Moduli Spaces of Stable (Complex) Curves and Stable Maps

http://www.math.wisc.edu/~robbin/chiconeTalk.pdf

A Colloquium for Carmen Chicone’s Birthday Conference May 20, 2006

1 Joel Robbin & Dietmar Salamon: A construction of the Deligne–Mumford orbifold, to appear in JEMS.

Joel Robbin, Yongbin Ruan, & Dietmar Salamon: The moduli space of regular stable maps, submitted to Math. Zeitschrift.

This material will eventually become a book on the foundations of quantum cohomology. 2 Moduli Spaces

A moduli space is the orbit space of a group action, or more gen- erally, of a groupoid. Various theories (Floer homology, quan- tum cohomology, etc) define topological invariants using them. Often the objects are stable, meaning that the isotropy (auto- morphism) groups are finite. We equip a moduli space with an an orbifold structure by constructing an etale groupoid with the same orbit space.

3 Apology

Of course, here I’m working with the moduli stack rather than the moduli space. For those of you who aren’t familiar with stacks, don’t worry: basically, all it means is that I’m allowed to pretend that the moduli space is smooth and that there’s a universal family over it.

Who hasn’t heard these words, or their equivalent, in a talk? And who hasn’t fantasized about grabbing the speaker by the lapels and shaking him until he says what –exactly– he means by them? But perhaps you’re now thinking that all that is in the past, and that at long last your’re going to learn what a stack is and what they do. 4 Apology (Continued)

Fat chance. Unless you’ve picked up this book for the first time and have opened it at random to this page, you must know better.

Joe Harris & Ian Morrison: Moduli of curves, Springer GTM 187 (1998) page 139.

5 Mg,n and Mg,n,d(M,J)

For a compact surface Σ (real dimension two) of genus g define the Riemann moduli space as ! n Mg,n = J (Σ) × Σn /Diff(Σ), Σn := Σ \ ∆ so a point of Mg,n is the isomorphism class [Σ, s∗, j] of a marked Riemann surface (Σ, s∗, j), i.e. j ∈ J (Σ) is a complex structure on Σ and s∗ = (s1, s2, . . . , sn) ∈ Σn is sequence of distinct points of Σ. For an almost complex manifold (M,J) and d ∈ H2(M)

Mg,n,d(M,J) = Bg,n,d(M,J)/Diff(Σ) where where an object in Bg,n,d(M,J) is a tuple (Σ, s∗, j, v) with (Σ, s∗, j) as above and v :Σ → M a (j, J)-holomorphic map such that v∗[Σ] = d. 6 Teichm¨uller space - I

We assume that the objects of J (Σ) × Σn are stable. This is the case precisely when

χ := 2 − 2g − n < 0.

In this case the identity component G := Diff0(Σ) of Diff(Σ) acts freely on P := J (Σ) × Σn and we have a principal fiber bundle G → P → B := P/G.

The base B is the Teichm¨uller space Tg,n. (Earle and Eells - 1969)

7 Teichm¨uller space - II

The tangent space to the (infinite dimensional) manifold P at a point (j, s∗) is = Ω0,1(Σ Σ) Σ Σ Σ T(j,s∗)P j ,T × Ts1 × Ts2 × · · · × Tsn . The linear operators defined by T(j,s∗)P → T(j,s∗)P (ˆ, ˆs1, ˆs2,..., ˆsn) → (jˆ, j(s1)ˆs1, j(s2)ˆs2, . . . , j(sn)ˆsn) determine an integrable complex structure on P. One can prove that the action of G := Diff0(Σ) on P admits a local (but not a global) holomorphic slice through every point. Thus Teichm¨uller space is a complex manifold. The action of Diff(Σ) is only semifree (the isotropy groups are finite) so the quotient

Mg,n := Tg,n/Γg,n, Γg,n := Diff(Σ)/Diff0(Σ) is a complex orbifold. (The discrete group Γg,n is the so-called mapping class group.) 8 The Universal Family

The associated fiber bundle

Q := P ×G Σ → B has fibers isomorphic to Σ and is equipped with n disjoint sections

Si := {[j, s1, s2, . . . , sn, z]: z = si}. It is a universal Riemann family as follows. Given any other marked Riemann family (P → A, R1,R2,...,Rn) and any fiber isomorphism

(P,R1,R2,...,Rn)a → (Q, S1, S2,..., Sn)b there is a unique extension

(P,R1,R2,...,Rn) → (Q, S1, S2,..., Sn) to a (sufficiently small) neighborhood of a in A. 9 ¯ ¯ Mg,n and Mg,n,d(M,J)

We will describe a compactification M¯ g,n of Mg,n called the Deligne-Mumford moduli space. It is a stratified space. The top stratum is Mg,n and the other strata are isomorphism classes [Σ, s∗, ν, j] of marked nodal Riemann surfaces (Σ, s∗, ν, j) of arith- metic genus g with n marked points s∗ as before.

¯ There is a similar compactification Mg,n,d(M,J) of Mg,n,d(M,J) due to Konsevitch consisting of isomorphism classes [Σ, s∗, ν, f] of stable marked nodal maps. The marked nodal map (Σ, s∗, ν, f) is stable if and only if each ghost component (=component on which f is constant) is a stable marked nodal Riemann surface.

10 Families

A Riemann family is a proper holomorphic map π : Q → B with no critical fibers and dimC(Q) = dimC(B) + 1. A critical point q is called nodal iff there are local coordinates near q such that

π(x, y, t1, . . . , tm−1) = (xy, t1, . . . , tm−1), x(q) = y(q) = 0. A nodal family is a Riemann family such that every critical point is nodal. A marked nodal family is a nodal family π : Q → B equipped with pairwise disjoint sections S1,...,Sn disjoint from the critical points. Each fiber is a marked nodal Riemann surface. A morphism is a holomorphic commutative diagram

(P,R∗) / (Q, S∗)

  A / B where the top horizontal arrow is an isomorphism on each fiber. 11 Universal Families

An unfolding of a marked nodal Riemann surface (Σ, s∗, ν, j) is a germ (π : Q → B,S1, . . . .Sn, b) at a point b ∈ B such that the fiber (π : Q → B,S1, . . . .Sn, b)b is isomorphic to (Σ, s∗, ν, j). An unfolding (Q → B,S∗, b) is universal iff for every other unfolding (P → A, R∗, a) every fiber isomorphism (P,R∗)a → (Q, S∗)b ex- tends uniquely to the germ of a morphism (P,R∗, a) → (Q, S∗, b). A universal family of type (g, n) is a marked nodal family such that

• each its unfoldings is universal,

• each fiber is of type (g, n), and

• each marked marked nodal Riemann surface of type (g, n) is isomorphic to at least one fiber. 12 Etale Groupoids

A groupoid is a category where each morphism is an isomor- phism. An etale groupoid is a groupoid where the objects and morphisms are manifolds, the structure maps are smooth, and the source and target maps are local diffeos. (Example: Open cover of a manifold.) Two etale groupoids are Morita equiv- alent iff they have a common refinement (and hence have the same orbit space.) An orbifold is a Morita equivalence class of etale groupoids.

Each universal family of type (g, n) determines a complex etale groupoid where the base is the manifold of objects and the fiber isomorphisms are the morphisms. Any two such are Morita equiv- alent. The orbifold thus determined is the Deligne-Mumford orbifold M¯ g,n. 13 Main Theorems

• Every stable marked nodal Riemann surface has a universal unfolding.

• If χ < 0, there is a universal family of type (g, n).

• The corresponding etale groupoid is proper (so its orbit space is Hausdorff).

• If χ < 0, the Deligne Mumford orbifold M¯ g,n is compact. (Mumford, Gromov.)

14 Remarks

To show that a stable marked nodal Riemann surface has a uni- versal unfolding we construct a universal unfolding for the domain of a desingularization using a slice. We add a parameter z for each node and insert xy = z with appropriate identifications. We then prove that the construction gives an unfolding (Q → B,S∗, b) which is “infinitesimally universal”. Essential ingredients are a difficult theorem asserting the equivalence of universality and infinitesimal universality and nonlinear Hardy decompositions.

15 Nonlinear Hardy Decompositions - I

The main problem is that a nodal unfolding is not locally triv- ial: the homotopy type of the fiber changes. Decompose the total space P of an unfolding (P → A, R∗, a0) into two manifolds intersecting in their common boundary, say P = P 0 ∪ P 00, where P 0 → A is trivial and P 00 is a small neighborhood of the nodal set. A holomorphic map defined on Pa is a map defined on the com- 0 00 mon boundary of Pa and Pa which extends holomorphically to 0 00 both parts Pa and Pa . The technical difficulty is the dependence on the parameter a.

16 Nonlinear Hardy Decompositions - II

Restrict to a fiber Σ = Pa. Decompose

Σ = Σ0 ∪ Σ00, Γ := Σ0 ∩ Σ00 = ∂Σ0 = ∂Σ00 where Σ00 is a small neighborhood (disjoint union of disks) of the nodal set. We define an open subset W of maps from Γ to Q and show that

U0 := {u|Γ: u ∈ Hol(Σ0,Q)}, U00 := {u|Γ: u ∈ Hol(Σ00,Q)} are submanifolds of W. If (Q → B,S∗, b) is an unfolding and u :Σ → Qb is an isomorphism, the unfolding is universal if and only if the submanifolds U0 and U00 intersect transversally in the single point u|Γ.

17 The Local Model

For a ∈ D and b ∈ C denote 2 2 Na := {(x, y) ∈ D : xy = a},Qb := {(x, y) ∈ C : xy = b}. A holomorphic map

(ξ, η): Na → Qb s determines a point (a, ξ, η, b) ∈ D×H ×H ×C where H = H (T, C). Each element of H is a Laurent series and the pair (ξ, η) satisfies ξ(x)η(a/x) = b. Theorem: The set of all such quadruples is a complex submanifold.

18 The Local Model for Maps

m The problem of holomorphic maps v : Nz → C is easier. The set n o s+1/2 m N := (ξ, η, z) ∃ v ∈ Hol (Nz, C ) s.t. ξ = v|∂1Nz, η = v|∂2Nz s m s m is a complex submanifold of H (T, C ) × H (T, C ) × D. If z 6= 0 then n (ξ, η, z) ∈ N ⇐⇒ η−n = z ξn for all n ∈ Z where X n X n ξ(x) =: ξnx , η(y) =: ηny . n∈Z n∈Z These conditions can be written as n n η−n = z ξn, ξ−n = z ηn, n ≥ 0 and are nice at z = 0. 19 Hyperbolic Rest Points

The equations n η−n = z ξn for all n ∈ Z t+iθ defining N say thatη ¯ = Φ (ξ) for a linear (R × T)-action Φ. The flow Φt has a hyperbolic restpoint at the origin. The { }t∈R analogous system for nonintegrable complex structures J also has a hyperbolic rest point at the origin. Does this give a proof that N is a submanifold in the nonintegrable case?

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