathbb{Z}}$, a decomposition of ${H_\mathbb{C}=H_\mathbb{Z}\otimes\mathbb{C}}$ into a direct sum ${H_\mathbb{C}=\oplus_{p+q=m}H^{p,q}}$ where ${H^{p,q}}$ is a complex subspace such that ${\overline{H^{p,q}}=H^{p,q}}$.

Let ${H_\mathbb{C}=\mathbb{C}^2}$ and ${\mathbb{Z}^2=\mathbb{Z} e_1\oplus \mathbb{Z} e_2}$. We can make a Hodge structure by ${H^{1,0}=\mathbb{C}(e_1-ie_2)}$ and ${H^{0,1}=\mathbb{C}(e_1+ie_2)}$, but not by trying to make ${H^{1,0}=\mathbb{C} e_1}$.

1.1. Hodge Structures of Weight 1

${H=H_\mathbb{Z}}$, then ${H_\mathbb{C}=H^{1,0}\oplus H^{0,1}}$. We have ${H_\mathbb{Z}\rightarrow H_\mathbb{R}\rightarrow H_\mathbb{C}=H^{1,0}\oplus H^{0,1}\rightarrow H^{0,1}}$ gives an isomorphism ${H_\mathbb{R}\cong H^{0,1}}$, and the image ${H_\mathbb{Z}\rightarrow H_\mathbb{R}}$ gies a lattice, so ${H^{0,1}/H_\mathbb{Z}}$ is a torus.

When ${X}$ is Kähler, ${H^{0,1}=H^1(X,\mathscr{O}_X)}$, we have an exact sequence of sheaves using the exponential map ${0\rightarrow \mathbb{Z}_X\rightarrow \mathscr{O}_X\rightarrow \mathscr{O}^*_X\rightarrow 0}$ which gives a long exact sequence including ${H^1(X,\mathbb{Z})\rightarrow H^1(X,\mathscr{O}_X)\rightarrow H^1(X,\mathscr{O}_X^*)\rightarrow H^2(X,\mathbb{Z})}$ where the last map takes line bundles to the first Chern class. So then ${H^1(X,\mathscr{O}_X)/H^1(X,\mathbb{Z})}$ is the kernel of Chern map, and this is a complex torus called the Picard torus.

1.2. Algebraic Operations on Hodge Structures

If ${H}$ and ${H'}$ are Hodge structures of the same weight then so is ${H\oplus H'}$.

${\hom_\mathbb{C}(H_\mathbb{C},\mathbb{C})}$ with ${\hom_\mathbb{Z}(H_\mathbb{Z},\mathbb{Z})}$ is a Hodge structure of weight ${-m}$.

If ${H}$ and ${H'}$ are Hodge structures of weight ${m}$ and ${m'}$, then ${H\otimes H'}$ is a Hodge structure of weight ${m+m'}$

${\bigwedge^nH}$ is a Hodge structure of weight ${mn}$

Take ${H_\mathbb{Z}=2\pi i \mathbb{Z}}$ and ${H_\mathbb{C}=\mathbb{C}=H^{-1,-1}}$ of weight ${-2}$.

We define ${T(m)=\otimes_m H_\mathbb{Z}}$, and this gives a Hodge structure of wieght ${-2m}$ with ${H_\mathbb{C}=H^{-m,-m}}$.

A Hodge structure of weight ${m}$ is defined by ${H_\mathbb{Z}}$ and a finite, decreasing filtration ${F}$ by subspaces ${H_\mathbb{C}\supset\ldots\supset F^p\supset F^{p+1}\supset\ldots}$ such that for all ${p}$, ${F^p\oplus \bar{F}^{m-p+1}=H_\mathbb{C}}$.

The two definitions of Hodge structure are equivalent.

Proof: Start with a decomposition. Define ${F^pH_\mathbb{C}=\oplus_{i\geq p} H^{i,m-i}\subset H_\mathbb{C}}$. Then ${\bar{F}^p=\oplus_{i\leq m-p} H^{i,m-i}}$, so the property of filtrations follows. For the other direction, define ${H^{p,q}=F^p\cap \bar{F}^q}$. $\Box$

A morphism of Hodge structures ${L:H\rightarrow H'}$ is a map defined on the abelian groups such that, after complexification, satisfies ${L(F^p)\subset {F'}^p}$.

1.3. Polarization

A polarization of a Hodge structure ${H}$ of weight ${m}$ is a bilinear form ${S:H_\mathbb{Q}\otimes H_\mathbb{Q}\rightarrow \mathbb{Q}}$ which is symmetric for ${m}$ even and skew-symmetric for ${m}$ odd such that the complex extension satisfies ${S(H^{p,q},H^{p',q'})=0}$ unless ${p=q'}$ and ${q=p'}$, and that ${i^{p-q}S(v,\bar{v})>0}$ for ${v\neq 0}$.

This gives a positive definite Hermitian form.

A mixed Hodge structure ${H}$ is defined to be a finitely generated group ${H_\mathbb{Z}}$, an increasing filtration ${W}$ on ${H_\mathbb{Q}}$ and a decreasing filtration ${F}$ such that ${\mathrm{Gr}^W_m H_\mathbb{Q}}$‘s complexification has a Hodge structure of weight ${m}$ induced from ${F}$.

2. Cattani 1 – Odds and ends

2.1. Riemann-Hodge Relations

Let ${\alpha,\beta\in H^{n-k}}$. We have ${\alpha\cup \beta\in H^{2n-2k}}$, and we define a number using ${(-1)^{(n-k)(n-k+1)/2}\int_M \alpha\cup\beta\cup\omega^k}$, then the Riemann-Hodge relations say that this gives a polarization on ${H_0^{n-k}(M,\mathbb{C})}$. We’ll denote this by ${Q(\alpha,\beta)}$.

In dimension one, we have ${H^1(M,\mathbb{C})=H^{1,0}\oplus H^{0,1}}$, and then ${Q(\alpha,\beta)=\int_M \alpha\wedge \beta}$, and ${iQ(\alpha,\bar{\alpha})>0}$ if ${0\neq \alpha}$.

Subset: ${G(g,H^1(M,\mathbb{C}))}$, example 1.16.

2.2. Connections

Look at the sequence ${0\rightarrow \mathbb{Z}\rightarrow \mathscr{O}_X\rightarrow \mathscr{O}_X^*\rightarrow 0}$, it gives the long exact sequence with the maps ${H^1(X,\mathscr{O}_X)\rightarrow H^1(X,\mathscr{O}_X^*)\rightarrow H^2(M,\mathbb{Z})}$, with the last map taking ${\{g_{\alpha\beta}\}}$ to ${n_{\alpha\beta\gamma}=\frac{1}{2\pi i} (\log g_{\alpha\beta}-\log g_{\alpha\gamma}+\log g_{\beta\gamma}}$.

We have ${0\rightarrow\mathbb{R}\rightarrow \mathcal{A}^0\stackrel{d}{\rightarrow} Z^1\rightarrow 0}$ and ${0\rightarrow Z^1\rightarrow \mathcal{A}^1\rightarrow Z^2\rightarrow 0}$, and ${\check{H}^2(M,\mathbb{R})\cong H^1(M,Z^1)\cong H^0(M,Z^2)/d(H^0(M,\mathcal{A}^1))}$ with the map taking ${n_{\alpha\beta\gamma}}$ to ${\frac{1}{2\pi i} d\log g_{\alpha\beta}}$.

Question: Can we find, in some natural way, 1-forms ${\{\theta_\alpha\in A^1(U_\alpha)\}}$ such that ${\frac{1}{2\pi i} d\log g_{\alpha\beta}=\theta_\beta-\theta_\alpha}$?

Let ${L\rightarrow M}$ be a line bundle, ${U_\alpha\rightarrow L}$ such that ${|\sigma_\alpha|^2=\rho_\alpha}$ and ${\rho_\beta=|g_{\alpha\beta}|^2\rho_\alpha}$.

${\frac{1}{2\pi i}\partial \log \rho_\beta=\frac{1}{2\pi i} (d\log(g_{\alpha\beta}+\partial\log\rho_\alpha)}$

Taking ${\theta_\beta=\frac{1}{2\pi} \partial \log\rho_\beta}$, we get ${-c(L)=\frac{1}{2\pi i}\bar{\partial}\partial \log \rho_\beta}$, and to get ${c(L)}$ we just exchange ${\partial}$ and ${\bar{\partial}}$.

Suppose you have a coframe, that is, a basis for sections of ${\mathscr{O}(U,E)}$, ${s_i}$. We want a derivative map ${D:\mathscr{O}(U,E)\rightarrow \Omega^1(U)\otimes \mathscr{O}(E)}$. We need ${D(fs)=df\otimes s+fDs}$, then we’ll have ${Ds_j=\sum_i \theta_{ij}\otimes s_j}$ where ${\theta_{ij}\in \Omega^1(U)}$.

These will need to satisfy some condition under change of coordinates in order to make sense. If ${g_{\alpha\beta}}$ is the matrix of transition functions, then if ${\theta'=g^{-1}_{\alpha\beta}dg{\alpha\beta}+g^{-1}_{\alpha\beta}\theta g_{\alpha\beta}}$. For any individual ${\theta_\alpha}$ we have ${\theta_\beta=g^{-1}_{\alpha\beta}dg_{\alpha\beta}+\theta_\alpha=d(\log g_{\alpha\beta})}$.

Finding ${\{\theta_\alpha\}}$ such that ${\theta_\beta-\theta_\alpha=\frac{1}{2\pi i}d\log g_{\alpha\beta}}$ is the same as having a connection.

3. Migliorini 1 – Hodge Theory of Maps

The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar constraints on algebraic maps?

Let ${f:X\rightarrow Y}$ a proper morphism. Our goal is going to be to linearize the problem.

For this lecture, we will assume ${f}$ is projective and smooth, to simplify the problem. In fact, we will assume ${X}$ and ${Y}$ are nonsingular.

So our map factors ${X\rightarrow \mathbb{P}^N\times Y\rightarrow Y}$ and ${df}$ is surjective. Equivalently, there exists a line bundle ${L}$ on ${X}$ which restricts on every fiber to an ample line bundle.

If ${X\subseteq\mathbb{P}^n}$ is a projective, nonsingular variety, tehn there’s a universal hyperplane section ${\{(x,H)|x\in H\}\subset X\times (\mathbb{P}^n)^*}$ and call it ${\mathcal{X}}$, we have a map ${\mathcal{X}\rightarrow (\mathbb{P}^n)^*}$. This is not smooth, and we call this map ${h}$. Define ${X^\vee}$ to be the set of hyperplanes tangent to ${X}$. Then ${h^{-1}((\mathbb{P}^n)^*\setminus X^\vee)\rightarrow (\mathbb{P}^n)^*\setminus X^\vee}$, we have a smooth projective morphism.

Lemma 1 (Ehresmann Lemma) Let ${M,N}$ be ${C^\infty}$ manifolds, ${N}$ connected, ${f:M\rightarrow N}$ a proper submersion. Then all of the fibers of ${f}$ are diffeomorphisc and ${f}$ is a ${C^\infty}$ locally trivial fibration.

In fact, if ${N}$ is contractible, then ${M}$ is just ${N\times F}$.

If ${\gamma}$ is a look in ${N}$ starting at ${n_0\in N}$, then you get in general a diffeomorphism of ${f^{-1}(n_0)}$ to itself that is not isotopic to the identity.

And example of this is to take a family of elliptic curves over ${\mathbb{C}^*}$, then a nontrivial loop in ${\mathbb{C}^*}$ gives a nontrivial (nonisotopically trivial, even) diffeomorphism ${E\rightarrow E}$.

As the diffeomorphisms depend only on the homotopy classes of looks, we get a natural representation ${\pi_1(N,n_0)\rightarrow \mathrm{Aut}(H^k(f^{-1}(n_0),\mathbb{Z}))}$.

Let ${f:M\rightarrow N}$. We define ${R^if_*\mathbb{Q}}$ to be the sheaf on ${N}$ associated to the presheaf which sends ${U}$ to ${H^i(f^{-1}(U),\mathbb{Q})}$. These are locally constant sheaves, but not necessarily constant, due to monodromy, that is, the representation of ${\pi_1}$ mentioned above.

So what does all this have to do with Hodge theory?

Consider ${S^1\rightarrow S^3\rightarrow S^2}$, the Hopf fibration. Here ${R^0f_*\mathbb{Q}=R^1f_*\mathbb{Q}=\mathbb{Q}}$.

Also look at ${S^2\times S^1\rightarrow S^2}$. Then ${R^0f_*\mathbb{Q}=R^1f_*\mathbb{Q}=\mathbb{Q}}$. The Künneth formula tells us that ${H^*(S^2\times S^1)=H^*(S^2)\otimes H^*(S^1)}$. But for the Hopf fibration, this fails: ${H^*(S^3)}$ is different.

In general, for a fibration, knowledge the the ${R^if_*\mathbb{Q}}$ does not suffice to compute the cohomology of the total space.

Now, if ${f:M\rightarrow N}$ is a fibration, there is a Leray Spectral Sequence with ${E_2^{pq}=H^p(N,R^qf_*\mathbb{Q})\Rightarrow H^{p+q}(M)}$.

If ${f:X\rightarrow Y}$ is projective and smooth with ${Y}$ connected, then the Leray spectral sequence degenerates at ${E_2}$.

If ${Y}$ is simply connected, then ${H^*(X)=H^*(Y)\otimes H^*(F)}$, as groups.