J-tamed inflation via tame to compatible deformations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911084968935424 |
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| author | Chakravarthy, Pranav Payette, Jordan Pinsonnault, Martin |
| author_facet | Chakravarthy, Pranav Payette, Jordan Pinsonnault, Martin |
| contents | We give a complete and self-contained exposition of the $J$-tame inflation lemma: Given any tame almost complex structure $J$ on a symplectic $4$-manifold $(M,ω)$, and given any compact, embedded, $J$-holomorphic submanifold $Z$, it is always possible to construct a deformation of symplectic forms $ω_t$ in classes $[ω_t]=[ω]+t\mathrm{PD}{Z}$, for $0\leq t$ less than an upper bound $0<T$ that only depends on the self-intersection $Z\cdot Z$. The original proofs of this fact make the unwarranted assumption that one can find a family of normal planes along $Z$ that is both $J$ invariant and $ω$-orthogonal to $TZ$ -- which amounts, in effect, to assuming the compatibility of $J$ and $ω$ along $Z$. We explain how the original constructions can be adapted to avoid this assumption when $Z$ has nonpositive self-intersection, and we discuss the difficulties with this line of argument in general to establish the full inflation when $Z$ has positive self-intersection. We overcome this problem by proving a `preparation lemma', which states that prior to inflation, one can isotope $ω$ within its cohomology class to a new form that still tames $J$ and which is compatible with $J$ along the submanifold $Z$. This preparation lemma can be regarded as an infinitesimal version of the "tamed-to-compatible" conjecture of S. K. Donaldson along an almost-complex submanifold $Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19110 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | J-tamed inflation via tame to compatible deformations Chakravarthy, Pranav Payette, Jordan Pinsonnault, Martin Symplectic Geometry Primary 53D35, Secondary 57R17, 53C15, 57R52 We give a complete and self-contained exposition of the $J$-tame inflation lemma: Given any tame almost complex structure $J$ on a symplectic $4$-manifold $(M,ω)$, and given any compact, embedded, $J$-holomorphic submanifold $Z$, it is always possible to construct a deformation of symplectic forms $ω_t$ in classes $[ω_t]=[ω]+t\mathrm{PD}{Z}$, for $0\leq t$ less than an upper bound $0<T$ that only depends on the self-intersection $Z\cdot Z$. The original proofs of this fact make the unwarranted assumption that one can find a family of normal planes along $Z$ that is both $J$ invariant and $ω$-orthogonal to $TZ$ -- which amounts, in effect, to assuming the compatibility of $J$ and $ω$ along $Z$. We explain how the original constructions can be adapted to avoid this assumption when $Z$ has nonpositive self-intersection, and we discuss the difficulties with this line of argument in general to establish the full inflation when $Z$ has positive self-intersection. We overcome this problem by proving a `preparation lemma', which states that prior to inflation, one can isotope $ω$ within its cohomology class to a new form that still tames $J$ and which is compatible with $J$ along the submanifold $Z$. This preparation lemma can be regarded as an infinitesimal version of the "tamed-to-compatible" conjecture of S. K. Donaldson along an almost-complex submanifold $Z$. |
| title | J-tamed inflation via tame to compatible deformations |
| topic | Symplectic Geometry Primary 53D35, Secondary 57R17, 53C15, 57R52 |
| url | https://arxiv.org/abs/2403.19110 |