Weinstein handlebodies for complements of smoothed toric divisors

Abstract

We study the interactions between toric manifolds and Weinstein handlebodies. We define a partially-centered condition on a Delzant polytope, which we prove ensures that the complement of a corresponding partial smoothing of the toric divisor supports an explicit Weinstein structure. Many examples which fail this condition also fail to have Weinstein (or even exact) complement to the partially smoothed divisor. We investigate the combinatorial possibilities of Delzant polytopes that realize such Weinstein domain complements. We also develop an algorithm to construct a Weinstein handlebody diagram in Gompf standard form for the complement of such a partially smoothed toric divisor. The algorithm we develop more generally outputs a Weinstein handlebody diagram for any Weinstein 4-manifold constructed by attaching 2-handles to the disk cotangent bundle of any surface F , where the 2-handles are attached along the co-oriented conormal lifts of curves on F . We discuss how to use these diagrams to calculate invariants and provide numerous examples applying this procedure. For example, we provide Weinstein handlebody diagrams for the complements of the smooth and nodal cubics in CP 2 . Contents 1. Introduction 2.1. Background on Weinstein domains 2.2. Weinstein Kirby calculus 2.3. Weinstein structures on the cotangent bundle 2.4. Morsification of the canonical Weinstein structure 3. Toric transformations 3.1. Algebraic viewpoint 3.2. Toric 4-manifolds 4. Weinstein structure on the complement of a partially smoothed toric divisor 4.1. Smoothing 4.2. The topology of the complements 4.3. The core, co-core, and attaching sphere from the toric perspective 4.4. An almost toric point of view 4.5. The global Weinstein structure on the complement 5. Combinatorial Possibilities for Delzant Polytopes and resulting Weinstein divisor complements 5.1. On the richness of Delzant family 5.2. Non-uniqueness of Weinstein manifolds from Delzant polytopes 5.3. The restrictiveness of the centeredness condition 6. Non-centered toric manifolds and obstructions 6.1. Exactness obstructions 6.2. Convexity obstructions 7. Cotangent bundles and handlebody diagrams 8. Lifting co-normals to Kirby diagrams 8.1. The procedure to obtain a standard Weinstein handlebody diagram

Smooth, Nonsymplectic Embeddings of Rational Balls in the Complex Projective Plane

“…Specifically the root of the tree is (1, 1, 1) which has a single child vertex (1, 1, 2). The vertex (1, 1, 2) also has a single child vertex (1,2,5). Any other vertex (𝑝 1 , 𝑝 2 , 𝑝 3 ) has two child vertices; the left child is (𝑝 2 , 𝑝 3 , 3𝑝 2 𝑝 3 − 𝑝 1 ), and the right child is (𝑝…”

“…We will give explicit handle descriptions corresponding to these almost toric pictures of ℂP 2 and transferring the cut (giving yet another proof that 𝑋 𝑝 1 ,𝑝 2 ,𝑝 3 is diffeomorphic to ℂ𝑃 2 ) in Section 5.3. In the literature the relationship between almost toric pictures and symplectic handlebodies has been studied for Weinstein domains [1], but not for closed symplectic manifolds. This handlebody description of transferring the cut is what lies behind our proof of Theorem 1.1 in Section 5.1.…”

Infinitely many planar fillings and symplectic Milnor fibers

“…For an example of an application of Theorem 1.6 and Corollary 5.1, see [2] where the authors showed that the Weinstein complement of certain divisors in toric 4-manifolds are not flexible.…”