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On Deligne's Monodromy Finiteness Theorem

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On Deligne's Monodromy Finiteness Theorem ON DELIGNE'S MONODROMY FINITENESS THEOREM (MIMUW AG SEMINAR, DEC 13, 2018) PIOTR ACHINGER Abstract. Let S be a smooth complex algebraic variety with a base point 0. If X=S is a family of smooth projective varieties, one obtains the associ- ∗ ated monodromy representation π1(S; 0) ! GL(H (X0; Q)) of the topological fundamental group of S on the rational cohomology of X0. Let us say that a representation π1(S; 0) ! GLN (Q) \comes from geometry" if it is a direct summand of such a monodromy representation. Representations coming from geometry are quite special: they factor through GLN (Z) up to conjugation, their monodromy groups are semisimple, and the corresponding local system on S underlies a polarizable variation of Hodge structures. Inspired by a theo- rem of Faltings, Deligne proved in the mid-80s that there are only finitely many such representations of rank N for fixed S and N. We explain the beautifully simple proof of this result, based on Griffiths’ study of the geometry of period domains. We also discuss the very recent results of Litt which in particular imply a suitable analog for the ´etalefundamental group. 1. Monodromy representations Ehresmann's theorem in differential geometry states that if f : X −! S is a proper submersion of smooth manifolds, then f is a fiber bundle. Consequently, n the cohomology groups of the fibers H (Xs; Q)(s 2 S) assemble into a local system n R f∗Q on S. If S is connected and 0 2 S, such a local system corresponds to a representation n σ : π1(S; 0) −! GL(H (X0; Q)); called the monodromy representation associated to f : X ! S (in degree n). Let now S be a smooth complex algebraic variety, and let f : X ! S be a smooth projective morphism. Then the fibers of f are not necessarily biholomorphic to one another, and the variation of the complex structure on the fibers is controlled by Kodaira{Spencer theory. However, the induced map of complex manifolds f an : Xan −! San is a proper submersion, and one gets the associated monodromy representation an n an σ : π1(S ; 0) −! GL(H (X0 ; Q)): an Definition 1.1. We say that a representation σ : π1(S ; 0) ! GLN (Q) comes from geometry if it is isomorphic to a subquotient (equivalently, a direct summand) of a monodromy representation attached to some smooth and projective f : X ! S. The aim of this talk is to discuss the proof of the following beautiful result of Deligne. 1 2 PIOTR ACHINGER Theorem 1 (Deligne 1987, [2]). For fixed (S; 0) and N ≥ 0, there exist only finitely an many isomorphism classes of representations σ : π1(S ; 0) ! GLN (Q) which come from geometry. What are the properties of representations coming from geometry which make them special (and make the above theorem true)? an n an n an (i) Since π1(S ; 0) ! GL(H (X ; Q)) factors through GL(H (X ; Z)=tors.), an a representation σ : π1(S ; 0) ! GLN (Q) coming from geometry has to be isomorphic to one which factors through GLN (Z) (that is, σ preserves a lat- tice). an n an (ii) The Zariski closure of the image of π1(S ; 0) ! GL(H (X ; Q)) is a semisim- ple algebraic group over Q [3]. Consequently, the full subcategory of repre- sentations spanned by those coming from geometry is semisimple (and in particular one can replace `subquotient' with `direct summand' in the defini- tion). n (iii) The local system R f∗Q underlies a polarized variation of Hodge structure (we will explain what this is in x3), and so does every local system associated to a representation coming from geometry. Theorem 1 will be deduced from a more general result in complex geometry: Theorem 2. Let (S; 0) be a pointed connected complex manifold such that π1(S; 0) is finitely generated, and let N ≥ 0. (a) There exist only finitely many isomorphism classes of Q-local systems of rank N on S underlying a polarizable integral variation of Hodge structures, up to semisimplification. (b) Suppose that S is compactifiable, i.e. that there exists a compact complex man- ifold S and a closed analytic subset Z ⊆ S such that S ' S n Z. Then there exist only finitely many isomorphism classes of Q-local systems of rank N on S which are a subquotient of a local system underlying a polarizable integral variation of Hodge structures. Here in (a), `up to semisimplification’ means that we identify two local systems if both can be equipped with filtrations such that the associated graded objects are isomorphic. In more concrete terms, this means that the associated characters (traces) are equal. And indeed this is how the theorem is proved: one proves a bound on these traces which depends only on N; since they are integral, they can take only finitely many values. Under the assumptions in (b), in fact the category of such subquotients is semisimple, so equality of traces implies isomorphism. In Section 2, we discuss Hodge structures and period domains. In Section 3, we discuss variations of Hodge structures, period maps, and formulate Deligne's semi-simplicity theorem. In Section 4, we prove Theorems 1 and 2. We use local systems coming from families of elliptic curves as our running example. In the last Section 5, we discuss a recent result of Litt [6] which provides analogs of the above results for the ´etalefundamental group. 2. Hodge structures We will now review some standard material from Hodge theory. We refer to Voisin's book [8] for details. DELIGNE'S MONODROMY FINITENESS 3 2.1. Hodge decomposition and Hodge structures. Let X be a smooth and projective complex algebraic variety. Then the cohomology groups Hn(X; C)1 have a canonical Hodge decomposition n ∼ M p;q p;q q p (2.1.1) H (X; C) = H ;H = H (X; ΩX ): p+q=n Since Hn(X; C) = Hn(X; R)⊗C, the group Gal(C=R) ' Z=2Z acts on Hn(X; C). Under this action, one has Hp;q = Hq;p. For reasons soon to become apparent, it is more convenient to consider the Hodge filtration i n M p;q n •≥i n • F H (X; C) = H = im H (X; ΩX ) ! H (X; ΩX ) ; p≥i one has Hp;q = F p \ F q and Hn(X; C) = F p ⊕ F q+1. This defines (2.1.1). We say that the above equips Hn(X; Z)=tors. with an integral Hodge structure in the sense of the definition below. Definition 2.2. Let n be an integer. (1) An integral Hodge structure of weight n is a finitely generated free Z-module V together with a decomposition M V ⊗ C = V p;q p+q=n satisfying V p;q = V q;p. Equivalently, V ⊗ C is equipped with a finite descending filtration F p satisfying F p ⊕ F q+1 = V ⊗ C. (2) The numbers hp;q = dim V p;q are the Hodge numbers of V . 1 1 2.3. Polarizations and Hard Lefschetz. Let now ! 2 H (X; ΩX ) be the image of an ample class L under 1 × 1 1 2 d log: Pic(X) = H (X; OX ) −! H (X; ΩX ) \ H (X; Z): The Hard Lefschetz theorem states that for n ≤ d = dim X the cup product map !d−n : Hn(X; C) −!∼ H2d−n(X; C); is an isomorphism. By basic linear algebra, this induces the Lefschetz decomposition n M k n−2k H (X; Q) = ! · Hprim (X; Q); k≥0 r where H (X; C)prim is the primitive cohomology r d−r+1 r 2d−r H (X; C)prim = ker ! : H (X; C) ! H (X; C) (for r > d we set this to be zero). This decomposition is compatible with Hodge structures because ! is of type (1; 1). Consider the bilinear intersection pairing Z d−n 1 d−n Q(α; β) = h! · α; βi = n ! ^ α ^ β; (2πi) X on Hn(X; C) which is symmetric for n odd and alternating for n even, and takes integral values for α; β 2 Hn(X; Z). The associated hermitian pairing Z H(α; β) = (2πi)nQ(α; β) = !d−n ^ α ^ β; X 1We will suppress the notation Xan from now on. 4 PIOTR ACHINGER has the property that the decomposition (2.1.1) is orthogonal with respect to H. Moreover, one has the Hodge{Riemann bilinear relation " p;q n (−1) H(α; α) > 0 for 0 6= α 2 H \ H (X; C)prim; where " = q + n(n − 1)=2. The hermitian pairing equal to (−1)k(k−1)=2+q+rH on r n−2r ! · H (X; C)prim is thus positive-definite. Definition 2.4. Let n be an integer. (1)A polarized integral Hodge structure of weight n is an integral Hodge struc- ture V of weight n endowed with a bilinear pairing Q(α; β): V × V ! Z such that the associated hermitian form H(α; β) = inQ(α; β) on V ⊗ C satisfies 0 0 i. H(V p;q;V p ;q ) = 0 if p 6= p0, ii.( −1)"H(α; α) > 0 for 0 6= α 2 V p;q, where " = q + n(n − 1)=2. (2) An integral Hodge structure V of weight n is polarizable if it admits a bilinear pairing as above. 1 1 Example 2.5. Let X be an elliptic curve. Then H (X; Q) = H (X; Q)prim has Hodge numbers h1;0 = 1 = h0;1.
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