Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1804.08013

Morse-Bott Split Symplectic Homology

Luís Diogo·Samuel T. Lisi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce a chain complex associated to a Liouville domain $(\overline{W}, dλ)$ whose boundary $Y$ admits a Boothby--Wang contact form (i.e. is a prequantization space). The differential counts cascades of Floer solutions in the completion $W$ of $\overline{W}$, in the spirit of Morse--Bott homology (as in work of Bourgeois, Frauenfelder arXiv:math/0309373 and Bourgeois-Oancea arXiv:0704.1039). The homology of this complex is the symplectic homology of the completion $W$. We identify a class of simple cascades and show that their moduli spaces are cut out transversely for generic choice of auxiliary data. If $X$ is obtained by collapsing the boundary along Reeb orbits and $Σ$ is the quotient of $Y$ by the $S^1$-action induced by the Reeb flow, we also establish transversality for certain moduli spaces of holomorphic spheres in $X$ and in $Σ$. Finally, under monotonicity assumptions on $X$ and $Σ$, we show that for generic data, the differential in our chain complex counts elements of moduli spaces that are transverse. Furthermore, by some index estimates, we show that very few combinatorial types of cascades can appear in the differential.

Explore related subjects

math.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Luís Diogo, Samuel T. Lisi. 2018-11-21. Morse-Bott Split Symplectic Homology. https://arxiv.org/abs/1804.08013

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Tightness of Chekanov's bound on displacement energy for some Lagrangian knots

By a classical theorem of Chekanov, the displacement energy, $e$, of a Lagrangian submanifold is bounded from below by the minimal area, $\hbar$, of pseudo-holomorphic disks with boundary on the Lagrangian. We compute $e$ and $\hbar$ for displaceable Chekanov tori in $\mathbb{C}P^n$, and for an infinite family of exotic tori in $\mathbb{C}^3$ constructed by Brendel. In these families, $e=\hbar$. We compare continuity properties of $e$ and $\hbar$ on the space of Lagrangians. This provides an example (suggested by Fukaya, Oh, Ohta, and Ono) where $e>\hbar$. Our calculations have further applications such as a new proof, inspired by work of Auroux, that Brendel's family of exotic tori consists of infinitely many distinct Lagrangians.

math.SG↗
arXiv2026

On intrinsic homological mirror symmetry for toric degenerations

This paper studies the Floer-theoretic aspects of homological mirror symmetry inspired by proposals of Perutz and Siebert and the Gross--Siebert intrinsic mirror symmetry program. Given a maximally unipotent degeneration of smooth projective Calabi--Yau manifolds over the punctured disk, we construct a ring using the fixed point Floer cohomology groups of the iterates of the monodromy of the degeneration equipped with the pair of pants product. Under the assumption that this ring is commutative, we can consider a candidate mirror family defined by the relative Proj construction. Further assuming that a smooth fiber $X_t$ contains a so-called tropical Lagrangian section, we construct a fully faithful embedding from the derived category of perfect complexes on our candidate mirror family into the Fukaya category of $X_t$. We verify both of these assumptions for certain Batyrev--Borisov toric degenerations, as well as some toric degenerations of Calabi--Yau threefolds coming from the Gross--Siebert reconstruction algorithm. These two geometric hypotheses are both phrased to support the general study of mirror symmetry for maximally unipotent degenerations of Calabi--Yau manifolds, largely reducing the symplectic inputs for proving homological mirror symmetry to the problem of constructing tropical Lagrangian sections.

math.SG↗
arXiv2026

Periodic Magnetic Geodesics with Every Low Energy: Existence and Localization

Let $(Q,g)$ be a Riemannian manifold equipped with a non-identically zero magnetic field represented by a closed $2$-form $β$. We allow $Q$ to be non-compact and do not assume that the metric $g$ is complete. We prove that if the magnetic strength, defined as the pointwise norm of $β$, attains a strict local maximum on a non-empty compact set $K$, then every sufficiently low energy level carries a contractible periodic magnetic geodesic of the pair $(g,β)$ localized near $K$. More precisely, such orbits exist in every neighborhood of $K$, and their lengths converge to zero with the energy. In particular, if $Q$ is compact, then every sufficiently small energy level carries a contractible periodic magnetic geodesic. We also show that, in general, neither the non-emptiness nor the compactness of $K$ can be omitted. Our proof relies on the calculus of variations of the Lagrangian action functional, and uses several new ideas. More precisely, we overcome: (i) the non-exactness of $β$ by restricting the minimax to the set of short loops; (ii) the non-completeness of $g$ by combining a compactification of $Q$ with Thom's Jet Transversality and a blow-up for sequences of magnetic geodesics with energy tending to zero; (iii) the possible non-compactness of Palais--Smale sequences by the positivity of the Ricci magnetic curvature for low energy established by the first-named author. Unlike previous work, Struwe's monotonicity argument cannot be used for our purposes, and we rely on a two-Lyapunov-function argument due to Abbondandolo and Majer.

math.SG↗