Hamiltonian Group Actions Today's Plan: • O(L) (Bundle Over CP 1

Home , CP1 (classification), Tautological bundle

Hamiltonian group actions Wednesday Mar.3, 2010; 12-2 pm

Today’s plan: •O(`) (bundle over CP1.) • Blowups and their eﬀect on topology and combinatorics • Cutting • First Chern class

The bundle O(`). (C∞) complex line bundles over CP1, up to isomorphism, are classiﬁed by the integers. In fact, the natural map

1 1 holomorphic line bundles over CP → complex line bundles over CP induces a bijection on equivalence classes. The bundle that corresponds to the integer ` is often called O(`), though strictly speaking this name denotes its space of holomorphic sections. The bundle can be constructed as follows.

2 ` × (C r {0}) ×C C−` 3 [z, w, v] = [zλ, wλ, λ v]

CP1 A holomorphic section of this bundle can be written as

` X j `−j v = f(z, w) = ajz w . j=0 Thus, the dimension of the space of holomorphic sections is ` + 1. A generic holomorphic section has ` simple zeros, of which each contributes 1 to the intersection number of this section with the zero section. Thus: ` = the self intersection number of the zero section.

The symplectic picture. Recall that the symplectic Delzant construction and the complex Delzant (Audin!) construction yield isomorphic manifolds. (We stated this without proof.) [Picture: the bottom part of a Hirzebruch trapezoid with right edge of slope −1/`.] We found a neighborhood of the preimage of the bottom edge that is isomorphict to O(`). Examples: [Picture of Hirzebruch trapezoid with each edge labelled by the self intersection of its preimage: from the right edge, counterclockwise: 0; −`; 0; `.] 1 2

[Same picture, with the bottom left corner chopped off. Labels of the edges, from the one on the right, counterclockwise: 0, −`, −1, −1, ` − 1. by the self intersection of its preimage: bottom edge – `; top edge – −`; left edge – 0; right edge – 0.] A concrete example: The tautological bundle over CP1, which we denote Taut, is O(−1). Let us see how. The fiber of Taut over [z, w] is C · (z, w). We have an isomorphism 2 × ∼ C r {0} ×C C1 = Taut via [z, w, v] 7→ the vector v · (z, w) in the fibre over [z, w]. (This is well defined: [zλ, wλ, λ−1v] maps to the same thing.) The map [z, w, v] 7→ v · (z, w) from Taut to C2 is a diffeomorphism above C2 r {0} and ∼ 1 2 takes the zero section (= CP ) to the origin. Through this map, Taut is the blowup of C at the origin. [Picture: a region bounded on the left by a vertical line, on the bottom by a horizontal line, and on the right by a line of slope 1.] Another concrete example: The tangent bundle of CP1 is O(2). Let us see how. Note that

1 2 T[z,w]CP = T(z,w)C /C · (z, w) 3 (u1, u2) O

2 = T(zλ,wλ)C /C · (zλ, wλ) 3 (λu1, λu2). Define 2 1 × C r {0} ×C C−2 → T CP by ( 2 (0, v/z) in T(z,w)C /C · (z, w) if z 6= 0 [z, w, v] 7→ 2 (−v/w, 0) in T(z,w)C /C · (z, w) if w 6= 0. Note: • This is unambiguous if both z and w are 6= 0; • This gets multiplied by λ (as it should) if we replace (z, w, v) by (zλ, wλ, λ2v). Symplectic picture: [Piece of a region bounded on the left by a vertical line, on the bottom by a horizontal line, and on the right by a line of slope = −1/2. Only shade a neighborhood of the “floor”.] or [similar, but the left line has slope 1 and the right line has slope −1.] Vectors along the edges that emanate from a corner are the weight vectors for the torus action. The standard S1 action on CP1 has weights 1 at the south pole and −1 at the north pole (with appropriate sign conventions). On the tangent bundle, at the origin of the tangent at the south and north poles, the weights are 1 and −1 respectively along the tangent to the zero section, and they are the same – 1 and −1 – along the fibre. (Indeed, 3 the tangent to the fibre at its origin is just another copy of the tangent to the zero section at that point.) There is also the fibrewise S1 action; its weights are +1 on each fiber and 0 in the tangent directions.

Symplectic toric blowup in (real) dimension 4 (Terminology: “toric surface”.)

[Picture of “corner chopping” of size δ.] Topologically: remove an open ball and collapse the boundary. Remarks. Under this operation, (1) The area decreases by δ2/2. (2) The perimeter decreases by δ. Before the blowup, there must have been two adjacent edges of lengths > δ. It follows that 1 the perimeter after the blowup ≥ the perimeter before the blowup. 2 (3) The length of an edge before the blowup is a nonnegative integer linear combination of the lengths of edges after the blowup.

Corollary: (Lemma 2.16 of the paper by Karshon-Kessler-Pinsonnault) Let ∆0,..., ∆s be the polygons obtained by a sequence of blowups of sizes δ1, . . . , δs. P 1 2 (1) area ∆s = area ∆0 − δ . 2 i P (2) perimeter ∆s = perimeter ∆0 − δi. s (3) For every i, δi < perimeter ∆i < 2 perimeter ∆s. (4) The length of every edge of ∆i is a nonnegative integer combination of the lengths of the edges of ∆s.

(Didn’t show in class today; will discuss later: Let M = M∆s and ω = ω∆s . + – Item (3), plus a lemma that uses the fact that b2 (M) = 1, give an a-priori bound on the numbers δi in terms of the symplectic manifold (M, ω). – This and the fact that δi is the area of the exceptional divisor imply that there exists a finite set D(M, ω) such that every δi must be in this set. – If ∆0 is a Hirzebruch trapezoid corresponding to integers a and b and slope `, from the numbers δi and from (M, ω) we can determine a + b and a · b and hence determine a and b. Given a and b, there are finitely many posisbilities for the slope `. This gives “finiteness”.) (A note made while the notes are being typed in the summer: recently, Dusa McDuff has a proof of “finiteness” of toric actions in *any* dimensions.)

Blowup - topologically: [Picture: region bounded by vertical line on left, segment of slope -1 on left/bottom, horizonal line on bottom.] 4

A neighborhood of the “exceptional divisor” is O(−1) (same picture). It is a disc bundle over CP1. This is the same as O(1) with the orientation reversed. In turn, O(1) can be identiﬁed with CP2 r {point}. So ... topologically, the blowup of M = CP2]M, where CP2 denotes CP2 with its orientation reversed and where ] denotes connected sum. This 1 = disc bundle over CP ∪ (M r {point}) , glued along an annulus bundle over CP1 (which is a neighbourhood of S3 in C2). Thus, Mf := the blowup of M = U ∪ V where 1 3 U ≈ CP ,V ≈ M r {point} ,U ∩ V ≈ S . By Van Kampen:

π1(U ∪ V ) = π1(U) ?π1(U∩V ) π1(V ) is the amalgamated free product

= (trivial) ?trivial π1(M r {point})

= π1(M). Thus,

π1(Mf) = π1(M). By Mayer Vietoris:

/ Hn(U ∩ V ) / Hn(U) ⊕ Hn(V ) / Hn(U ∪ V )

BCED / Hn−1(U ∩ V ) / ... GF When n = 2,@A we get 3 1 3 H2(S ) −−−→ H2(CP ) ⊕ H2(M r {point}) −−−→ H2(Mf) −−−→ H1(S ) . | {z } | {z } | {z } | {z } =0 =Z =H2(M) =0 We deduce that H2(Mf) = Z ⊕ H2(M) where the inclusion H2(M) ,→ H2(Mf) is obtained by choosing cycles away from the surgery ∼ 1 and where the new generator is the exceptional divisor = [CP ]. Symplectic cutting [Picture: momentum map for an S1 action ....] Topologically, 1 (1) Mcut = {Φ > 0} t {Φ = 0}/S . 5

The ﬁrst term on the right, {Φ > 0}, with its symplectic structure restricted from M, is 1 a symplectic open subset of Mcut. The second term on the right, {Φ = 0}/S , with the reduced symplectic structure induced from M, is a symplectic submanifold of Mcut. – What is the smooth structure on this disjoint union? Take the S1 momentum map M × C / R

(m, z) / Φ(m) − |z|2/2. Set 2 1 Mcut = (M × C)red = Φ(m) = |z| /2 /S . We will now show how to express this as the disjoint union of an open subset and a codimension two submanifold as in (1), with their symplectic structures induced from M as required. We start with the inclusion in Mcut of the open subset of M. {Φ(m) = |z|2/2} 6 ei+ nnn nnn nnn nnn {Φ > 0} / Mcut i+

i+ p m / [m, 2Φ(m)]. We have ∗ ∗ i+ωcut = ei+(ω ⊕ ωC) = ω. We now show the inclusion in Mcut of the symplectic reduction of M.

eired {Φ = 0} / {Φ(m) = |z|2/2} m7→(m,0)

π πcut Mred / Mcut. ired We have ∗ ∗ ∗ ∗ π iredωcut = eiredπcutωcut ∗ = eired(ω ⊕ ωC) from the deﬁnition of the cut space as a reduction of M × C = i∗ω ∗ = π ωred from the deﬁnition of ωred as a reduction of M. A symplectic blowup is a special case of symplectic cutting. 6

Reminders on the ﬁrst Chern class. Let L → Σ be a complex line bundle over an oriented closed two-manifold. Then 2 c1(L) = its Euler class ∈ H (Σ) = the Poincare dual of the zero set of a generic section or trivialize the bundle over every half of the base and take the degree of the transition function. [picture] E.G. for the bundle L over CP1 whose space of sections is O(k), Z c1(L) = k. 1 CP If E → M is a complex vector bundle, then c (E) = c (∧topE). This is unchanged by a 1 1 C continuous deformation of the complex bundle. ∗ ∗ For an arbitrary base, c1(E) is characterized by the i c1(E) = c1(i E) for every pullback diagram i∗E / E

/ Σ i B.

Now, for a symplectic vector bundle, Take c1 with respect to any choice of a ﬁbrewise compatible Hermitian structure. (Recall that the space { ﬁbrewise compatible Hermitian structures } is contractible.) Let Σ2 ⊂ M 4 be a smoothly embedded symplectic two-manifold.

TM = T Σ ⊕ νΣ.

c1(νΣ) = Σ · Σ.

c1(T Σ) = 2 − 2 genus Σ. So c1(TM)(Σ) = 2 − 2 genus Σ + Σ · Σ.

Later we will see that, in a toric surface, the preimages of edges generate H2. Also, X P.D. (c1(TM)) = [Cj] j where Cj denotes the preimage of jth face.