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}!

2.4. Associated Graded

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?

About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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