Abstract
We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn–Pham type. We focus on dimensions 2 and 3, though the constructions generalise to higher ones. The techniques give significant latitude in controlling the homology class, Maslov class and monotonicity constant of the Lagrangian, and a range of possible diffeomorphism types; they are also explicit enough to be amenable to calculations of pseudo-holomorphic curve invariants. Applications include infinite families of monotone Lagrangian S^1 times Sigma _g in {mathbb {C}}^3, distinguished by soft invariants for any genus g ge 2; and, for fixed soft invariants, a range of infinite families of Lagrangians in Brieskorn–Pham hypersurfaces. These are generally distinct up to Hamiltonian isotopy. In specific cases, we also set up well-defined counts of Maslov zero holomorphic annuli, which distinguish the Lagrangians up to compactly supported symplectomorphisms. Inter alia, these give families of exact monotone Lagrangian tori which are related neither by geometric mutation nor by compactly supported symplectomorphisms.
Highlights
We present techniques for constructing families of compact monotone Lagrangians, including exact ones, in affine varieties
The diffeomorphism classes of closed orientable 3-manifolds admitting monotone Lagrangian embeddings into C3 are understood, by work of Evans and Kedra [19, Theorem B] building on Fukaya [25] and Damian [15]: if L is such a manifold, L is diffemorphic to S1 × g, where g is a surface of genus g; the S1 factor in a monotone S1 × g ⊂ C3 must have Maslov index two
For any sufficiently large r and s, we can construct an infinite family of homologous monotone Lagrangian S1 × g in Yr,s = {x3 + y3 + zr + ws = 1}, with fixed arbitrary Maslov class and monotonicity constant, distinct up to compactly supported symplectomorphisms of Yr,s
Summary
We present techniques for constructing families of compact monotone Lagrangians, including exact ones, in affine varieties. Our prefered setting will be Brieskorn–Pham hypersurfaces, i.e. affine varieties of the form. Cm+1, though our constructions will carry over to e.g. any affine variety which contains a suitable (truncated) Brieskorn–Pham hypersurface as a Stein submanifold. The article focuses on complex dimensions 2 and 3, though the techniques extend to give constructions in higher dimensions, which will briefly be discussed
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