Locality in the Fukaya category of a hyperkähler manifold

Solomon, Jake P.; Verbitsky, Misha

doi:10.1112/s0010437x1900753x

Cited by 11 publications

(7 citation statements)

References 37 publications

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“…Since the relative grading is zero, for generic almost complex structures, the moduli space of pseudo-holomorphic curves is empty; • Even if the two J-complex Lagrangians do not intersect transversely, for a generic value of θ ∈ R, if we consider the complex structure K(θ) = cos(θ)K + sin(θ)I, then there are no K(θ)-holomorphic strips with boundary on the Lagrangians; cf. [65]. Note that K(θ) is ω 3 -tame for θ close to 0.…”

Section: Conditions On Intersections Let Us Recall the Definition Of ...mentioning

confidence: 96%

A sheaf-theoretic model for SL(2,C) Floer homology

Abouzaid,

Manolescu

2017

Preprint

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Given a Heegaard splitting of a three-manifold Y , we consider the SL(2, C) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible representations. On that subset, the intersection of the Lagrangians is an oriented d-critical locus in the sense of Joyce. Bussi associates to such an intersection a perverse sheaf of vanishing cycles. We prove that in our setting, the perverse sheaf is an invariant of Y , i.e., it is independent of the Heegaard splitting. The hypercohomology of the perverse sheaf can be viewed as a model for (the dual of) SL(2, C) instanton Floer homology. We also present a framed version of this construction, which takes into account reducible representations. We give explicit computations for lens spaces and Brieskorn spheres, and discuss the connection to the Kapustin-Witten equations and Khovanov homology.

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Section: Conditions On Intersections Let Us Recall the Definition Of ...mentioning

confidence: 96%

A sheaf-theoretic model for SL(2,C) Floer homology

Abouzaid,

Manolescu

2017

Preprint

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“…Then for all but a countable number of complex structures L ∈ H, all compact complex subvarieties of X L are trianalytic. This result was often applied in symplectic geometry to infer that a general almost complex structure on X does not support complex curves ( [EV], [SV2]). Proposition 1.3 follows directly from Proposition 1.5.…”

Section: Trianalytic Subvarieties In Hyperkähler Manifoldsmentioning

confidence: 99%

Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds

Anna¹,

Verbitsky²

2021

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A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere S 2 of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold M and a generic complex structure L ∈ S 2 , the complex manifold (M, L) has algebraic dimension 0. A stronger result is proven when the hypercomplex nilmanifold is abelian. Consider the Lie algebra of left-invariant vector fields of Hodge type (1,0) on the corresponding nilpotent Lie group with respect to some complex structure I ∈ S 2 . A hypercomplex nilmanifold is called abelian when this Lie algebra is abelian. We prove that all complex subvarieties of (M, L) for generic L ∈ S 2 on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.

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“…A strong form of this fact will play a crucial role for us here: Theorem 4. [46] Let W be a complete hyperkähler manifold, and L 1 , L 2 , . .…”

Section: Floer Theory In Hyperk äHler Manifoldsmentioning

confidence: 99%

Microsheaves from Hitchin fibers via Floer theory

Shende¹

2021

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Consider a smooth component of the moduli space of stable Higgs bundles on which the Hitchin fibration is proper. We record in this note the following corollaries to existing results on Floer theory and Fukaya categories. (1) Any any smooth Hitchin fiber determines a microsheaf on the global nilpotent cone. (2) Distinct fibers give rise to orthogonal microsheaves. (3) The endomorphisms of the microsheaf is isomorphic to the cohomology of the Hitchin fiber.Any sheaf on the moduli stack of bundles which is microsupported in the nilpotent cone restricts (by definition) to a microsheaf on the locus of stable and nilpotent Higgs bundles. Conversely, such microsheaves can be restricted to the moduli of stable bundles, and then extended to the stack of all bundles. We expect (but do not prove) that our microsheaves should restrict from, and extend to, the Hecke eigensheaves of the geometric Langlands programme. HITCHIN FIBERS AND HECKE EIGENSHEAVESLet us recall what Hecke eigensheaves are, and why one might hope to get them from fibers of the Hitchin system. More detailed discussions can be found in e.g. [7,32,15]. This section is purely motivational and logically unrelated to the results presented in the remainder of the article.Let C be a smooth compact complex curve, G a reductive group, and Bun G (C) the moduli of G-bundles on C. We write [X] for a category of sheaves on X, whose precise nature is irrelevant here. There are 'Hecke operators'given by convolution with respect to a correspondence parameterizing pairs of bundles which are isomorphic away from a single point c ∈ C, and whose difference at this point is controlled by a cocharacter µ of G.The geometric Langlands conjecture asserts, in particular, the existence of certain F ∈ [Bun G (C)] which are 'Hecke eigensheaves' in the sense that H µ (F ) = F ρ µ (χ), where χ is a G ∨ local system on C, and ρ µ is the representation corresponding to the character µ of G ∨ .The Hecke action transforms microsupports (or characteristic cycles) by setwise convolution with the conic Lagrangian ss(H µ ) ⊂ T * Bun G (C) × T * Bun G (C) × T * C; which is roughly the conormal to the image of the Hecke correspondence. This conormal respects the fibers of Hitchin's integrable system h :is the spectral cover corresponding to the point b ∈ B and representation µ.The above geometry suggests that any sufficiently functorial procedure relating Hitchin fibers to sheaves on Bun G (C) -and spectral curves to sheaves on C -may be expected to yield Hecke eigensheaves. Quantization is one such procedure: the semiclassical limit of a quantum state on M determines a Lagrangian submanifold of T * M (see e.g. [6]). Taking M = Bun G (C), and the corresponding Lagrangians to be the Hitchin fibers, quantizing the Hitchin system may be expected to yield Hecke eigensheaves, and indeed does [7]. Related approaches include [5,14,32,9,15]. This note explains how to use Floer theory to produce sheaves on Bun G (C) from Hitchin fibers.

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Locality in the Fukaya category of a hyperkähler manifold

Cited by 11 publications

References 37 publications

A sheaf-theoretic model for SL(2,C) Floer homology

A sheaf-theoretic model for SL(2,C) Floer homology

Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds

Microsheaves from Hitchin fibers via Floer theory

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