## ICTP Day 15 – The End

Today the conference ended, and I’ve got an early plane to catch. I’ll be away from the blog for a bit. Here’s the notes from the last day, which, I admit, are a bit sketchier than most of the others, but they’re what I have:

1. Cattani – Mixed Hard Lefschetz and Mixed Hodge-Riemann Bilinear Relations

This is work with Kaplan and Schmid

Started out with quite a bit of review.

Let ${\omega_1,\ldots,\omega_k}$ be Kähler classes, then ${L_{\omega_1}\ldots L_{\omega_k}:H^{n-k}(M,\mathbb{C})\cong H^{n+k}(M,\mathbb{C})}$.

In fact, we can also define primitive cohomology this way, and the Lefschetz decomposition, etc. We get the H-R Bilinear relations this way also, just slightly modified to having the form be ${i^{p-q}(-1)^{(n-k)(n-k+1)/2}\int_M \alpha\wedge\bar{\alpha}\wedge\omega_1\wedge\ldots\wedge\omega_k>0}$.

This is very natural, but it first appears in a 1990 paper by Gromov. He proved it for (1,1). It also appeared in 2006 by Dinh-Nguyen.

In between, there was work on cohomology algebras for polytopes, by Timorin.

Now, ${\ker(L_{\omega_1})\cap \mathrm{Im}(L_{\omega_2})\cap H^n(M)=\{0\}}$, and now, being purely primitive doesn’t guarantee the sign. Look at ${H^{2,2}=H_0^{2,2}\oplus L_\omega H_0^{1,1}\oplus\oplus L_\omega^2 H_0^{0,0}}$, has signs ${+,-,+}$.

We care about things like this to prove things about intersection cohomology.

${MV(P_1,\ldots,P_d)^2\geq MV(P_1,\ldots, P_d) MV(P_2,P_2,\ldots,P_d)}$

Now, look at some integral polytopes. They give ${f_i=\sum_{\alpha\in P_i\cap \mathbb{Z}^d} u_\alpha t^\alpha}$ and we set ${Z}$ to be the common zero locus ${f_3=\ldots=f_d=0}$ as a subset of ${(\mathbb{C}^*)^d}$

Then BRK proved that ${MV(P_1,\ldots,P_d)}$ is the number of points in ${f_1=\ldots=f_d=0}$.

A polarized Hodge-Lefschetz module of weight ${k}$ is the following data:

1. a ${\mathbb{Z}}$-graded vector space over ${\mathbb{Q}}$, ${V_*=\oplus V_\ell}$
2. ${Q}$ a nondegenerate bilinear form of parity ${(-1)^k}$
3. ${\mathfrak{g}=Lie(\mathrm{Aut}(V_\mathbb{R}),\mathbb{Q})}$ and ${\mathfrak{a}\subset\mathfrak{g}}$ an abelian subspace
4. ${N_0\in\mathfrak{a}}$

satisfying:

1. There exists a bigrading ${V_\mathbb{C}=\oplus_{0\leq p,q\leq k} V^{p,q}}$ with ${\bar{V}^{p,q}=V^{q,p}}$ and ${(V_\ell)_\mathbb{C}=\oplus_{p+q=\ell} V^{p,q}}$
2. ${T(V^{p,q})\subset V^{p-1,q-1}}$ for all ${T\in\mathfrak{g}}$
3. For ${\ell\geq 0}$, ${N_0^\ell:V_\ell\rightarrow V_{-\ell}}$ is an isomorphism
4. For ${\ell\geq 0}$, the induced HS on ${P_\ell(N_0)=\ker(N_0^{\ell})}$ is polarized by ${Q(\cdot,N_0^\ell\cdot)}$

Let ${\mathcal{C}}$ be the polarizing cone ${\{T\in \mathfrak{a}|\mbox{3 and 4 hold}\}}$.

Let ${V=H^*(X,\mathbb{C})}$ for a compact Kähler manifold, and ${V_\ell=H^{k-\ell}(X,\mathbb{C})}$.

Let ${T\in\mathcal{C}}$, and set ${\tilde{V}=TV}$ and ${\tilde{V}_\ell=TV_{\ell+1}}$, ${\tilde{Q}(Tu,Tv)=Q(u,Tv)}$, and let the Lie algebras act on this.

This data is a PHLM of weight ${k-1}$ and bigrading ${\tilde{V}^{p,q}=V^{p+1,q+1}}$.

This is a PHL module, and every PHLM arises in this way.

2. Cautis – sl(2) actions and Hodge modules (on Grassmannians)

2.1. Basic sl(2) review

Let ${V}$ be a ${\sl_2}$ rep (finte dimensional). It breaks up into ${h=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)}$ eigenspaces (the weight spaces) ${V(-N)}$, ${V(-N+2)}$, through ${V(\lambda), V(\lambda+2), V(N)}$. with ${e}$ and ${f}$ the raising and lowering elements. If we restrict ${h}$ to ${V(\lambda)}$ it is just ${\lambda\mathrm{id}}$ for some ${\lambda\in\mathbb{Z}}$.

Remark: if ${s=\left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right)\in SL_2(\mathbb{C})}$, it gives an isomorphism ${V(-\lambda)\rightarrow V(\lambda)}$.

2.2. Categorical sl(2) actions

The data consists of additive categories ${\mathcal{C}(-N),\ldots,\mathcal{C}(\lambda)\ldots \mathcal{C}(N)}$ with functors ${e,f,h}$ such that ${ef=fe+h}$ such that on ${\mathcal{C}(\lambda)}$, we have ${EF\cong FE\oplus \mathrm{id}^\lambda}$, and ${FE\cong EF\oplus\mathrm{id}^{-\lambda}}$.

Take ${\mathcal{C}(\lambda)=MHM(G(k,N))}$ as ${k}$ varies. That is, teh category of mixed Hodge modules on grassmannians, and set ${\lambda=2k-N}$. Then given ${W}$ the collection of flags ${Fl(k,k+1;N)}$, we have forgetful maps to ${G(k,N)}$ and ${G(k+1,N)}$, and we get ${E}$ and ${F}$ by pulling back and pushing forward along these maps.

We claim that this is a categorical sl(2) action.

For ${k\leq N/2}$, we compute (by convolving) that ${EF}$ is given by taking the collection of ${k-1,k}$ flags with the ${k-1}$ in common, and forgetting it, getting a subset of ${G(k,N)\times G(k,N)}$.

Then, ${EF}$ is given by the intersection cohomology, and this is an simple application of the decomposition theorem.

2.3. Equivalences

Given a “strong” categorical ${\mathfrak{sl}_2}$-action, one can form ${T:=E^{(\lambda)}\rightarrow E^{(\lambda+1)}F\rightarrow\ldots}$ natural transofmrations, and ${T}$ induces an equivalence ${\mathcal{HC}(-\lambda)\rightarrow \mathcal{HC}(\lambda)}$ of homotopy categories.

There is an iso ${MHM(G(k,N))\rightarrow MHM(G(N-k,N))}$

In this case, this equivalence is given by ${j_*\mathscr{O}_U\in MHM(G(k,N)\times G(N-k,N))}$ where ${U}$ is the open subset where we have ${\dim V\cap V'=0}$.

In fact, ${Gr^W_\ell(j_*\mathscr{O}_U)}$ gives the functor ${E^{(\lambda+\ell)}F^{(\ell)}}$, and the inverse is ${j_!\mathscr{O}_U}$!

Look at the filtered module of differential operators on ${X}$. MHM’s are filtered modules in a way compatible with this. Now, ${gr \mathscr{D}_X\cong \mathrm{Sym}^* T_X}$.

Now, if ${M\in MHM(X)}$, then ${gr(M)}$ is a ${\mathrm{Sym}^*T_X}$-module, which is also an object of ${Coh(T^*X)}$.

The idea is to use the ${gr}$ fucntor to get the ${\mathfrak{sl}_2}$ action on ${\oplus_k DCoh(T^*G(k,N))}$.

(I stopped understanding it around here)

3. Migliorini – Topology of Hitchin systems and Hodge theory of character varieties (joint work with M. de Cataldo and T. Hausel)

Let ${C}$ be a nonsingular projective curve ${g>1}$.

The Narasimhan-Seshadri correspondence (special case, rank 2 and degree 1) is ${N_D=\{(A_1,\ldots,A_g,B_1,\ldots,B_g)\in U(2)^{2g}|\prod [A_i,B_i]=-I\}/U(2)}$ and ${N_D}$ be the moduli space of stable rank 2 degree 1 holomorphic vector bundles on ${C}$. There exists a natural real analytic diffeomorphism ${N_B\cong N_D}$.

Hitchin complexified this. Let ${M_B=\{(A_1,\ldots,A_g,B_1,\ldots,B_g)\in GL_2(\mathbb{C})^{2g}|\prod [A_i,B_i]=-I\}/GL_2(\mathbb{C})}$, and ${M_D}$ the moduli space of stable rank 2, degree 1 Higgs bundles on ${C}$. There exists a natural real analytic diffeomorphism ${M_B\cong M_D}$.

Now, ${M_B}$ is affine, but ${M_D}$ is covered by mid-dim compact algebraic subvarieties.

Hitchin’s map ${h:M_D\rightarrow \mathbb{C}^{4g-3}}$ takes ${(E,\phi)}$ to ${(T_i\phi, \det \phi)}$, where ${\mathbb{C}^{4g-3}}$ is the sum of the spaces of abelian and quadratic differentials.

Also, ${M_D}$ has sympletic holomorphic, and a ${\mathbb{C}^*}$ action ${\lambda(E,\phi)=(E,\lambda\phi)}$.

The map ${h}$ is proper (in fact, projective!), is ${\mathbb{C}^*}$-equivariant, and is a Lagrangian fibration. To find the fibers of ${h}$, take ${(\alpha_1,\alpha_2)\in \mathbb{C}^{4g-3}}$, and then look at ${C_s\rightarrow C}$ the spectral curve over ${C}$ given by ${y^2-\alpha_1 y+\alpha_2=0}$.

If ${F}$ is a rank 1 torsion free sheaf on ${C_s}$, then ${\pi_*F}$ is a rank 2 locally free sheaf on ${C}$, so

${h^{-1}(\alpha_1,\alpha_2)=\overline{JC}(C_s)}$, the compactified Jacobian.

Back to ${M_B}$, which is a smooth affine variety. MHS on ${H^*(M_B)}$ studied by Hausel and Rodriguez-Villegas, has the property that it is split over ${\mathbb{Q}}$ and of Hodge-Tate type. They also discovered “very strange symmetry” which is a “Curious Hard Lefschetz”…that is, there exists a class ${\alpha\in H^2}$ of type ${(2,2)}$, such that ${\alpha^k:Gr^@_{d-2k}\rightarrow Gr^W_{d+2k}}$ is an isomorphism.

Then, there’s the nonabelian Hodge theorem, which says that ${H^*(M_B)=H^*(M_D)}$ and this gives a pure HS on ${H^*(M_D)}$. But also, ${h}$ defiens several filtartions on ${H^*(M_D)}$ (Leray filtration, and modificulations ${P_*}$) and it was introducted in BBD by setting ${P_\ell H=\mathrm{Im} (H(\tau_{\leq \ell} Rh_*\mathbb{Q})\rightarrow H(Rh_* \mathbb{Q}))}$.

Now, ${P_*}$ associated with a spectral sequence degenerating in the ${E_2}$ page, and if the map is smooth, it is the leray filtration. Also, if ${f:X\rightarrow Y}$ is projective and ${\eta}$ relatively ample, then ${\eta^\ell:Gr^P_{d-\ell}\cong Gr^P_{d+\ell}}$.

Cohomological characterization of ${P_*}$: If ${f:X\rightarrow Y}$ projective, and assume ${Y}$ is affine, then we have ${P_kH^a(X)=\ker( H^a(X)\rightarrow H^a(f^{-1}(\Lambda_{a-k-1})}$ the generic ${a-k-1}$ dimensional linear subvar of ${Y}$.

Under the NAHT, the weight filtration ${W_*}$ on ${H^*(M_B)}$ is sent to ${P_*}$ on ${H^*(M_D)}$. The plitting of ${W}$ corresponds to Deligne’s splitting of ${P_*}$.

The same result holds for ${PGL_2}$ and ${SL_2}$, but much harder

What about rank 2 and degree 0? The guess is the that same holds for intersection cohomology. What about other groups? ${M_{dR}}$? How good is this?