# Geometry of moduli spaces

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1 Geometry of moduli spaces 20. November / 45

2 (1) Examples: C: compact Riemann surface C = P 1 (C) = C { } (Riemann sphere) E = C / Z + Zτ (torus, elliptic curve) 2 / 45

3 (2) Theorem (Riemann existence theorem, Chow s theorem): Every compact Riemann surface C is an algebraic curve, i.e. admits a holomorphic embedding i : C P n = P n (C) and i(c) is the set of solutions of finitely many homogeneous polynomial equations. Conclusion: It makes no difference wether one considers the classification problem for compact Riemann surfaces or for smooth projective algebraic curves (over C). 3 / 45

4 (3) Question: How can one classify algebraic curves? The topological classification of orientable compact connected 2-dimensional real surfaces is given by their genus (= # of holes) g = 0 g = 1 genus g 4 / 45

5 (4) Question: How many algebraic (= holomorphic = conformal) structures exist on a topological surface of genus g? g = 0: The algebraic structure is unique, i.e. every algebraic curve of genus 0 is isomorphic to P 1. g = 1: Every curve of genus 1 arises as E τ = C / Z + τz, Im τ > 0. τ 1 + τ τ H 1 = {z C; Im z > 0}. 1 Question: When is E τ = Eτ? 5 / 45

6 (5) The group SL(2, Z) acts on H 1 by ( ) a b : τ aτ + b c d cτ + d. One shows that E τ = Eτ τ τ modulo SL(2, Z). I.e. we consider the quotient of H 1 by SL(2, Z) and obtain M 1 = SL(2, Z) H 1 = {elliptic curves}/ = Using the j-function j : H 1 C one shows that j : M 1 = SL(2, Z) H 1 = C. 6 / 45

7 (6) Curves of genus g 2 Riemann (1857): Curves of genus g 2 depend on 3g 3 moduli. Then the following holds: M g = {algebraic curves of genus g}/ = M g carries the structure of a quasi-projective variety dim M g = 3g 3 M g is irreducible M g has at most finite quotient singularities. 7 / 45

8 (7) An important property of M g is the following: Let f : X U be a family of smooth projective curves, i.e. Cu = f 1 (u) u for every u U the fibre C u = f 1 (u) is a smooth projective curve of genus g. Then the map U φ U : U M g u [C u ] is a morphism (and M g is minimal with this property). Formally one describes this in the language of representations of functors and (Deligne Mumford) stacks. 8 / 45

9 (8) Compactifications of M g M g is a quasi-projective variety. Question: Is there a geometrically meaningful compactification of M g? M g := moduli space of stable curves of genus g Definition: A projective algebraic curve is stable if it has at most nodes as singularities (but it need not be irreducible) and Aut(C) < g = 2 g = / 45

10 The geometry of M g (1) Fact: M g carries the structure of a projective variety containing M g as a (Zariski-)open set. g = 0 g = 1 g 2 M 0 = {pt} M 1 = M 1 {pt} = C {pt} = P 1 Question: What can we say about the geometry of the variety M g? 10 / 45

11 (1) X = projective manifold, dim C X = n. T X = tangent bundle of X Ω 1 X = TX = cotangent bundle of X. The sections of Ω 1 X are 1-forms, i.e. locally of the form n Ω = f i (z 1,..., z n ) dz i. i=1 11 / 45

12 (2) ω X := Λ n Ω 1 X = det Ω 1 X = canonical bundle The sections of ω X are n-forms, i.e. locally of the form ω = f (z 1,..., z n ) dz 1... dz n. = ω X... ω X (k-th power of the canonical bundle, k 1) ω k X The sections are k-fold pluricanonical forms and locally of the form ω = f (z 1,..., z n ) (dz 1... dz n ) k. 12 / 45

13 (3) X = projective manifold, dim C X = n. Definition: The k-th plurigenus of X is defined by P k (X ) := dim C H 0 (X, ω k X ) i.e. the number of independent global k-fold pluricanonical forms. 13 / 45

14 (4) Definition: of X is defined as if P k (X ) = 0 for all k 1 0 if P κ(x ) = k (X ) = 0 or 1 for all k 1 and there is at least one k 0 1 with P k0 (X ) = 1 κ if P k (X ) c k κ + l.o.t. (c > 0). Remark: κ(x ) {, 0, 1,..., dim X }. Definition: X is of general type if κ(x ) = dim X. 14 / 45

15 (5) In the case of curves the situation is as follows: κ(c) = g(c) = 0 C = P 1 κ(c) = 0 g(c) = 1 C = elliptic curve κ(c) = 1 g(c) 2 is a rough birational invariant for algebraic. 15 / 45

16 Geometry of M g (2) Question: What is the Kodaira dimension of (a smooth model of) M g? Theorem (Harris Mumford, Eisenbud Harris): M g is of general type for g 24. Theorem (Farkas): M 22 is of general type. Theorem: κ(m g ) = for g 16. This result is due to Severi, Sernesi, Chang, Ran, Bruno, Verra,... Question: κ(m g ) =? for 17 g 21, 23? Proposition (Farkas): κ(m 23 ) / 45

17 (1) Abelian are higher-dimensional generalisations of elliptic curves E = C / Zτ + Z = C / Λ τ A g-dimensional torus is given by (Λ τ = Zτ + Z). A = C g / Λ (Λ = lattice of rank 2g in C g ). Clearly A is a compact complex abelian Lie group (and every such Lie group is of this form). Definition: A torus A is called an abelian variety if A is projective, i.e. an embedding A P N exists (for some N). Remark: This is automatic for g = 1, but does not hold in general. 17 / 45

18 (2) Many aspects of the theory of elliptic curves carry over to abelian. The Siegel upper half plane of genus g is defined as H g = {τ Mat(g g, C); τ = t τ, Im τ > 0}. The integral symplectic group is defined by where Remark: For g = 1 Sp(g, Z) = {M GL(2g, Z); t MJM = J} J = ( 0 1g 1 g 0 ). Sp(1, Z) = SL(2, Z). 18 / 45

19 (3) The group Sp(g, Z) acts on H g by ( ) A B : τ (Aτ + B)(Cτ + D) 1. C D The quotient A g = Sp(g, Z) H g = moduli space of principally polarized abelian of dimension g. A polarized abelian variety is a pair (A, L ) where A is an abelian variety und L is an ample line bundle on A. A polarization is called principal if L g = g!. 19 / 45

20 (4) A g has the following properties: A g is quasi-projective, irreducible dim A g = 1 2g(g + 1) A g has only finite quotient singularities. 20 / 45

21 (5) Question: What is the Kodaira dimension of (a smooth projective model of) A g? Theorem: κ(a g ) = for g 5. g 3 : g = 4 : g = 5 : classical Clemens Donagi; Mori, Mukai; Verra Theorem: A g is of general type for g 7. g 9 : g = 8 : g = 7 : Tai Freitag Mumford Problem: κ(a 6 ) =? 21 / 45

22 (6) The Siegel domain H g is the hermitian symmetric space H g = Sp(g, R)/U(g) where U(g) is the (up to conjugation) unique maximal compact subgroup. It can also be realised as a bounded complex domain via the Cayley transformation H g D g = {Z Mat(g g, C); Z = t Z, t ZZ < 1 g } τ (τ i 1 g )(τ + i 1 g ) 1. Example: H 1 = D 1 = {z C; z < 1}. 22 / 45

23 (7) Modular forms Definition: A modular form of weight k with respect to Sp(g, Z) is a holomorphic function such that F : H g C F (M(τ)) = (det(cτ + D)) k F (τ) ( ) A B for all M = Sp(g, Z). C D Remark: For g = 1 one needs to add a condition of holomorphicity at infinity (the cusp). 23 / 45

24 (8) Definition: A modular form is a cusp form if it vanishes at. S k (Sp(g, Z)) = {F ; F is a cusp form of weight k}. dim C S k (Sp(g, Z)) <. 24 / 45

25 (9) Modular forms and differential forms Let F be a modular form of weight k(g + 1) with respect to Sp(g, Z). Define ω F = F (dτ 11 dτ dτ gg ) k. Fact: ω F is invariant w.r.t. Sp(g, Z). Hence we can view ω F as a k-fold pluricanonical form on (an open part of) A g. More precisely A g = A g {fixed points of Sp(g, Z)} Then ω F H 0 (A g, ω k A g ). 25 / 45

26 S: smooth projective surface, minimal The Kodaira classification classifies surfaces according to Kodaira dimension. κ(s) = κ(s) = 0 S is rational or a geometrically ruled surface S is one of the following: abelian surface K3 surface Enriques surface bielliptic surface κ(s) = 1 κ(s) = 2 S is elliptic fibration S is of general type. 26 / 45

27 (1) Definition: A K3 surface is a compact complex surface with the following properties ω S = O S π 1 (S) = {0}. Examples: (1) Quartic surfaces in P 3 : S = {(x 0 : x 1 : x 2 : x 3 ) P 3 ; f 4 (x 0,..., x 3 ) = 0} where f 4 is homogeneous of degree 4, general. E.g. S = {x0 4 + x1 4 + x2 4 + x3 4 = 0}. (2) Complete intersections of degree (2, 3) in P 4 : S = {f 2 = f 3 = 0} P 4. Remark: There exist non-algebraic K 3 surfaces. 27 / 45

28 (2) The second cohomology group H 2 (S, Z) of a K3 surface has the structure of a lattice, i.e. it carries a non-degenerate symmetric bilinear form (given by the intersection form). More precisely where and U = H 2 (S, Z) = 3U 2E 8 ( 1) ( ) (hyperbolic plane) E 8 ( 1) = non-degenerate, even, unimodular lattice of rank 8. The second cohomology group with complex coefficients has a Hodge decomposition H 2 (S, C) = H 20 H 11 H 02 (H 20 = H 02 ). (2,0)-forms (1,1)-forms (0,2)-forms 28 / 45

29 (3) Since ω S = O S it follows that dim C H 20 = dim C H 0 (S, ω S ) = dim C H 0 (S, O S ) = 1. I.e. we have a 1-dimensional subspace C = H 20 H 2 (S, C) = C 22. This inclusion carries all information on S (Torelli theorem). L K3 = 3U + 2E 8 ( 1) (sign(l K3 ) = (3, 19)). A marking of a K3 surface is an isomorphism φ: H 2 (S, Z) = L K3. This defines a 1-dimensional subspace φ(c ω S ) = φ(h 20 ) L K3 C. 29 / 45

30 (4) With respect to the intersection form on H 2 (S, C) one has (ω S, ω S ) = 0, (ω S, ω S ) > 0. This leads one to define Ω K3 = {[x] P(L K3 C); (x, x) = 0, (x, x) > 0}. dim C Ω K3 = 20 Given a marking φ: H 2 (S, Z) = L K3 of a K3 surface one defines the period point φ(ω S ) = [C ω S ] Ω K3. 30 / 45

31 (5) O(L K3 ) = {g; g is an orthogonal transformation of L K3 }. This group acts on Ω K3. Theorem (Torelli): The K 3 surface S can be reconstructed from its period point. I.e. the quotient M K3 = O(L K3 ) Ω K3 is the moduli space of. Remark: The group action is badly behaved, e.g. the quotient M K3 is not hausdorff. 31 / 45

32 (6) The situation improves when one restricts to the algebraic case, i.e. to polarized K 3 surfaces. A polarized K3 surface is a pair (S, L ) where L is an ample line bundle. Instead of L it suffices to consider The degree of L (resp. h) is h = c 1 (L ) H 2 (S, Z). deg L = c 1 (L ) 2 = h 2 = 2d > 0. Question: How can we describe moduli of polarized K 3 surfaces (of given degree 2d)? 32 / 45

33 (7) If h L K3 is a primitive element with h 2 = 2d, then hl K3 = 2U 2E8 ( 1) 2d =: L 2d. As before we consider Ω L2d = {[x] P(L 2d C); (x, x) = 0, (x, x) > 0}. Then dim Ω L2d = 19 Ω L2d = D L2d D L 2d where D L2d is a homogeneous symmetric domain of type IV. 33 / 45

34 (8) Let and set Õ(L 2d ) = {g O(L K3 ); g(h) = h} F 2d = Õ(L 2d) Ω L2d. Theorem (Torelli): The quotient F 2d is the moduli space of pseudo-polarized K 3 surfaces of degree 2d. 34 / 45

35 (9) The action of Õ(L 2d) on Ω L2d is properly discontinuous. The following holds: dim F 2d = 19 F 2d has only finite quotient singularities F 2d is quasi-projective (Baily Borel). There exist different compactifications of F 2d : F2d BB : Baily Borel compactification : toroidal compactifications F tor 2d 35 / 45

36 (10) Question: What is the Kodaira dimension of F 2d? Theorem (Mukai): F 2d is unirational (and hence κ(f 2d ) = ) if 1 d 10, d = 12, 16, 17 and 19. Theorem (Gritsenko, H., Sankaran, 2007): F 2d is of general type for d > 61 and d = 46, 50, 54, 57, 58, 60. Remark (A. Peterson): F 2d is also of general type for d = / 45

37 (1) Recall that Ω L2d = {[x] P(L 2d C); (x, x) = 0, (x, x) > 0} = D L2d D L 2d. Let and (L 2d C) {0} D L 2d = affine cone over D L2d P(L 2d C) O + (L 2d ) = {g O(L 2d ); g(d L2d ) = D L2d }. 37 / 45

38 (2) Definition: Let Γ O + (L 2d ) be a group of finite index. A modular form (automorphic form) of weight k with respect to Γ and with a character χ is a holomorphic function such that F : DL 2d C (1) F (tz) = t k F (Z) (t C ) (2) F (γz) = χ(γ)f (Z) (γ Γ). 38 / 45

39 (3) Fact: S 19k (Õ+ (L 2d ), det k ) F F (dz) k H 0 (F2d, ω k F ) 2d suitable volume form where F 2d = F 2d {fixed points of Õ+ (L 2d )}. Question: When can one extend these forms to pluricanonical forms on a smooth projective model F 2d of F 2d? Obstructions: (1) elliptic obstructions (from singularities of F 2d ) (2) reflective obstructions (from quasi-reflections in Õ+ (L 2d )) (3) parabolic obstructions (dz picks up poles along the boundary). We will concentrate on (3) in the sequel. 39 / 45

40 (4) Low weight cusp form trick Assume we have a cusp form F a of low weight a (i.e. a < 19 = dim Ω L2d ), i.e. Let k be even and consider F a S a (Õ+ (L 2d ), det ε ); ε = 0 or 1. F Fa k S k(19 a }{{} )(Õ+ (L 2d )) S 19k (Õ+ (L 2d )) = S 19k (Õ+ (L 2d ), det k ). >0 Then has no poles along the boundary. ω F = F (dz) k Question: Construction of low weight cusp forms? 40 / 45

41 (1) Let L 2,26 = 2U 3E 8 ( 1) = 2U Λ Then Borcherds has constructed a modular form of weight 12. Φ 12 : DL 2,26 C Idea: Restrict this form to D L 2d. (Λ = Leech lattice). Choose l E 8 ( 1), l 2 = 2d < 0. This defines embeddings L 2d = 2U E 8 ( 1) 2d L 2,26 resp. Ω L2d Ω L2, / 45

42 (2) Let R l := {r E 8 ( 1); r 2 = 2, (r, l) = 0} N l := R l. We define the quasi-pullback of Φ 12 to D L 2d F l := Proposition: If N l > 0 then Φ 12(z) (r, z) D ±r R l L2d. by 0 = F l S N (Õ+ 12+ l (L 2d ), det). 2 This gives us low weight cusp forms provided 2 N l / 45

43 (3) Question: When can we find l E 8 ( 1) with l 2 = 2d which is orthogonal to at least 2 and at most 12 roots? Proposition: Such l exists if 4N E7 (2d) > 28N E6 (2d) + 63N D6 (2d) where N L (2d) is the number of representations of the integer 2d in the lattice L. One can show that ( ) holds for d > 143. ( ) The remaining small d in the theorem can be done by a computer search. 43 / 45

44 (1) This technique can also be applied to other moduli problems. Definition: An irreducible symplectic manifold is a compact complex manifold X with the following properties: (1) X is Kähler (2) There exists a (up to scalar) unique non-degenerate 2-form ω X H 0 (X, Ω 2 X ) ( ω X = O X ) (3) X is simply connected. Examples (1) dim X = 2: X = S = K3 surface (2) X is a deformation of Hilb n S (3) X is a deformation of a generalized Kummer variety (4) O Grady s sporadic examples in dimension 6 and / 45

45 (2) In this case Torelli does not hold (in the strict form). However one still has a finite dominant map M Γ D L component of a moduli space of polarized irreducible symplectic manifolds where L is a lattice of signature (2, n) and Γ is a suitable group. General type results have been obtained in the case X def Hilb 2 S, split polarization (Gritsenko, H., Sankaran). Work in progress X def Hilb n S, general polarization O Grady s 10-dimensional sporadic case. 45 / 45

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