ICTP Day 12 – End of the Summer School

The summer school is over. Tomorrow begins the conference, including talks from such luminaries as Arapura, Illusie, Griffiths…and that’s just the first three talks tomorrow. I’ll be talking on Thursday at 17:05, and until then, it’s polishing time for my talk, and working on a new lead on the research it’s expositing. Meanwhile, today’s notes are somewhat sparse. I’d lost track of the Shimura Varieties thread days ago, and today I lost the Beilinson-Bloch thread, sadly. Hopefully I’ll work it out later, but my lack of arithmetic background has caught up with me, I couldn’t keep everything straight…well. Here’s the notes from today, tomorrow’s should be more modular:

1. Kerr – Deligne’s Theorem on Abelian Varieties, Part II

Let {A} is a CM abelian variety, that is, an abelian variety such that {MT(H^1(A))} is abelian.

Now, if {t\in H^{2p}(A^{an},\mathbb{Q})\cap F^pH^{2p}_{dR}(A)} and {\sigma\in \mathrm{Aut}(\mathbb{C}/\mathbb{Q})} then we want to show that {t^\sigma\in F^pH^{2p}_{dR}(A^\sigma)} lives in {H^{2p}(A^{an,\sigma},\mathbb{Q})}.

Let {E/\mathbb{Q}} be a CM field of degree {2e} such that {E} is totally imaginary and there exists {p\in Gal(E/\mathbb{Q})} with {p^2=\mathrm{id}}, {\phi\circ p=\bar{\phi}} for all {\phi\in \hom(E,\mathbb{C})}.

Now, take {F} to be the totally real fixed field, and {\xi} such that {E=F(\xi)}, and {\xi^2\in F} and {\sqrt{-1}\phi_i(\xi)>0} for {i=1,\ldots,e} with {\hom(E,\mathbb{C})} generated by {\Phi=\{\phi_1,\ldots,\phi_e,\bar{\phi}_1,\ldots,\bar{\phi}_e\}}. We call {(E,\Phi)} the CM type of {E}.

Now, consider {A/\mathbb{C}} an abelian variety with {E\rightarrow \mathrm{End}(A)\otimes\mathbb{Q}=\mathscr{E}}. Then {V=H^1(A,\mathbb{Q})} is an {E}-vector space of even dimension {d} and {\dim A=ed=D}.

Now, {V} is self-dual, and so {E} acts on {V^\vee} and we have natrual quotient map {\bigwedge^d V^\vee\rightarrow \bigwedge_E^d V^\vee}, and the dual is an inclusion defined over {E}.

{E} is a {\mathbb{Q}}-vector space of dimension {2e} and it acts on {E\otimes_\mathbb{Q}\mathbb{C}=\oplus_{\phi\in \hom(E,\mathbb{C})} \mathbb{C}_\phi} and similarly for {V}, and we have

\displaystyle \begin{array}{ccccc}(\bigwedge^d_E V)_\mathbb{C}\cong \oplus \bigwedge_\mathbb{C}^d V_{\phi_i} & & \to & & \oplus_{\sum d_i=d} (\otimes_i \bigwedge_\mathbb{C}^{d_i} V_{\phi_i})\cong (\bigwedge^d V)_\mathbb{C} \\ \uparrow &&&& \uparrow\\ \bigwedge^d_E V & & \to & & \bigwedge^d V\end{array}

The HS on {V} may be viewed as {\phi:\mathbb{U}\rightarrow GL(V)} taking {zz} to the {\mathbb{C}}-linear endomorphism of multiplciation by {z^{1-0}} on {V^{1,0}} and {z^{0-1}} on {V^{0,1}} and this must commute with {v(E)}.

Therefore, {V_{\phi_i}=(V_{\phi_i}\cap V^{1,0})\oplus (V_{\phi_i}\cap V^{0,1})=V^{1,0}_{\phi_i}\oplus V^{0,1}_{\phi_i}}, and are of dimension {a_i} and {b_i} with {a_i+b_i=d_i}.

So the Hodge type of {\bigwedge^d_\mathbb{C} V_{\phi_i}\cong \bigwedge_\mathbb{C}^{a_i}V^{1,0}_{\phi_i}\otimes \bigwedge^{b_i}_\mathbb{C} V^{0,1}_{\phi_i}} is {(a_i,b_i)}.

Conclusion: If {\dim(V^{1,0}_{\phi_i})=d/2} for each {i=1,\ldots,2e}, then {\bigwedge^d_E V\subset \bigwedge^d_\mathbb{Q} V} consists of Hodge classes (the Weil classes).

If {A_0} is an abelian variety of dimension {d/2} and {A=A_0\otimes_\mathbb{Q} E=A_0\times\ldots\times A_0} {2e} times, this is then {\mathbb{C}^{d/2}\otimes \mathbb{C}^{2e}/\Lambda\otimes\mathscr{O}_E}. Let {V=H^1(A,\mathbb{Q})}, this is just {H^1(A_0,\mathbb{Q})\otimes_\mathbb{Q} E}, and so taking {E} to act on the factor of {E}, we get {V_{\phi_i}\cong V_0\otimes \mathbb{C}_{\phi_i}\cong V_{i,\mathbb{C}}}.

This gives us that {\bigwedge^d V_{\phi_i}\cong \bigwedge^d V_{i,\mathbb{C}}=H^d(A_0,\mathbb{C})\cong H^{d/2,d/2}(A_0)}.

Moreover, {\mathrm{Aut}(\mathbb{C})} changes neither the product structure on {A}, the endomorphisms (which are defined by cycles in {A\times A}) nor the class of {[p]} on {A_0}. Thus, {\bigwedge^d_E V} in this cases consists of absolute Hodge clases.

Now, think of {V} as a fixed {\mathbb{Q}}-vector space of dimension {D} with nondegenerate alternating form {Q:V\times V\rightarrow \mathbb{Q}}.

Let {\phi} be any weight 1 Hodge structure on {V} polarized by {Q} and {E\rightarrow \mathrm{End}(V,\phi)} an isomorphism (in such a way that {Q} gives {V^{1,0}_{\phi_i}} and {V^{0,1}_{\phi_i}}. We impose the condition that {\dim V^{1,0}_{\phi_i}=d/2} for all {i}.

Then there exists a unique {E}-Hermitian form {\psi:V\times V\rightarrow E} with {Q=\mathrm{tr}_{E/\mathbb{Q}}(\mathscr{E}\cdot \psi)} and {\phi} stabilizes {\psi} and commutes with {i(E)}. Hence, {M_\phi\subset \mathrm{Aut}_E V\cap Sp(V,\mathbb{Q})=\mathrm{Res}_{F/\mathbb{Q}} U_E(V,\psi)} and {X=M_\phi(\mathbb{R})^+}, {\phi\subset h^D} is a MT domain which precisely classifies the abelian varieties (or HS’s) satisfying the above conditions which are precisely that the HS for which {\bigwedge^d_EV\subset\bigwedge^d V} consists of Hodge classes.

Now, {\mathcal{A}\rightarrow \Gamma\backslash X} a torsion free congruence subgroup is by the Baily-Borel theorem a quasi-projective algebraic variety parameterizing such {A}.

Applying Principle B again leads to

Weil classes on “Veil algebraic varieties” are absolute Hodge

The rest of Deligne’s proof: Let {M} be cut out by {Hg'_A} and {\check{M}} be cut out by {{AH_g}'_A} (the Hodge and absolute Hodge tensors) then

If a tensor {t\in T^{k,\ell} H^1(A,\mathbb{Q})} is fixed by {\check{M}}, then it is absolute Hodge.

For CM abelian varieties, Deligne shows that {\check{M}\supseteq M} is an equality by producing enough absolute Hodge classes to push {\check{M}} inside {M}. He does this by looking at endomorphisms of the CM field, {A_{\sigma\Phi}\rightarrow A_\Pi} and Weil Hodge classes.

This is dense on {\prod_{\Phi_i} A_{\Phi_i}}.

2. Green 4

Today we’re going to look at cycles over {k} on a variety {X} defined over {\mathbb{Q}}.

Let {X} be defined over {\mathbb{Q}}. Then {CH^p(X(\mathbb{Q}))_\mathbb{Q}} is captured by cycles classes and {\mathcal{AJ}^p_X\otimes\mathbb{Q}}.

That is, {F^2=0}.

The Conjecture in fact says that {F^m=0} for {m\geq} the transcendence degree of {k}, plus two, for {X} defind over {k}.

Now, if {X} is defined over {\mathbb{Q}} and {k} a finitely generated extension of {\mathbb{Q}}, we can find {S} with {k=\mathbb{Q}(S)}, and set {\mathcal{X}=X\times S}, and for any cycle {Z\in Z^p(X(k))} we can spread it to {\mathcal{Z}\in Z^p(X\times S(\mathbb{Q}))}, and there will exist a {W\subset S} a proper subvariety, defined over {\mathbb{Q}} of lower dimension with {\mathcal{W}\in Z^{p-1}(X\times W)} such that {\mathcal{Z}\rightarrow \mathcal{Z}+\mathcal{W}}.

If the conjecture on varieties over {\mathbb{Q}} is ok, then {Z\cong 0} in rational equivalence over {\mathbb{Q}} for some {\mathcal{W}}, so {[\mathcal{Z}+\mathcal{W}]} is torsion, and so its Abel-Jacobi image is also zero after tensoring with {\mathbb{Q}}.

(I lost track of the lecture here)

3. Kerr 5

No notes taken

4. Green 5

No notes taken


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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