## ICTP Day 5

Today was the last day of “basic Hodge theory”…so we talked about intersection cohomology and the decomposition theorem and the like. I admit to losing the thread of the lectures a couple of times (marked in the notes) and not including all of the examples in the second deCataldo lecture. I’m spending the weekend in Venice and working on the talk I’m giving on the 1st, day 6 commences on Monday!

1. ElZein 4

Let ${X}$ be a smooth complex algebraic variety. Then the cohomology groups carry a functorial MHS.

You cannot define ${F}$ on ${\mathbb{Q}}$, but you REALLY need to define ${W}$ on ${\mathbb{Q}}$, before you get to ${\mathbb{C}}$.

Consider ${Y}$ a normal crossing divisor in a smooth proper algebraic variety ${X}$ such that ${X^*\cong X\setminus Y}$. For ${y\in Y}$, there exists ${U\ni y}$ in ${X}$ with ${z_1,\ldots,z_n}$ such that ${z_1\ldots z_r=0}$ is a local equation for ${Y}$.

We can write ${Y=\cup_i Y_i}$ with the indices coming from an ordered set and the ${Y_i}$ are irreducible. Let ${\sigma}$ by ${\sigma_1<\ldots<\sigma_a}$ in ${I}$, then ${Y_\sigma=Y_{\sigma_1}\cup\ldots\cup Y_{\sigma_a}}$ be a smooth proper algebraic variety.

Let ${(\mathscr{E}^*_{X^*},F)}$ be the filtered complex of differential forms on ${X*=X\setminus Y}$ then take the global sections and compute the cohomology. This ${F}$ will be cpute the correct Hodge filtration!

The deRham complex with logarithmix singularities along ${Y}$, ${\Omega_X^*\langle Y\rangle=\Omega_X^*(\log Y)\subset \gamma_*\Omega^*_{X^*}}$ where ${\gamma:X^*\rightarrow X}$ is the inclusion.

Locally, ${\Omega^k_X(\log Y)|_U}$ is the free ${\mathscr{O}_U}$-module generated by ${\frac{dz_{i_1}}{z_{i_1}}\wedge\ldots\wedge\frac{dz_{i_\ell}}{z_{i_\ell}}\wedge dz_{j_1}\wedge\ldots\wedge dz_{j_k}}$ where ${i_r\leq r}$, ${j_s>r}$ and ${\ell+m=k}$.

The weight filtration ${W}$ with ${\sigma}$ of length ${a}$ form a set ${S^a}$ with ${Y^a=\coprod_{\sigma\in S^a} Y_a}$ and ${W_m\Omega^p_X(\log Y)=\sum_{\sigma\in S^m}\Omega^{p-m}_X\wedge \frac{dz_{\sigma_1}}{\sigma_1}\wedge\ldots\wedge \frac{dz_{\sigma_m}}{z_{\sigma_1}}}$ contains a maximum of ${m}$ of the ${\frac{dz_i}{z_i}}$.

${W_m}$ is an increasing filtration by subcomplexes, by construction.

Now, look at the residue map ${\mathrm{Res}:\mathrm{Gr}^W_m\Omega_X^i(\log Y)\rightarrow \prod_\alpha \Omega^{i-m}_{Y^m}}$ for each ${\sigma}$. ${\alpha\wedge\frac{z_{\sigma_1}}{\sigma_1}\wedge\ldots\wedge \frac{dz_{\sigma_m}}{z_{\sigma_m}}\rightarrow \alpha|_{Y^\sigma}}$.

Lemma 1 ${\mathrm{Res}}$ is an isomorphism

For any ${\sigma}$, there’s an inverse ${\rho_\sigma}$ taking ${\alpha}$ to wedgeing it with the ${\frac{dz_i}{z_i}}$ in ${\sigma}$.

Now, we get a map ${\mathrm{Res}:(\mathrm{Gr}^W_m\Omega_X^*(\log Y),F)\rightarrow (\pi_*\Omega^*_{Y_m}[-m],F[-m])}$ is a filtered isomorphism.

The associated gradeds are also isomorphic, then.

1. ${H^p(\mathrm{Gr}^W_m\Omega_X^*(\log Y))\cong\begin{cases}\pi_* \mathbb{C}_{Y^m}&p=m\\ 0&\mbox{else}\end{cases}}$
2. ${H^p(W_m\Omega_X^*(\log Y))\cong\begin{cases}\pi_* \mathbb{C}_{Y^p}&p\leq m\\ 0&\mbox{else}\end{cases}}$
3. ${H^p(\Omega_X^*(\log Y))\cong \pi_*\mathbb{C}_{Y^p}}$ for all ${p}$

${0\rightarrow W_{m-1}\Omega_X^*(\log Y)\rightarrow W_m(\Omega_X^*(\log Y)\rightarrow \mathrm{Gr}^W_m\Omega_X^*(\log Y)\rightarrow 0}$ is exact.

Now, we claim that for ${y\in Y}$ we have ${H^p(W_m\Omega^*_X(\log Y))_y=H^p(U(y)\cap X^*,\mathbb{C})}$ for ${p\leq m}$.

((Shamefully, I lost the thread here))

There is a mixed Hodge complex with ${K_\mathbb{Z}=\mathbb{R}\Gamma(X,\mathbb{R} j_*\mathbb{Z}_{X\setminus Y})}$ where ${j:X\setminus Y\rightarrow X}$ is the inlcusion, with ${(K_\mathbb{Q},W)=(\mathbb{R}\Gamma(X,\mathbb{R} j_*\mathbb{Q}_{X\setminus Y}),T=W)}$ where ${T}$ is the truncation functor, and with ${(K_\mathbb{C},W,F)=(\mathbb{R}\Gamma(X,\Omega_X^*(\log Y),W,F)}$.

And then ${(\mathrm{Gr}^W_m(K^\mathbb{Q}))}$ and ${(\mathrm{Gr}^W_m(K^\mathbb{C}),F)}$ form a Hodge complex of weight ${m}$, and ${(\pi_*\Omega^*_{Y^m}[-m],F[-m])}$ is a Hodge complex of weight ${m}$.

2. Cattani 4

Last time, we defined an abstract VHS, starting from a local system ${\mathcal{L}_\mathbb{Z}}$, and the Gauss-Manin connection, which is the flat connection associated with a local system underling a PVHS.

2.1. Period Map

Let ${\mathbb{V}\rightarrow (\Delta^*)^r\times \Delta^{n-r}}$ be a PVHS.

Linear algebra:

1. ${V_\mathbb{Z}}$ a lattice, ${V_\mathbb{Q}=V_\mathbb{Z}\otimes_\mathbb{Z} \mathbb{Q}}$
2. An integer ${k}$
3. A collection of hodge numbers ${h^{p,q}}$ with ${p+q=k}$ and ${\sum h^{p,q}=\dim V_\mathbb{C}}$, ${h^{p,q}=h^{q,p}}$ and ${f^p=\sum_{a\geq p} g^{a,k-a}}$.
4. ${Q}$ a non-degenerate bilinear form over ${\mathbb{Z}}$ of parity ${(-1)^k}$.

Now, let ${D}$ (of all this data) be the space of all Hodge structures with ${V_\mathbb{C}=\oplus_{p+q=k} V^{p,q}}$ polarized by ${Q}$ and with ${\dim V^{p,q}=h^{p,q}}$.

Now, define ${\check{D}}$ to be the space of all filtrations ${F}$ on ${V_\mathbb{C}}$ with ${\dim F^p=f^p}$ and ${Q(F^p,F^{k-p+q})=0}$.

Look at weight 1. Then ${\check{D}=\{F\subset G(n,\mathbb{C}^{2n})|Q(F,F)=0\}}$.

Now, let ${G_\mathbb{C}=\{M\in\mathrm{GL}(V_\mathbb{C})|Q(Mu,Mv)=Q(u,v)\}}$. Then ${G_\mathbb{C}}$ acts naturally on ${\check{D}}$. In fact, it acts transitively.

Now look on ${V_\mathbb{C}=V^{2,0}\oplus V^{1,1}\oplus V^{0,2}}$, then ${-Q(u,\bar{u})>0}$ for ${u\in V^{2,0}}$ and ${Q(v,\bar{v})>0}$ for ${v\in V^{1,1}}$, so we can write ${Q}$ in the form ${\left(\begin{array}{ccc}&&-I\\ &I&\\-I&&\end{array}\right)}$.

That actually has a real structure, and if we look at ${G=G_\mathbb{C}\cap \mathrm{GL}(V_\mathbb{R})}$, then this group acts on ${D}$.

Now, by basic Lie theory, we have ${\check{D}=G_\mathbb{C}/B}$ and ${D=G/B\cap G}$.

So what are these groups? For weight 2, ${Q}$ is positive definite on ${V^{1,1}}$ and negative definite on ${V^{2,0}}$, so we have ${G=O(2h^{2,0},h^{1,1})}$. The subgroup fixing a given Hodge structure is ${U(h^{2,0})\times O(h^{1,1})}$, and so ${D=SO(2h^{2,0},h^{1,1})/U(h^{2,0})\times SO(h^{1,1})}$, which sits inside:

$\displaystyle SO(2h^{2,0})/U(h^{2,0})\rightarrow SO(2h^{2,0},h^{1,1})/U(h^{2,0})\times SO(h^{1,1})$ $\displaystyle\rightarrow SO(2h^{2,0},h^{1,1})/SO(h^{2,0})\times SO(h^{1,1})$

and it turns out that every Hermitian symmetric space arises in this manner, though we don’t always get something that nice.

So we have ${G/V=D\subset \check{D}=G_\mathbb{C}/B}$, and have ${B\rightarrow G_\mathbb{C}\rightarrow \check{D}}$, with ${B}$ acting on ${\mathfrak{g}/\mathfrak{b}}$ and ${T^h(\check{D})=\check{D}\times_B \mathfrak{g}/\mathfrak{b}}$.

If ${V_\mathbb{C}=\oplus V^{p,q}}$, then we can define ${\mathfrak{g}^{a,-a}=\{X\in\mathfrak{g}|X(V^{p,q})\subset V^{p+a,q-a}\}}$ a weight 0 Hodge strcuture.

Let ${b=\oplus_{a\geq 0} \mathfrak{g}^{a,-a}}$. Then ${B}$ preserves ${b\oplus \mathfrak{g}^{-1,1}}$ and ${T^{-1,1}(\check{D})=\check{D}\times_B (b\oplus \mathfrak{g}^{-1,1})}$.

If we have a PVHS, then parallel translation to a fixed fiber defines a holomorphic map ${B\rightarrow \Gamma \backslash D}$ where ${\Gamma}$ is the monodromy. We call this the period map.

the HS differential takes values on a horizontal subbundle.

We’re going to discuss it from the largely analytic point of view.

So now, lets look at the local situation, a PVHS over ${(\Delta^*)^r}$. We have ${\pi_1((\Delta^*)^r)=\mathbb{Z}^r}$ with generators ${c_1,\ldots,c_r}$, which give us elements ${\gamma_1,\ldots,\gamma_r\in \mathrm{GL}(V_\mathbb{Z})}$. We know that these are quasi-unipotent, and we’ll assume that they’re actually unipotent.

If we set ${N_i}$ to be the log of the monodromy, we’ll have ${\gamma_j=e^{N_j}}$ by definition, but then we’ll have ${[N_i,N_j]=0}$ and ${N_i^{k+1}=0}$. So we can define ${\psi(z_1,\ldots,z_r)=\exp(-\sum z_i N_i)\tilde{\phi}(z_1,\ldots,z_r)}$ where we have ${\phi:(\Delta^*)^r\rightarrow \Gamma \backslash D}$ and ${\tilde{\phi}}$ lifting it onto the PVHS to ${D}$.

The map, ${\psi}$ takes the PVHS to ${\check{D}}$ and we call the induced map on ${(\Delta^*)^r}$ also ${\psi}$.

The map ${\psi:(\Delta^*)^r\rightarrow \check{D}}$ extends holomorphically to the origin.

Note, this is not necessarily a Hodge structure! We just have ${\check{D}\ni\psi(0)}$, and call it the limit Hodge filtration.

Now, we have that ${\exp(\sum z_j N_j)\psi(0)\in D}$ if the imaginary parts of ${z_j}$ are large enough. We call these period maps Nilpotent orbits.

This afternoon, we’ll see how the limiting Hodge filtration along with the weight filtration from the monodromy determines a polarized MHS.

3. de Cataldo 1 – Hodge Theory of Maps

The goal of these two lectures is to state the decomposition theorem and to give examples.

Let ${f:X\rightarrow Y}$ be a proper morphism of complex algebraic varieties. The direct image complex of the intersection cohomology complex of ${X}$ splits into a direct sum of intersection cohomology complexes on ${Y}$.

Now, let ${f:X\rightarrow Y}$ a morphism and ${F}$ a complex of sheaves on ${X}$. Then let ${F\rightarrow I}$, where ${I}$ is a complex of injective sheaves on ${X}$ such that ${\mathscr{H}^iF=\mathscr{H}^i I}$. Now, define ${Rf_*F=f_*I}$.

So, if ${c:X\rightarrow pt}$, we can define ${H^i(Rc_*F)=H^i(X,F)}$.

So then we have ${R^if_*F=\mathscr{H}^i Rf_*F}$.

${(R^if_*F)_y=\varinjlim_{U\ni y} H^i(f^{-1}(U),F)}$

Now, ${Rf_*F}$ contains information about the topology of the map, and we can extract it using the Leray Spectral Sequence.

${H_{sing}(X,\mathbb{Z})=\check{H}(X,\mathbb{Z})=H(X,\mathbb{Z}_X)}$

${\Delta^*\rightarrow \Delta}$. Then we have ${\mathbb{Z}_\Delta\rightarrow Rf_*\mathbb{Z}_{\Delta^*}\rightarrow H^1(\Delta^*)_\sigma [-1]\rightarrow 0}$ doesn’t split.

${\Delta\rightarrow\Delta}$ by ${z\mapsto z^2}$. Then ${R^if_*\mathbb{Z}_X=0}$ for ${i>0}$, and we have ${0\rightarrow \mathbb{Z}_Y\rightarrow f_*\mathbb{Z}_X\rightarrow P\rightarrow 0}$ doesn’t split.

However, with ${\mathbb{Q}_X}$, it does split, because the obstruction is divison by 2. So then we have that ${f_*\mathbb{Q}_X}$ is a direct sum of a local system and something which isn’t a local system.

Take a rule nodal cubic, ${f}$ the normalization and ${j}$ the inclusion of the smooth locus. Then we have ${0\rightarrow \mathbb{Z}_Y\rightarrow f_*\mathbb{Z}_X\rightarrow P_\sigma\rightarrow 0}$ and ${P}$ is a skyscraper sheaf at the singular point. We can show that this doesn’t split.

However, we have ${0\rightarrow \mathbb{Z}_Y\rightarrow f_*\mathbb{Q}_X\rightarrow\mathbb{Q}^3\rightarrow 0}$ where ${\mathbb{Q}^3}$ is a three dimensional vector space at the origin.

The decomposition theorem is far too much to hope for in anything other than complex algebraic geometry. For instance, look at the Hopf surface, which is fibered over ${\mathbb{C}\mathbb{P}^1}$ by algebraic curves, but which is not algebraic. Then we don’t have ${E_2}$ degeneration, even though the map is a submersion and ${R^if_*\mathbb{Q}}$ is constant on ${\mathbb{C}\mathbb{P}^1}$, with ${R^1f_*\mathbb{Q}\cong \mathbb{Q}^2}$.

Now, let ${C\subset\mathbb{C}\mathbb{P}^2}$, and look at the cone over the curve in ${\mathbb{C}^3}$. Blow it up at the origin, and we get a smooth surface which is the total space of ${L=\mathscr{O}_{\mathbb{P}^2}(-1)|_C}$.

But instead of looking at the blowup, look at the cone minux the origin (note, over ${\mathbb{C}}$, this doesn’t disconnect the space). Call this map ${j:U\rightarrow Y}$. Then ${Rj_*\mathbb{Q}_U}$ has cohomology equal to ${H(U)}$, and to compute this we use the spectral sequence, and at that ${E_2}$ page is ${\begin{array}{ccc}\mathbb{Q}&\mathbb{Q}^{2g}&\mathbb{Q}\\ \mathbb{Q} & \mathbb{Q}^{2g} &\mathbb{Q}\end{array}}$ and all others zero, so the only differential is from the upper left to the bottom right, and is an isomorphism,and all other differentials vanish, so we get that the cohomology sheaves are ${\mathbb{Q}_Y}$, ${\mathbb{Q}^{2g}_0}$, ${\mathbb{Q}^{2g}_0}$ and ${\mathbb{Q}_0}$, where the subscript ${0}$ means that we have skyscraper sheaves supported at the origin.

Now, ${Rf_*\mathbb{Q}_X=\tau_{\leq 1} Rj_*\mathbb{Q}_U\oplus H^2(C)_0[-2]}$. Why do we throw things away? We’ll see later. This is an example of the decomposition theorem. The first summand is the intersection complex of the cone, ${I_Y}$ and the latter is ${I_\sigma [-2]}$.

Look at ${C\times\mathbb{C}}$ and then contract the curve. This isn’t a holomorphic map! It’s a real algebraic map, though. The decomposition theorem fails. We can tell by looking at the second term, because in the top space, the curve’s cohomology class is trivial, because the bundle is trivial.

4. ElZein 5

We’re going to be working with the derived category, now. For ${D^+(X)}$, we take a complex and an injective resolution, then apply derived functors. We set ${D^+F(X)}$ to be the derived category of filtered complexes, so we require ${F^pK\rightarrow F^pI}$ qis and ${\mathrm{Gr}^p_F K\rightarrow \mathrm{Gr}^p_F I}$ an iso. Finally, we define ${D^+F^q(X)}$ the derived category of bifiltered complexes.

4.1. The Weight Filtration

Now ${(\Omega_X^*(\log Y),W,F)}$ is a MHS. Let ${\Gamma:\mathcal{A}\rightarrow \mathcal{B}}$ be a functor of abelian categories, we require that the functor be left exact. Let ${(K,F)}$ be a filtered complex, then we have ${_F E_1^{pq}=R^{p+q}\Gamma(\mathrm{Gr}^p_F K)=H^{p+q}(\mathbb{R}\Gamma(\mathrm{Gr}^p_F K))}$.

Now, we look at ${_W E_1^{pq}=H^{p+q}(\mathbb{R}\Gamma \mathrm{Gr}^W_{-p}K)}$ , but instead, we can use hypercohomology, and get ${_W E_1^{pq}=\mathbb{H}^{p+q}(X,\mathrm{Gr}^W_{-p}\Omega_X^*(\log Y)=H^{p+q}(X,\pi_* \mathbb{C}_{Y^{-p}}[p])}$, which is finally ${\oplus_\sigma H^{2p+q}(Y^\sigma,\mathbb{C})}$ where ${Y^\sigma=Y_{\sigma_1}\cap\ldots\cap Y_{\sigma_k}}$ with the induced Hodge filtration.

So, this has a Hodge structure of weight ${q}$.

${_W E_1^{pq}=\oplus_{|G|=-p} H^{2p+q}(Y^\sigma)(P)}$ has weight ${q}$, and we have ${d_1}$ which maps ${H^{2p+q}(Y^\sigma)\rightarrow \oplus H^{2p+q+2}(Y^{\sigma'})(p+1)}$, and if ${Y^{\sigma'}\supset Y^\sigma}$ then there is ${s\in [1,-p]}$ such that the Gysin map ${i_s}$ (the Poincaré dual of the restriction map) allows us to compute that ${E_2^{pq}=\mathrm{Gr}^W_{-p}H^{p+q}(X-Y,\mathbb{C})}$, which is a HS of weight ${q}$.

And this gives us ${W[n]}$ on ${H^n}$.

4.2. Simplicial Resolutions

Let ${X}$ be a variety (in particular, a topological space).

A simplicial variety ${X_*}$ over ${X}$ is a family of varieties ${\pi_n:X_n\rightarrow X}$ for all ${n\in\mathbb{N}}$ such that for each increasing map ${f:[0,n]\rightarrow [0,m]}$ we define a morphis ${X(f):X_m\rightarrow X_n}$ over ${X}$ satisfying the natural composition laws.

A sheaf ${F}$ on ${X_*}$ is a family of sheaves ${F_n}$ on ${X_n}$ satisfying that ${F_*(f):F_n\rightarrow X_*(f)_*F_m}$ for all increasing functions ${[0,m]\rightarrow [0,n]}$ satisfying the natural conditions.

So now, we take a complex ${\pi_*F_0\rightarrow \ldots\rightarrow \pi_*F_n\stackrel{d_n}{\rightarrow}\pi_*F_{n+1}\rightarrow\ldots}$ with ${d_n=\sum_{i=0}^{n+1}(-1)^i\delta_i}$ where ${\delta_i:[0,n]\rightarrow [0,n+1]}$ is increasing function skipping ${i}$.

So now, let ${K^*}$ be a complex of sheaves on ${X_*}$. This is a family of complex ${K^{*,n}}$ for each ${n}$. A resolution is ${K^{*,n}\rightarrow I^{*,n}}$ which must be compatible.

${\mathbb{R}\pi_*K=s(\pi_*I^{*,*})}$ with differential ${d}$. ${(\mathbb{R}\pi_*K^*)^n=\oplus_{p+q=n} \pi_*^qI^{pq}}$ and ${d(x^{pq})=d_I(x^{pq})+(-1)^p\sum_{i=0}^{n+1}\delta_i(x^{pq})}$.

Let ${\pi:X_*\rightarrow X}$ be a simplicial variety over ${X}$. Then ${\pi}$ is of cohomological descent if for all ${F}$ on ${X}$ we have ${F\stackrel{\cong}{\rightarrow} \mathbb{R}\pi_*(\pi^*F)}$.

For each separated complex variety ${X}$ there exists a simplicial variety ${X_*}$ which is proper and smooth and a normal crossing divisor ${Y_*\subset X_*}$ and a map ${\pi:U_i=X_i-Y_i\rightarrow X}$ satisfying the cohomological descent property.

So by the descent property, we have ${H^*(U_*,\mathbb{Z})\cong H^*(X,\mathbb{Z})}$ and we can give the former an MHS, so it gives the latter one.

5. Cattani 5

We have local monodromy and we have the logarithms of the monodromy, ${N_1,\ldots,N_r}$. The monodromy theorem tells us that the period map can be written as ${\exp \left(\sum \frac{\log b_j}{2\pi i}N_j\right)\psi(t_1,\ldots,t_r)}$ where ${\psi}$ extends holomorphically to ${\psi:\Delta^r\rightarrow \check{D}}$, with ${F_{lim}=\psi(0)}$.

But them we also have ${(t_1,\ldots,t_r)\mapsto \exp\left(\sum \frac{\log b_j}{2\pi i} N_j\right) F_{lim}}$. So we want to find out what kinds of holomorphic maps we can take.

We take the weight filtration, it comes from the Jordan decompositions, and includes the fact that ${N^\ell:\mathrm{Gr}^W_\ell\rightarrow \mathrm{Gr}^W_{-\ell}}$. But what about when there are more than one monodromy operator?

The weight filtration determined by ${\sum_{\lambda_j>0} \lambda_jN_j}$ is unique, and we denote it by ${W(C)}$, for the weight filtration of the cone. Moreover, ${(W(C)[-k],F_{lim})}$ is a MHS. In fact, each ${N}$ is a ${(-1,-1)}$ morphism of MHS. Additionally, the converse holds.

So a nilpotent orbit is just a MHS, with the weight filtration from the nilpotent cone, and with the polarization the obvious thing to try.

When you happen to have a MHS that splits over ${\mathbb{R}}$, then you should be able to extend everything to an ${\mathfrak{sl}_2}$-invariant picture.

There exists a canonical (Schmid) splitting of the MHS ${(W(N),F_{lim})}$. More precisely, there is a natural way to produce another Hodge filtration ${F_0}$ such that ${(W(C),F_0)}$ splits over ${\mathbb{R}}$.

Look at ${i\in\mathbb{H}_1}$. Then our space is ${SL_2(\mathbb{C})/B}$, and we have an induced Hodge filtration ${\mathfrak{g}_\mathbb{C}=\mathfrak{sl}_2(\mathbb{C})=\mathfrak{sl}_2(\mathbb{C})^{-1,1}\oplus \mathfrak{sl}_2(\mathbb{C})^{0,0}\oplus \mathfrak{sl}_2(\mathbb{C})^{1,-1}}$. If ${y,n_-,n_+}$ are the generators of the Lie algebra, then the first summand is generated by ${(iy+n_-+n_+)}$ and the second by ${(n_+-n_-)}$.

In the several variables situation, the key is to understand how to two splitting relate.

MHS’s come in two types: split and nonsplit. Nonsplit give you nilpotent orbits, split give you ${\mathfrak{sl}_2}$ modules.

Now, look at ${\mathfrak{g}_\mathbb{C}=I^{p,q}\mathfrak{g}=\{X\in\mathfrak{g}|X(I^{a,b}\subset I^{a+b,b+q}\}}$. Let ${b=\oplus I^{a,b}_{a\geq 0}\mathfrak{g}}$.

Now, ${\psi(t_1,\ldots,t_r)=\exp \Gamma(t_1,\ldots,t_r) F_{lim}}$ where ${\Gamma:\Delta^r\rightarrow \mathfrak{g}}$ has ${\Gamma(0)=0}$. Set ${t_i=\exp(2\pi i z_i)}$, then ${\phi(z_1,\ldots,z_r)=\exp(\sum z_jN_j)\exp \Gamma(t_1,\ldots,t_r)F_{lim}}$. Denote by ${E(z)=\exp X(z)}$.

Then horizontality is ${E^{-1}dE=dX_{-1}}$, and we call this Griffiths differential equation.

We have then that ${d(E^{-1}dE)=0}$, as ${dX_{-1}\wedge dX_{-1}=0}$. And so ${X_{-1}(z)=\sum z_j N_j+\Gamma_{-1}}$, where ${\Gamma_{-1}:\Delta^r\rightarrow \oplus_b I^{-1,b} \mathfrak{g}}$.

6. de Cataldo 2

Define ${I_Y}$ to be ${\tau_{d-1} Rj_*\mathbb{Q}_U}$ where ${U}$.

Let ${Y=\mathbb{C}^d}$ and ${U}$ the complement of the origin. Then we have ${0\rightarrow\mathbb{Q}_{\mathbb{C}^d}\rightarrow Rj_*\mathbb{Q}_U\rightarrow \mathbb{Q}_0[-(2d-1)]\rightarrow 0}$.

If ${Y}$ is smooth, then ${I_Y=\mathbb{Q}_Y}$. But for a nodal curve, if ${j}$ is the inclusion of the smooth locus, then ${I_Y}$ is ${j_*\mathbb{Q}}$.

On the other hand, cusps aren’t seen, topologically.

Now let ${C}$ be a curve of genus ${g}$ and ${C}$ be the cone over it, and ${U}$ the complement of the origin. Then look at ${\tau_{\leq 1} Rj_*\mathbb{Q}_U}$, it is an extension of ${\mathbb{Q}^{2g}_0[-1]}$ by ${\mathbb{Q}_Y}$, but this doesn’t split.

Recall the example ${\Delta\rightarrow\Delta}$ with ${z\mapsto z^2}$. At the origin, something happens: the preimage remains connected! So ${f_*\mathbb{Q}/\mathbb{Q}=P}$ is a local system, but it has stalk ${0}$ at the origin, but is ${-1}$ over ${\Delta^*}$.

So why do we truncate at ${d-i}$ where ${d}$ is the dimension?

This is actually due to Poincaré duality! We have ${H_c^{d-i}\cong (H^{d+i})^*}$. So Poincaré duality implies that ${b_{d-i}=b_{d+i}}$.

Let ${K}$ be a complex on ${Y}$. Then there exists another complex, the dual ${K^*}$, such that ${H^i(U,K^*)\cong H^{-i}_c(U,K)^*}$.

The complex ${\mathbb{Q}[d]}$ is self-dual for a smooth variety, but NOT for a singular one! We set ${IC_Y=I_Y[d]}$, then ${IC_Y}$ is self-dual, for all varieties.

The rank of ${\mathscr{H}^i(IC_Y^*)_0}$ is equal to the rank of ${H^{-i}_c(V_0,IC_Y)}$.

So ${\tau_{\leq d-1} Rj_*\mathbb{Q}_U}$ is motivated by Poincaré duality.

${IH^{d-i}\cong (IH_c^{d+i})^*}$

Here, we’ll find that the Lefschetz hyperplane theorem, Hard Lefschetz theorem, Hodge decomposition, primitive Lefschetz decomposition, and the Hodge-Riemann bilinear relations all hold for intersection cohomology groups.

If ${f:X\rightarrow Y}$ is a proper morphism of complex algebraic varieties, then ${Rf_*I_Y=\oplus_{b\in B} I_{\bar{Z}_b}(L_b)[\ell_b]}$ where ${B}$ is a finite set, ${Z_b\subset Y}$ are locally closed nonsingular varieties, ${L_b}$ are simple local systems and ${\ell_b\in\mathbb{Z}}$.