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Hodge Theory and Boundaries of Moduli Spaces

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Hodge Theory and Boundaries of Moduli Spaces Hodge theory and boundaries of moduli spaces Phillip Griffiths∗ Based on correspondence and discussions with Mark Green, Colleen Robles and Radu Laza ∗Institute for Advanced Study 1/82 I The general theme of this talk is the use of Hodge theory to study questions in algebraic geometry. I This is a vast and classical subject; here we shall focus on one particular aspect. I In algebraic geometry a subject of central interest is the study of moduli spaces M and their compactifications M. I Points in @M = MnM correspond to singular varieties X0 that arise from degenerations X ! X0 of a smooth variety X . 2/82 I It is well known and classical that X0 may be \simpler" than X and understanding X0 may shed light on X . For an example, any smooth curve X may be degenerated to a stick curve 1 X0 = = configuration of P 's and deep questions about the geometry of a general X may frequently be reduced to essentially combinatorial ones on X0. 3/82 I The central thesis of this talk is Hodge theory may be used to guide the study of degenerations X ! X0. Underlying this thesis are the points ∗ I the polarized Hodge structure on H (X ; Z) (mod torsion) is the basic invariant of a smooth projective variety X ; I the study of degenerations (V ; Q; F ) ! (V ; Q; W ; Fo) of polarized Hodge structures is fairly highly developed; I this understanding may be used to suggest what the possible degenerations X ! X0 should be and where to look for them. 4/82 This thesis is well known and has been effectively used in the classical case when the Hodge structures are parametrized by a Hermitian symmetric domain and 1-parameter degenerations are used. It has been utilized in the non-classical case mainly for 1-parameter degenerations of Calabi-Yau threefolds, but to my knowledge not so much in other non-classical cases or for several parameter degenerations other than for the case of algebraic curves. 5/82 I the structure of this talk will be I discussion of some of the basic aspects of Hodge theory; I the example of algebraic curves, which indicates how the study might go in the classical case; I discussion of the first non-classical case, namely that of an algebraic surface X of general type with h2;0(X ) = 2, 1;0 2 h (X ) = 0 and KX = 2; this is based on correspondence with Mark Green, and also with Colleen Robles and Radu Laza, and reflects developments in a very early stage. 6/82 I A final general point: a schematic of how Hodge theory interacts with algebraic geometry for the above surface is canonical 0 2 2;0 ! H (ΩX ) = H (X ) series jKX j bi-canonical variation of Hodge structure ! 1 series j2KX j associated to H (C); C 2 jKX j . which arises from KX = KC C. 7/82 Definition A polarized Hodge structure (PHS) of weight n is given by (V ; Q; F •) where I V is a Q-vector space (in practice we will have V ⊃ VZ); I Q · V ⊗ V ! Q is a non-degenerate form with Q(u; v) = (−1)nQ(v; u); • n n−1 0 I F is a decreasing filtration F ⊂ F ⊂ · · · ⊂ F = VC satisfying p n−p+1 ∼ F ⊕ F −! VC; 0 5 p 5 n; p n−p+1 I Q(F ; F ) = 0; I Q(v; Cv¯) > 0 for v 6= 0, where C is the Weil operator. 8/82 q q;p Setting V p;q = F p \ F = V , the third condition above is p;q p;q q;p VC = ⊕V where V = V and Cv = i p−q on V p;q. Example Let X ⊂ PN be a smooth projective variety of dimension d and L 2 H2(X ; Q) the restriction of the generator of a hyperplane class in H2(PN ; Z). Then multiplication by L gives a nilpotent operator on H∗(X ; Q) m m+2 L : H (X ; Q) ! H (X ; Q): 9/82 The Hard Lefschetz theorem k m−k ∼ m+k L : H (X ; Q) −! H (X ; Q) implies that the action of L may be uniquely extended to the action of an sl2-triple fL; Y ; Λg where m Y = (m − d)Id on H (X ; Q) ∗ is the grading. This sl2-action decomposes H (X ; Q) into irreducibles whose basic building blocks are the lowest weight spaces k+1 m−k n m−k L m+k+2 o H (X ; Q)prim = ker H (X ; Q) −−! H (X ; Q) = V of the irreducible summands. 10/82 Setting Q(u; v) = (Lk u · v)[X ] and using the decomposition n−k H (X ; C)prim induced by the Hodge decomposition Hm−k (X ; C) = ⊕ Hp;q(X ) gives a PHS of weight p+q=m−k m − k. Instead of the usual Hodge diamond, we will picture the groups in the first quadrant in the (p; q) plane. q dim X = 1 L r r p "r 0 1 H (ΩX ) = holomorphic 1-forms on the compact Rieman surface X . 11/82 dim X = 2 and X is regular (H1;0(X ) = 0) 0 LH (X ) H rr HHj 1;1 H (X )prim r r 0 r 2 H (ΩX ) <o> 12/82 dim X = 1 The PHS on H1(X ) for X a compact Riemann surface is classically represented by the matrix of periods γ 0 1 ! , , ! 2 H ΩX ˆδ,γ ··· δ Even more classically, if X is given as the Riemann surface associated to an algebraic curve, e.g., the one of genus g given in C2 by 2g+2 2 Y f (x; y) = y − (x − ai ) = 0; ai distinct i=1 g(x) dx 2g(x) dx ! = = ; 0 5 deg g(x) 5 g − 1 y fy (x; y) 13/82 the periods are δ ::: a1 a2 a3 a4 γ The closure in P2 of the above affine curve has a singular point at [0; 0; 1] and the condition deg g(x) 5 g − 1 is the one that gives a holomorphic 1-form on X . 14/82 dim X = 2 The surfaces we shall be considering are minimal, of general type and have 2;0 1;0 2 h (X ) = 2; h (X ) = 0; KX = 2: They are extremal; i.e., they have the maximum h2;0(X ) subject to the other conditions above. They were analyzed classically by Castelnuovo and more recently by Horikawa. A general one is the minimal desingularization of the quartic 4 surface on P given in affine coordinates x1; x2; x3; x4 by ( f (x) = h(x) − k(x)2 = 0 2 g(x) = x1 − x2 = 0 where h(x); k(x) are a general cubic, quadric. 15/82 This surface has as singularity a double curve with eight pinch points. Think of it as an analogue of the smooth curve with affine equation 2g+2 2 Y y − (x − ai ) = 0 i=1 encountered above. It is the first regular, non-classical extremal surface. We will say more about it later. 16/82 The holomorphic 2-forms are `(x)dx ^ dx ! = 3 4 ; deg `(x) 1 f (x)g (x) 5 x1 x2 and where `(x) vanishes on the double conic so that ! is holomorphic on X . With these conditions we find that h2;0(X ) = 2. Think of ! as the analogue of g(x) dx where fy (x) deg g(x) 5 g − 1 above. One has b2(X ) = 32, so that 1;1 h (X )prim = 31 and the PHS is given by the matrix of periods. 17/82 The surface is more conveniently given in the homogenous coordinate [t0; t1; x3; x4] in the weighted projective space P(1; 1; 2; 2) by 2 2 2 2 2 2 t0 G(t0 ; t0t1; t1 ; x3; x4) = F (t0 ; t0t1; t1 ; x3; x4) : For fixed [t0; t1] we obtain genus 3 plane quartic curves which constitute the pencil jKX j. <o> 18/82 Definition D = period domain given by f(V ; Q; F •); dim V p;q = hp;qg Dˇ = compact dual given by f(V ; Q; F •); Q(F p; F n−p+1) = 0g. If G = Aut(V ; Q), then D = GR=H; H compact \ ˇ D = GC=P; P parabolic and H = P \ GR: 19/82 Example ∼ t D = Hg = fZ = g × g matrix; Z = Z and Im Z > 0g = Sp(2g; R)=U(g) ˇ 2g 2g D = GrassL(g; C ) = Lagrangian g-planes in (C ; Q) 0 I where Q = −I 0 . The embedding D ,! Dˇ is Z Z ! span of the columns of : Ig 1 For g = 1 we have H ,! P and GR = SL2(R) acting by linear fractional transformations. 20/82 Example ˇ D = GrassQ (2; VC) = f2-planes E : Q(E; E) = 0g [ D = 2-planes E with Q(E; E) < 0: The GR = SO(4; h) and D = GR=U(2) × SO(h) is not an Hermitian symmetric domain (HSD). Definition A variation of Hodge structure (VHS), or period mapping, is given by a holomorphically varying family of PHS's parametrized by a complex manifold S and satisfying a differential constraint. In order to have the PHS's occurring in a fixed vector space (V ; Q) we need to use the universal covering Se ! S. 21/82 Example S = ∆∗ = ft : 0 < jtj < 1g, Se = H = fz : Im z > 0g, and t = e2πiz . We may take products of this to have S = ∆∗`. Definition A VHS is given by Φ Se e / D Φ S / ΓnD where we have the monodromy representation ρ : π1(S) ! GZ and where Φe is an equivariant holomorphic mapping satisfying (i) Φ(e γz) = ρ(γ)Φ(e z) for γ 2 π1(S), z 2 Se; p dFz p−1 (ii) dz ⊂ Fz (infinitesimal period relation = IPR).
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