arXiv subjects

Ziming Nikolas Ma

Publications and source records attributed to Ziming Nikolas Ma.

14 recordsLinked to original sources

arXiv2026

A perturbative construction of primitive forms from log Landau-Ginzburg mirrors of toric manifolds

We introduce the notion of a logarithmic Landau-Ginzburg (log LG) model, which is essentially given by equipping the central degenerate fiber of the family of Landau-Ginzburg (LG) models mirror to a projective toric manifold with a natural log structure. We show that the state space of the mirror log LG model is naturally isomorphic to that of the original toric manifold. Following Li-Li-Saito, we give a perturbative construction of primitive forms by studying the deformation theory of such a log LG model, which involves both smoothing of the central degenerate fiber and unfolding of the superpotential. This yields a logarithmic Frobenius manifold structure on the base space of the universal unfolding. The primitive forms and flat coordinates we obtained are computable and closely related to the bulk-deformed Lagrangian Floer superpotential of a projective toric manifold, at least in the semi-Fano case.

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arXiv2024

Smoothing, scattering, and a conjecture of Fukaya

In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold $\check{X}$ and the multi-valued Morse theory on the base $\check{B}$ of an SYZ fibration $\check{p}: \check{X}\to \check{B}$, and the other between deformation theory of the mirror $X$ and the same multi-valued Morse theory on $\check{B}$. In this paper, we prove a reformulation of the main conjecture in Fukaya's second correspondence, where multi-valued Morse theory on the base $\check{B}$ is replaced by tropical geometry on the Legendre dual $B$. In the proof, we apply techniques of asymptotic analysis developed in our previous works to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety introduced in another of our recent work. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part $X_{\text{sf}} \subseteq X$ allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions.

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arXiv2024

Symmetry in Deformation quantization and Geometric quantization

In this paper, we explore the quantization of Kähler manifolds, focusing on the relationship between deformation quantization and geometric quantization. We provide a classification of degree 1 formal quantizable functions in the Berezin-Toeplitz deformation quantization, establishing that these formal functions are of the form $f = f_0 - \frac{\hbar}{4π}(Δf_0 + c)$ for a certain smooth (non-formal) function $f_0$. If $f_0$ is real-valued then $f_0$ corresponds to a Hamiltonian Killing vector field. In the presence of Hamiltonian $G$-symmetry, we address the compatibility between the infinitesimal symmetry for deformation quantization via quantum moment map and infinitesimal symmetry on geometric quantization acting on Hilbert spaces of holomorphic sections via Berezin-Toeplitz quantization.

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arXiv2022

Tropical Lagrangian multi-sections and smoothing of locally free sheaves over degenerate Calabi-Yau surfaces

We introduce the notion of tropical Lagrangian multi-sections over a $2$-dimensional integral affine manifold $B$ with singularities, and use them to study the reconstruction problem for higher rank locally free sheaves over Calabi-Yau surfaces. To certain tropical Lagrangian multi-sections $\mathbb{L}$ over $B$, which are explicitly constructed by prescribing local models around the ramification points, we construct locally free sheaves $\mathcal{E}_0(\mathbb{L},{\bf{k}}_s)$ over the singular projective scheme $X_0(B,\mathscr{P},s)$ associated to $B$ equipped with a polyhedral decomposition $\mathscr{P}$ and a gluing data $s$. We then find combinatorial conditions on such an $\mathbb{L}$ under which the sheaf $\mathcal{E}_0(\mathbb{L},{\bf{k}}_s)$ is simple. This produces explicit examples of smoothable pairs $(X_0(B,\mathscr{P},s),\mathcal{E}_0(\mathbb{L},{\bf{k}}_s))$ in dimension 2.

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arXiv2021

Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties

Given a degenerate Calabi-Yau variety $X$ equipped with local deformation data, we construct an almost differential graded Batalin-Vilkovisky (dgBV) algebra $PV^{*,*}(X)$, producing a singular version of the extended Kodaira-Spencer differential graded Lie algebra (dgLa) in the Calabi-Yau setting. Assuming Hodge-to-de Rham degeneracy and a local condition that guarantees freeness of the Hodge bundle, we prove a Bogomolov-Tian-Todorov--type unobstructedness theorem for smoothing of singular Calabi-Yau varieties. In particular, this provides a unified proof for the existence of smoothing of both $d$-semistable log smooth Calabi-Yau varieties (as studied by Friedman and Kawamata-Namikawa and maximally degenerate Calabi-Yau varieties (as studied by Kontsevich-Soibelman and Gross-Siebert). We also demonstrate how our construction yields a logarithmic Frobenius manifold structure on a formal neighborhood of $X$ in the extended moduli space by applying the technique of Barannikov-Kontsevich.

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arXiv2020

Geometric quantization and quantum moment maps on coadjoint orbits and Kähler-Einstein manifolds

Deformation quantization and geometric quantization on Kähler manifolds give the mathematical description of the algebra of quantum observables and the Hilbert spaces respectively, where the later forms a representation of quantum observables asymptotically via Toeplitz operators. When there is a Hamiltonian $G$-action on a Kähler manifold, there are associated symmetries on both the quantum algebra and representation aspects. We show that in nice cases of coadjoint orbits and Kähler-Einstein manifolds, these symmetries are strictly compatible (not only asymptotically).

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arXiv2020

Smoothing pairs over degenerate Calabi-Yau varieties

We apply the techniques developed in our previous work with Leung to study smoothings of a pair $(X,\mathfrak{C}^*)$, where $\mathfrak{C}^*$ is a bounded perfect complex of locally free sheaves over a degenerate Calabi-Yau variety $X$. In particular, if $X$ is a projective Calabi-Yau variety admitting the structure of a toroidal crossing space and with the higher tangent sheaf $\mathcal{T}^1_X$ globally generated, and $\mathfrak{F}$ is a locally free sheaf over $X$, then we prove, using the recent results of Felten-Filip-Ruddat, that the pair $(X,\mathfrak{F})$ is formally smoothable when $\text{Ext}^2(\mathfrak{F},\mathfrak{F})_0 = 0$ and $H^2(X,\mathcal{O}_X) = 0$.

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arXiv2020

Tropical counting from asymptotic analysis on Maurer-Cartan equations

Let $X = X_Σ$ be a toric surface and $(\check{X}, W)$ be its Landau-Ginzburg (LG) mirror where $W$ is the Hori-Vafa potential. We apply asymptotic analysis to study the extended deformation theory of the LG model $(\check{X}, W)$, and prove that semi-classical limits of Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to tropical disks in $X$ with Maslov index 0 or 2, the latter of which produces a universal unfolding of $W$. For $X = \mathbb{P}^2$, our construction reproduces Gross' perturbed potential $W_n$ which was proven to be the universal unfolding of $W$ written in canonical coordinates. We also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of $W_n$ across walls of the scattering diagram formed by the Maslov index 0 tropical disks originally observed by Gross (in the case of $X = \mathbb{P}^2$).

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arXiv2020

Fukaya's conjecture on Witten's twisted $A_\infty$-structures

Wedge product on deRham complex of a Riemannian manifold $M$ can be pulled back to $H^*(M)$ via explicit homotopy, constructed using Green's operator, to give higher product structures. We prove Fukaya's conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of deRham differential from a statement concerning homology to one concerning real homotopy type of $M$. Various applications of this conjecture to mirror symmetry are also suggested by Fukaya.

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arXiv2019

Refined scattering diagrams and theta functions from asymptotic analysis of Maurer-Cartan equations

We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer-Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich-Soibelman, Gross-Siebert and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer-Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of flat sections of the Maurer-Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with non-degenerate skew-symmetrized Euler forms.

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arXiv2019

Fukaya's conjecture on $S^1$-equivariant de Rham complex

Getzler-Jones-Petrack introduced $A_\infty$ structures on the equivariant complex for manifold $M$ with smooth $\mathbb{S}^1$ action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of $A_\infty$ structures. We extend and prove Fukaya's conjecture relating this Witten's deformed equivariant de Rham complexes, to a new Morse theoretical $A_\infty$ complexes defined by counting gradient trees with jumping which are closely related to the $\mathbb{S}^1$ equivariant symplectic cohomology proposed by Siedel.

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arXiv2019

Scattering diagrams from asymptotic analysis on Maurer-Cartan equations

Let $\check{X}_0$ be a semi-flat Calabi-Yau manifold equipped with a Lagrangian torus fibration $\check{p}:\check{X}_0 \rightarrow B_0$. We investigate the asymptotic behavior of Maurer-Cartan solutions of the Kodaira-Spencer deformation theory on $\check{X}_0$ by expanding them into Fourier series along fibres of $\check{p}$ over a contractible open subset $U\subset B_0$, following a program set forth by Fukaya in 2005. We prove that semi-classical limits (i.e. leading order terms in asymptotic expansions) of the Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to consistent scattering diagrams, which are tropical combinatorial objects that have played a crucial role in works of Kontsevich-Soibelman and Gross-Siebert on the reconstruction problem in mirror symmetry.

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arXiv2018

SYZ mirror symmetry from Witten-Morse theory

This is a survey article on the recent progress in understanding the Strominger-Yau-Zaslow (SYZ) mirror symmetry conjecture, especially on the effect of quantum corrections, via Witten-Morse theory using the program first depicted by Fukaya to obtain an explicit relation between differential geometric operations, e.g. wedge product of differential forms and Lie-bracket of the Kodaira-Spencer complexes, with combinatorial structures, e.g. Morse $A_\infty$ structures and scattering diagrams.

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arXiv2013

Lattice point counting via Einstein metrics

We obtain a growth estimate for the number of lattice points inside any Q-Gorenstein cone. Our proof uses the result of Futaki-Ono-Wang on Sasaki-Einstein metric for the toric Sasakian manifold associated to the cone, a Yau's inequality, and the Kawasaki-Riemann-Roch formula for orbifolds.

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