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!) con- struction 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.