Let $M$ be a smooth projective variety and $\mathbf{D}$ an ample normal crossings divisor. From topological data associated to the pair $(M,\mathbf{D})$, we construct, under assumptions on Gromov–Witten invariants, a series of distinguished classes in symplectic cohomology of the complement $X=M\setminus \mathbf{D}$. Under further ‘topological’ assumptions on the pair, these classes can be organized into a log(arithmic) PSS morphism, from a vector space which we term the logarithmic cohomology of $(M,\mathbf{D})$ to symplectic cohomology. Turning to applications, we show that these methods and some knowledge of Gromov–Witten invariants can be used to produce dilations and quasi‐dilations (in the sense of Seidel–Solomon [Geom. Funct. Anal. 22 (2012) 443–477]) in examples such as conic bundles. In turn, the existence of such elements imposes strong restrictions on exact Lagrangian embeddings, especially in dimension 3. For instance, we prove that any exact Lagrangian in any complex three‐dimensional conic bundle must be diffeomorphic to a product $S^1\times {\mathrm{\Sigma }}_g$ or a connect sum ${\#}^n{S}^1\times S^2$.

中文翻译：

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对数PSS态射及其在Lagrangian嵌入中的应用

让 $\mathrm{中号}$ 成为平滑的投射变体，并且 $\mathbf{d}$一个正常的交叉口除数。从与该对关联的拓扑数据$（\mathrm{中号}，\mathbf{d}）$，我们在Gromov–Witten不变量的假设下，构造了补码的辛同调性的一系列杰出类 $X=\mathrm{中号}\setminus \mathbf{d}$。在该对上的进一步“拓扑”假设下，可以将这些类从向量空间组织为对数（算术）PSS态射，我们称其为对数同调。$（\mathrm{中号}，\mathbf{d}）$辛同调。谈到应用，我们表明，这些方法和格罗莫夫-威滕不变量的一些知识可以用于生产扩张术和准扩张术（在赛德尔所罗门的意义[的Geom。功能该分析。22（2012）443-477] ），例如锥形束。反过来，这些元素的存在对精确的拉格朗日嵌入（尤其是在维度3上）施加了严格的限制。例如，我们证明了在任何复杂的三维圆锥曲线束中的任何精确的拉格朗日必须对产品求微分。${\mathrm{小号}}^{\mathrm{1个}}\times {\mathrm{\Sigma }}_G$ 或连接总和 ${＃}^{ñ}{\mathrm{小号}}^{\mathrm{1个}}\times {\mathrm{小号}}^2$。