相关论文: Reconstruction of $T_{\mathbb{P}^2}$ via tropical …
The SYZ conjecture suggests a folklore that "Lagrangian multi-sections are mirror to holomorphic vector bundles". In this paper, we prove this folklore for Lagrangian multi-sections inside the cotangent bundle of a vector space, which are…
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…
We introduce the notion of tropical Lagrangian multi-sections over a fan and study its relation with toric vector bundles. We also introduce a "SYZ-type" construction for toric vector bundles which gives a reinterpretation of Kaneyama's…
In this note we give a complete description of all the hyperplane section of the projective bundle associated to the tangent bundle of $\mathbb{P}^2$ under its natural embedding in $\mathbb{P}^7.$ As an application one obtains a description…
The Hamilton-Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence…
We present a geometric realization of the duality between skeleta in $T^*\mathbb P^n$ and collars of local surfaces. Such duality is predicted by combining two auxiliary types of duality: on one side, symplectic duality between $T^*\mathbb…
Our purpose is to show the existence of a Calabi-Yau structure on the punctured cotangent bundle $T^{*}_{0}(P^2\mathbb{O})$ of the Cayley projective plane $P^{2}\mathbb{O}$ and to construct a Bargmann type transformation from a space of…
On a compact connected Riemann surface $C$ of genus at least $2$, we construct Lagrangian correspondences between moduli spaces of rank-$n$ Higgs bundles (respectively, holomorphic connections) and the Hilbert schemes of points on $T^\ast…
We construct a Lagrangian submanifold, inside the cotangent bundle of a real torus, which we call a Lagrangian pair of pants. It is given as the graph of the differential of a smooth function defined on the real blow up of a Lagrangian…
We show that any locally planar tropical curve $\Gamma \subset \mathbb{R}^n$ (with unit edge weights) can be realized as the limit of the rescaled moment map images of a family of special Lagrangian submanifolds in $T^*T^n$ with respect to…
In this work, we present a novel approach for reconstructing shape point clouds using planar sparse cross-sections with the help of generative modeling. We present unique challenges pertaining to the representation and reconstruction in…
This article is a continuation of the work "Tropical Lagrangian multi-sections and smoothing of locally free sheaves over degenerated Calabi-Yau surfaces". We generalize the notion of tropical Lagrangian multi-sections to any dimensions.…
A tropical expansion is a degeneration of a toroidal embedding, induced by a polyhedral subdivision of its tropicalisation. Each irreducible component of a tropical expansion admits a collapsing map down to a stratum of the original…
Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic $L^2$ torsion, which lies in the determinant line of the twisted $L^2$ Dolbeault cohomology and represents a volume element there.…
We construct special Lagrangian pair of pants in general dimensions, inside the cotangent bundle of $T^n$ with the Euclidean structure.
The cotangent bundle $T^*X$ of a smooth intersection $X$ of two quadrics admits a Lagrangian fibration determined by the intrinsic geometry of $X$. We show that this fibration is actually the Hitchin morphism if we endow $X$ with a…
The Tango bundle T over P^5 is proved to be the pull-back of the twisted Cayley bundle C(1) via a map f : P^5 --> Q_5 existing only in characteristic 2. The Frobenius morphism F factorizes via such f. Using f the cohomology of T is computed…
Given a measure space ${\mathcal X}$, we can construct a number of induced structures: eg. its $L^2$ space, the space ${\mathcal P}({\mathcal X})$ of probability distributions on ${\mathcal X}$. If, in addition, ${\mathcal X}$ admits a…
The algebraic foundation of tropical polynomial algebra provides the framework for the geometric construction of the supplement and the reversal of tropical varieties, thereby inducing a duality of reduced tropical varieties; for classes of…
We study the logarithmic vector bundles associated to arrangements of smooth irreducible curves with small degree on the blow-up of the projective plane at one point. We then investigate whether they are Torelli arrangements, that is, they…