Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.15103

Estimate of Periodic Orbits of Degenerate Hamiltonians

Hao Jiao
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Arnold conjecture is a classic and important conjecture in the field of symplectic geometry, which describes the estimation of the number of periodic solutions of the Hamiltonian quantity on any symplectic manifold, which is equivalent to the symplectic version of Morse's theory. In the past development process, Andreas Floer was the first to propose Floer's theory, and solved the situation of non-degenerate Hamiltonian quantities under monotonic symplectic manifolds, and then Kenji Fukaya and Kaoru Ono solved the estimation of non-degenerate Hamiltonian quantities under rational coefficients in 1999. In 2023, Bai-Xu solved the estimation under the integer coefficient, and the non-degenerate version of Arnold's conjecture was completely resolved, but the degenerate version of Arnold conjecture did not progress so smoothly. In this paper, we have compiled some work on the degenerate version of Arnold conjecture, and based on these works, a new proof idea is proposed. By combining the action of quantum cohomology on Floer cohomology, combined with a universal energy estimation of the trajectories of $bar{partial}_H$-holomorphic in the paper, we can prove the degenerate Arnold conjecture in some special cases under the framework of Floer's theory.

Explore related subjects

math.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hao Jiao. 2026-06-13. Estimate of Periodic Orbits of Degenerate Hamiltonians. https://arxiv.org/abs/2606.15103

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

$b^k$-Symplectic Manifolds and $[Q,R]=0$

We study the geometric quantization of $b^k$-symplectic manifolds using the integrability of Lie algebroids. Using a groupoid index, we define a quantization for $b^k$-symplectic manifolds whose singular locus is a normal crossing divisor and which carry a Hamiltonian action of a compact connected Lie group, generalizing Guillemin--Miranda--Weitsman in a few directions. Firstly, we show this quantization is the index of a $\spinc$-Dirac operator, answering a question of theirs. In particular, it is a finite-dimensional virtual representation for every $k$, whereas their formal quantization is infinite-dimensional when the modular degrees are even. Secondly, our symplectic form can have singularities along hypersurfaces which can have normal crossings. Finally, we prove that quantization commutes with reduction for the Hamiltonian action of a possibly non-abelian compact connected Lie group, when the modular degrees are odd. In the case when the modular degrees are not odd, we give an example when $[Q,R]=0$ fails.

math.SG↗