# 1 Moduli spaces of polarized Hodge structures.

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1 1 Moduli spaces of polarized Hodge structures. First of all, we briefly summarize the classical theory of the moduli spaces of polarized Hodge structures. 1.1 The moduli space M h = Γ\D h. Let n be an integer, and let h be a sequence of positive integers (h n,0, h n 1,0,, h 0,n ) satisfying h p,q = h q,p, called the Hodge numbers. Let H Z be a free abelian group of rank h p,n p, with a non-degenerate bilinear form Q : H Z H Z Z, which is symmetric (resp. anti-symmetric) if n is even (resp. odd). Let G Z be the group functor Aut(H Z, Q) on rings, sending a ring R to the group of automorphisms on the free R-module H R := H Z R preserving the bilinear form Q. It is clearly a group scheme over Z. Let Γ be an arithmetic subgroup of G Z (Z) ([5], 3). The set of Hodge structures of weight n on H R with prescribed Hodge numbers h, such that Q induces a polarization on H R (i.e. it induces a morphism H R H R R( n) of Hodge structures, and the bilinear form Q C (u, v) := Q(u, Cv), where C is the Weil operator, is symmetric and positive definite), is parameterized by the homogeneous space D h = G R /K, where K is the stabilizer group of a fixed polarized Hodge structure F 0 on H R. This homogeneous space D = D h = G R /K has a complex structure defined as follows. It is clear that Q : H R H R R( n) is a morphism of Hodge structures if and only if Q(F p, F n p+1 ) = 0 for all p. Let f p = r p hr,n r, and let D, the compact dual of D, to be the subspace of the product of the Grassmannians p Gr(f p, H R ) consisting of all flags F F p+1 F p such that Q(F p, F n p+1 ) = 0. Then D = G C /P, where P is a parabolic subgroup corresponding to a fixed flag. This gives D a complex structure. We see that D D is the locus of flags satisfying (i) F p F n p+1 = 0 (so that F p F n p+1 = HC ), and (ii) Q(u, Cu) > 0 for u 0 in H C. They are both open conditions, so D D is an open complex submanifold. The group Γ acts on D h properly discontinuously, and the quotient M h = Γ\D h is the moduli space of Γ-equivalence classes of Q-polarized Hodge structures on H C with Hodge type h. See ([3], 0.3.6, 0.3.7). 1.2 Variations of Hodge structure. Definition Let S be a complex manifold. A variation of Hodge structure H of weight n on S is given by a local system H Z of free abelian groups of finite rank on S; a finite decreasing filtration F H O of the vector bundle H O := H Z Z O S by holomorphic sub-bundles, such that the following conditions are satisfied: 1) (Griffiths transversality) the natural flat connection = d id HZ : H O Ω 1 S H O takes F p H O into Ω 1 S F p 1 H O, for every p; 2) for each point s S, the fiber F (s) over s is a Hodge structure of weight n. A polarization of the variation of Hodge structure H is a locally constant bilinear form Q : H Z H Z Z such that on each fiber over s S, it induces a polarization of the fiber Hodge structure. 1

2 Suppose we have a polarized family of Hodge structures (H, Q : H Z H Z Z) of weight n on S, and a global section of the sheaf Γ\Isom((H Z, Q), (H Z, Q)), where H Z is regarded as a constant sheaf on S, and assume that all the monodromies of this family of Hodge structures on S are contained in Γ. Then there is a period map ϕ : S M h inducing this family of Hodge structures. This map is locally liftable to D h. If f : X S is a projective smooth morphism between quasi-projective complex algebraic manifolds, with a relative hyperplane section η H 0 (S, R 2 f Z), then the family of the primitive part P n (X s, Z) of the cohomology groups H n (X s, Z) modulo torsion form a polarized variation of Hodge structure of weight n on S, and it induces a period map S M h. To be precise, the family of H n (X s, C) s are the stalks of R n f (f 1 O S ), and the Hodge filtration on R n f (f 1 O S ) is given by the degenerate spectral sequence E pq 1 = Rq f Ω p X/S = Rp+q f (f 1 O S ), which is induced from the resolution Ω X/S of f 1 O S (the relative holomorphic Poincaré lemma, see ([1], 3.4)). Since η is a global section, the primitive part form a variation of sub-hodge structures on S. 2 Logarithmic Hodge structures. One can ask the following question. Let f : X S be a family of projective manifolds, and let S be the complement of a normal crossing divisor D in some compact manifold S, and suppose one can extend the family f to a family f : X S which is log smooth (here S has the log structure induced by the divisor D). Is it possible to enlarge the moduli space M h to some M h so that the period map extends to ϕ : S M h? To study the degenerations of Hodge structures, Kato and Usui introduced the notion of logarithmic Hodge structures. 2.1 The ringed space X log. Let (X, α : M X O X ) be an fs log analytic space over C (for instance the C-points of an fs log scheme over C), and let X log be the set of pairs (x, u), where x X and u : M X,x S 1 is a homomorphism of monoids, such that u(f) = f(x)/ f(x) for f OX,x M X,x. Here S 1 is the unit circle in the complex plane. Let τ : X log X be the function (x, u) x. For any open U X and f M X (U), there is a function arg(f) : τ 1 (U) S 1 sending (x, u) u(f). We give X log the weakest topology such that the functions τ and arg(f) are continuous. Over the open set X X where the log structure is trivial, the map τ is a homeomorphism, and the section j log : X X log is a homotopy equivalence. The map τ is proper, with fibers τ 1 (x) compact tori (S 1 ) m, where m is the rank of M gp X,x. One can define a sheaf of rings O X log on X log. Roughly speaking, this is the subsheaf of rings of j log O X on X log generated over τ 1 O X by log(q), for all q M gp X. See ([3], 2.2.4) for the precise definition. For example, if x X and y τ 1 (x), and the free abelian group M gp X,x has rank m and is generated by f 1,, f m M gp X,x, then the stalk O X log,y is isomorphic to the polynomial 2

3 ring O X,x [log(f 1 ),, log(f m )]. This shows that in general, (X log, O X log) is not a locally ringed space. Let Ω 1 X be the sheaf of log differential forms on the fs log analytic space X, i.e. Ω 1 X = (Ω 1 X (O X Z M gp X ))/{( dα(f), α(f) f) f M X}. For a morphism f : X Y of fs log analytic spaces, define Ω 1 X/Y = Coker(f Ω 1 Y Ω 1 X). They are both coherent O X -modules. Let Ω r X/Y be the r-th exterior power of Ω1 X/Y, and let Ω r X log /Y = τ Ω r log X/Y = τ 1 Ω r X/Y τ 1 O X O X log. One can define differential maps and have the log de Rham complex (Ω X/Y, d) (resp. (Ω, d)) on X (resp. X log ). X log /Y log For y X log and x = τ(y) X, let sp(y) be the set of all ring homomorphisms s : O X log,y C that extend the evaluation map ev x : O X,x C. Since O X log,y is isomorphic to the polynomial ring over O X,x generated by log of a basis for M X,x, if we fix an s 0 sp(y), then we have a bijection: s (f s(log(f)) s 0 (log(f))) : sp(y) 2.2 Log variations of polarized Hodge structure. Hom group (M gp X,x, C). Definition Let X be an fs log analytic space. A log variation of polarized Hodge structure of weight n on X is given by a local system of free abelian groups of finite rank H Z on X log, a bilinear form Q : H Z H Z Z, a finite decreasing filtration F H O of H O := H Z O X log by O X log-submodules, such that the following conditions are satisfied: 1) there exist a locally free O X -module E and a finite decreasing filtration F E by O X - submodules, such that Gr p (E ) is locally free for each p, and F p H O = τ F p E = τ 1 F p E τ 1 O X O X log; 2) for y X log and x = τ(y) X, let s sp(y) and let f 1,, f r M X,x O X,x generate the monoid M X,x. If the exp(s(log(f i ))) are sufficient small for all i, then (H Z,y, Q, F (s)) is a polarized Hodge structure of weight n; 3) the connection d id : H O Ω 1 X log OX log H O takes F p H O into Ω 1 X log F p 1 H O. Here F (s), the specialization of F at s, is the decreasing filtration of H C,y := C Z H Z,y defined by F p (s) = C s,ox log,y F p H O. For a fixed point y X log, the family (H Z,y, Q, F (s)) s sp(y) is called a polarized log Hodge structure on the log point (x, M X,x ); this is the same as a log variation of polarized Hodge structure on the log point (x, M X,x ). Log variations of polarized Hodge structure arise from geometry in the following way. Let f : X Y be a projective log smooth morphism between fs log analytic spaces, and we fix a line bundle on X which is relatively ample over Y. By a theorem of Kajiwara and Nakayama, for every integer n, the sheaf R n f log Z is a local system on Y log. We take H Z to be R n f log Z modulo torsion, take Q to be the pairing induced by the fixed ample line bundle, take E to be R n f (Ω X/Y ), with filtration F p E = R n f (Ω p X/Y ) E, and take F p H O to be τ F p E. Then by a theorem of Kato, Matsubara and Nakayama, this is a log variation of polarized Hodge structure on Y. 3

4 3 Kato-Usui spaces. We fix n, h, H Z, Q, G Z, D and D as in (1.1). Let g R = Lie(G R ). A subset σ g R is called a nilpotent cone if it is a cone n σ = R 0 N i generated by mutually commutative nilpotent operators N i g R End(H R ). Let Γ be a neat subgroup of G Z (Z), i.e. for every element γ Γ, its eigenvalues on H C generate a torsion-free subgroup of C. 3.1 Nilpotent orbits. Definition Let σ = i R 0N i be a nilpotent cone. A subset Z D is called a σ-nilpotent orbit, if there exists an F 0 D such that Z = exp( i CN i)f 0, NF p 0 F p 1 0 for all p Z and N σ, exp( i z in i )F 0 D if Im(z i ) 0 for all i. We also call the pair (σ, Z) a nilpotent orbit. Let Σ be a fan in g Q, i.e. Σ is a set of rational nilpotent cones in g R (namely, those generated by nilpotent operators in g Q ) such that if σ Σ, then all faces of σ are in Σ, for σ, σ Σ, the intersection σ σ is a face of both σ and σ, for every σ Σ, we have σ ( σ) = 0. One can then define the set D h,σ (or just D Σ, if there is no confusion) of nilpotent orbits in the directions in Σ to be the set of nilpotent orbits (σ, Z) where σ Σ. There is a natural injection F (0, {F }) : D D Σ. 3.2 The moduli space M Σ. Let Σ be a fan in g Q and let Γ G Z (Z) be a subgroup. Then we say that Γ is compatible with Σ if for every γ Γ and σ Σ, we have Ad(γ)(σ) Σ. In this case, there is an action of Γ on D Σ given by (σ, Z) γ (Ad(γ)(σ), γz). We say that Γ is strongly compatible with Σ if every cone σ Σ is generated by elements in log Γ. Kato and Usui showed that when Γ is strongly compatible with Σ and the arithmetic subgroup Γ is neat, the quotient set Γ\D Σ can be given the structure of a log locally ringed space over C, in fact a log manifold (see ([3], 3.5.7)). Roughly speaking, a log manifold is a log locally ringed space over C, which is locally isomorphic to the zero locus of some log differential forms on a log smooth analytic space. Informally speaking, Kato and Usui proved the following. First, there is a one-to-one correspondence between D Σ and the set of polarized log Hodge structures of the given type. Second, if X S is a log smooth family extending the projective smooth family X S, where S S is the complement of a normal crossing divisor, then the period map extends to S M Σ. We briefly explain the first part in the following. We shall show how to get a nilpotent orbit from a polarized log Hodge structure on a log point ([3], ). Let x be an fs log point with log structure M x. Then M x is a sharp fs monoid and M gp x if a free abelian group of finite rank, say r. Fix y x log. We have x log = Hom(M gp x, S 1 ) (S 1 ) r and hence π 1 (x log ) = Hom(M gp x, Z) Z r. Let 4

5 π 1 + (xlog ) π 1 (x log ) be the subset consisting of those homomorphisms a : M gp x Z that take M x into N; this subset is an fs monoid. Let (H Z, Q, F H O ) be a polarized log Hodge structure on x. Let (h i ) n be a family of generators for π 1 + (xlog ) and s 0 sp(y). Let z 1,, z r be complex numbers, and let s sp(y) be such that ( log(f) ) s 2πi ( log(f) ) s 0 = 2πi r z i h i (f), Let N i : H Q,y H Q,y be the logarithm of h i. Then we have ( n F (s) = exp z i N i )F (s 0 ), for f M gp x. which shows that (F (s)) s sp(y) is an orbit of filtrations under exp(σ C) for σ = i R 0N i. Moreover, the condition 2) in (2.2.1) implies that F (s) D if Im(z i ) 0 for all i, and the condition 3) in (2.2.1) implies that NF (s 0 ) p F (s 0 ) p 1 for all p Z and N σ. In other words, the family (F (s)) s is a σ-nilpotent orbit. References [1] Pierre Deligne, Travaux de Griffiths, Séminaire N. Bourbaki, , exp. NO. 376, p [2] Phillip Griffiths, Logarithmic Hodge structures: Report on the work of Kato- Usui, Notes from a talk given at MSRI on March 18, 2009, available at abrmovic/griffiths-log.pdf [3] Kazuya Kato, Sampei Usui, Classifying Spaces of Degenerating Polarized Hodge Structures, Annals of Mathematics Studies, Number 169, Princeton University Press, Princeton and Oxford, [5] James Milne, Introduction to Shimura Varieties, 2004, available at 5

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