Symplectic fillings of asymptotically dynamically convex manifolds II-\(k\)-dilations (Q2161291)

The paper under review is the second in a series of two articles by the author on symplectic fillings of asymptotically dynamically convex (ADC) contact manifolds. As a generalization of a notion of dynamical convexity given by positivity of index of Reeb orbits [\textit{H. Hofer} et al., Ann. Math. (2) 148, No. 1, 197--289 (1998; Zbl 0944.37031)], \textit{O. Lazarev} [Geom. Funct. Anal. 30, No. 1, 188--254 (2020; Zbl 1436.53055)] defined the class of \textit{asymptotically dynamically convex contact structures} (or ADC for short). For a contact manifold \((Y,\xi)\) with \(c_1(\xi)=0\) and any contact form \(\alpha\) of \(\xi\), let \(\mathcal{P}^{<D}(\alpha)\) denote the set of contractible Reeb orbits of period smaller than \(D\). After choosing a global trivialization of the bundle \(\xi\) there is Conley-Zehnder index \(\mu_{XZ}(x)\in\mathbb{Z}\) associated with each contractible non-degenerate Reeb orbit \(x\). The (SFT) \textit{degree} of \(x\) is defined to be \(\deg(x) :=\mu_{CZ}(x)+(\dim Y+ 1)/2-3\). A contact manifold \((Y,\xi)\) with \(c_1(\xi)=0\) is called \textit{asymptotically dynamically convex} if there exists a sequence of non-increasing contact forms \(\alpha_1\ge\alpha_2\ge\alpha_3\cdots\) for \(\xi\) and increasing positive numbers \(D_1<D_2<D_3\cdots\) going to infinity such that all elements of \(\mathcal{P}^{<D_k}(\alpha_k)\) are nondegenerate and have positive degree. In order to define a `\(q\)-analogue' of the classical Picard-Lefschetz formula for the action of a Dehn twist on the middle-dimensional cohomology of a variety via a `categorification' of the Picard-Lefschetz formula by Lagrangian Floer cohomology, \textit{P. Seidel} and \textit{J. P. Solomon} [Geom. Funct. Anal. 22, No. 2, 443--477 (2012; Zbl 1250.53078)] introduced the notion of the symplectic dilation. A degree-one element of the symplectic cohomology \(SH^\ast(W)\) is called a \textit{symplectic dilation} if it is mapped to the canonical identity in \(SH^0(W)\) by the BV (Batalin-Vilkovisky or loop rotation) operator \(\triangle:SH^\ast(W)\to SH^{\ast-1}(W)\). For a ADC contact manifold \(Y\) with an exact filling \(W\) with vanishing first Chern class, the first one in this series [\textit{Z. Zhou}, J. Topol. 14, No. 1, 112--182 (2021; Zbl 1491.57024)] constructed two structure maps on the positive symplectic cohomology \(SH^\ast_+(W)\) of \(W\), a Floer-theoretic invariant of Liouville domains, proved that the existence of \textit{symplectic dilations} are properties independent of the filling for ADC manifolds, and used their properties to obtain various topological applications on symplectic fillings, including the uniqueness of diffeomorphism types of fillings for many contact manifolds. In order to generalize Seidel's results on the homological essentiality of odd-dimensional Lagrangian spheres to a more general class of Weinstein manifolds \(W\), \textit{Y. Li} [``Exact Calabi-Yau categories and odd-dimensional Lagrangian spheres'', Preprint, \url{arXiv:1907.09257}] gave a generalization of the symplectic dilation. He called a cohomology class \(\tilde{b}\) in the degree-one \(S^1\)-equivariant symplectic cohomology \(SH^1_{S^1}(W)\) a \textit{cyclic dilation} if \(\mathbf{B}(\tilde{b})=h\in SH^0(W)^{\times}\), where the map \(\mathbf{B}:SH^\ast_{S^1}(W)\to SH^{\ast-1}(W)\) is the connecting map in Gysin's long exact sequence \[ \cdots\to SH^{\ast-1}(W)\stackrel{\mathbf{I}}{\longrightarrow}SH^{\ast-1}_{S^1}(W) \stackrel{\mathbf{S}}{\longrightarrow} SH^{\ast+1}_{S^1}(W)\stackrel{\mathbf{B}}{\longrightarrow}\cdots \] [\textit{F. Bourgeois} and \textit{A. Oancea}, J. Topol. Anal. 5, No. 4, 361--407 (2013; Zbl 1405.53121)], where the composition \(\mathbf{B}\circ\mathbf{I}\) gives the BV operator \(\triangle\). The cyclic dilation has an algebraic counterpart, the so-called exact Calabi-Yau structure on a homologically smooth \(A_\infty\)-category. Yin Li showed that an exact Calabi-Yau structure on the wrapped Fukaya category \(\mathcal{W}(W)\) of a Liouville manifold \(W\) is related to a cyclic dilation \(\tilde{b}\in SH^1_{S^1}(W)\) via the cyclic open-closed string map defined by \textit{S. Ganatra} [``Cyclic homology, \(S^1\)-equivariant Floer cohomology, and Calabi-Yau structures'', Preprint, \url{arXiv:1912.13510}]. In particular, the wrapped Fukaya category \(\mathcal{W}(W)\) of a nondegenerate Liouville manifold \(W\) admits the structure of an exact Calabi-Yau structure if and only if there exists a cyclic dilation \(\tilde{b}\in SH^1_{S^1}(W)\). The concept of \(k\)-dilation (and more general \(k\)-semi-dilation) for Liouville domains introduced in the paper under review is a structural generalization of the symplectic dilation, and so closely related to the cyclic dilation. In fact, for a Liouville manifold \(W\), the vanishing of symplectic cohomology of \(W\) is equivalent to the existence of a \(0\)-dilation, and the symplectic dilation is equivalent to a \(1\)-dilation. The existence of a cyclic dilation for \(h=1\) is equivalent to the existence of a \(k\)-dilation for some \(k\). Proposition 5.1 shows that the cotangent bundle of an oriented, rationally-inessential and spin compact manifold admits a \(k\)-dilation for some \(k\). The paper's first main theorem (Theorem A) concludes that the Milnor fiber of the singularity \(x_0^k+\cdots+x_k^m=0\) admits a \((k-1)\)-dilation but not a \((k-2)\)-dilation for \(m\ge k\) over \(\mathbb{Q}\). Therefore the concept of \(k\)-dilation is a non-trivial extension of symplectic dilation. For a Liouville domain \(W\), the \textit{order of dilation \(\mathrm{D}(W)\)} of it is the minimal number \(k\) such that \(W\) admits a \(k\)-dilation over \(\mathbb{Q}\); and the \textit{order of semi-dilation} \(\mathrm{SD}(W)\) of \(W\) is the minimal number \(k\) such that \(W\) admits a \(k\)-semi-dilation over \(\mathbb{Q}\). \(\mathrm{D}(W)\) and \(\mathrm{SD}(W)\) are \(\infty\) if the structure does not exist. These two numerical invariants serve as measurements of the complexity of Liouville domains, and are independent of certain fillings if the contact boundary are ADC. Actually, the existence of \(k\)-(semi)-dilation is a uniruled condition. Theorem 3.27 concludes that an exact domain \(W\) admitting a \(k\)-semi-dilation is \((1, \Lambda)\)-uniruled for some \(\Lambda>0\). Here \(W\) is said to be \textit{\((1, \Lambda)\)-uniruled} for \(\Lambda>0\) if for any almost complex structure \(J\) that is convex near \(\partial W\) and any point \(e\in \mathrm{Int}(W)\) there exists a proper holomorphic curve \(u:S\to\mathrm{Int}(W)\) passing through \(e\) such that \(\int_S u^\ast\omega\le\Lambda\) and \(H_1(S;\mathbb{Q})=0\). The existence of \(k\)-(semi)-dilation is also closely related to the Gutt-Hutchings capacity defined by \textit{J. Gutt} and \textit{M. Hutchings} [Algebr. Geom. Topol. 18, No. 6, 3537--3600 (2018; Zbl 1411.53062)]. In particular, a Liouville domain \(W\) has finite \(c_1^{GH}(W)\) if and only if \(W\) admits a cyclic dilation for \(h = 1\) or equivalently a \(k\)-dilation for some \(k\). In order to study persistence of \(k\)-dilations, the author uses the \(u\)-adic spectral sequence to construct a sequence of structural maps \(\Delta^k_+:\ker \Delta_+^{k-1}\to \operatorname{coker}\Delta_+^{k-1}\) and \(\Delta^k_{\partial}: \ker \Delta_+^{k}\to \operatorname{coker}\Delta_{\partial}^{k-1}\), where \(\Delta^0_+=0:SH^\ast_+(W)\to SH^\ast_+(W)\) and \(\Delta^0_{\partial}\) is the composition \(SH^\ast_+(W)\to H^{\ast+1}(W)\to H^{\ast+1}(Y)\). These structural maps may be used to reinterpret \(k\)-(semi)-dilation. Indeed, when \(c_1(W)=0\), Proposition 3.8 concludes that \(W\) carries a \(k\)-dilation iff \(1\in\operatorname{im}\Delta^k\) and \(W\) carries a \(k\)-semi-dilation iff there exists an element \(x\in H^\ast(\partial W)\) such that \(\pi_0(x)=1\) and \(x\in\operatorname{im}\Delta^k_{\partial}\), where \(\pi_0\) denotes the projection \(H^\ast(W)\to H^0(W)\) as well as the projection \(H^\ast_{S^1}(W)\to H^0(W)\). For a \textit{topologically simple} filling \(W\) of \(Y\), i.e., \(c_1(W)=0\) and \(\pi_1(Y)\to\pi_1(W)\) is injective, this paper's second main theorem (Theorem B) concludes that if \(Y\) is a ADC contact manifold, then \(\Delta^k_+\) and \(\Delta^k_{\partial}\) are independent of topologically simple Liouville fillings. As a corollary (Corollary C) the persistence of \(k\)-dilations is obtained if \(Y\) is an ADC contact manifold, that is, 1) the existence of \(k\)-semi-dilation is a property independent of topologically simple Liouville fillings \(W\); and 2) the existence of \(k\)-dilation is independent of Weinstein fillings \(W\) if \(\dim Y\ge 4k+1\) or \(\frac{1}{2}\dim W\) is odd. Finally, we mention an application (Corollary D): For two simply connected contact manifolds \(Y_1\), \(Y_2\) with topologically simple exact fillings \(W_1\), \(W_2\), respectively, if \(Y_1\) is ADC and \(\mathrm{SD}(W_2)>\mathrm{SD}(W_1)\), then there is no exact cobordism from \(Y_2\) to \(Y_1\) with vanishing first Chern class.