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Algebraic capacities

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Abstract

We study invariants coming from certain optimisation problems for nef divisors on surfaces. These optimisation problems arise in work of the author and collaborators tying obstructions to embeddings between symplectic 4-manifolds to questions of positivity for (possibly singular) algebraic surfaces. We develop the general framework for these invariants and prove foundational results on their structure and asymptotics. We describe the connections these invariants have to embedded contact homology (ECH) and the Ruelle invariant in symplectic geometry, and to min–max widths in the study of minimal hypersurfaces. We use the first of these connections to obtain optimal bounds for the sub-leading asymptotics of ECH capacities for many toric domains.

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Notes

  1. That is, a symplectic 4-manifold such that \(\partial X=Y\) and \(\omega |_{\partial X}=\lambda \).

  2. Two regions \(\Omega ,\Omega '\subseteq {\mathbb {R}}^2\) are affine equivalent if there exists a map \(A\in {\text {GL}}_2({\mathbb {Z}})\) such that \(A(\Omega )\) is a lattice translate of \(\Omega '\).

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Acknowledgements

I would like to thank Julian Chaidez, David Eisenbud, Michael Hutchings, Dan Cristofaro-Gardiner, Vinicius Ramos, Jonathan Lai, Mengyuan Zhang, Antoine Song, Ɖan Daniel Erdmann-Pham, Tara Holm, and Stefano Filipazzi for many insightful and supportive conversations. I am especially glad for the range of specialisms represented by these researchers, and grateful to each of them for bearing with me as I attempted to translate some aspect of algebraic capacities into their world and back again.

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  1. Washington University, St. Louis, USA

    B. Wormleighton

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Correspondence to B. Wormleighton.

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Wormleighton, B. Algebraic capacities. Sel. Math. New Ser. 28, 9 (2022). https://doi.org/10.1007/s00029-021-00718-2

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