Related papers: Calabi-Yau Structures, Spherical Functors, and Shi…
A construction of Calabi-Yaus as quotients of products of lower-dimensional spaces in the context of weighted hypersurfaces is discussed, including desingularisation. The construction leads to Calabi-Yaus which have a fiber structure, in…
In this paper, we generalize construction of Seidel's long exact sequence of Lagrangian Floer cohomology to that of compact Lagrangian submanifolds with vanishing Malsov class on general Calabi-Yau manifolds. We use the framework of…
We compare different constructions of mirrors of del Pezzo surfaces, focusing on degree $d \leq 3$. In particular, we extract Lefschetz fibrations, with associated exceptional collections, from the mirrors obtained via the Hori-Vafa and…
This is the first of a series of papers about \emph{quantization} in the context of \emph{derived algebraic geometry}. In this first part, we introduce the notion of \emph{$n$-shifted symplectic structures}, a generalization of the notion…
We generalize Calabi-Yau 3-folds from the special Lagrangian perspective. More precisely, we study SU(3)-structures which admit as "nice" a local special Lagrangian geometry as the flat $\mathbf{C}^3$ or a Calabi-Yau structure does. The…
In this article we study a differential algebra of modular-type functions attached to the periods of a one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of quintic threefolds. Such an algebra is…
Using tropical geometry we propose a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for Calabi-Yau hypersurfaces and by Batyrev…
We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of…
Complex Ricci-flat (i.e., Calabi-Yau) hypersurfaces in spaces admitting a maximal (toric) $U(1)^n$ gauge symmetry of general type (encoded by certain non-convex and multi-layered multitopes) may degenerate, but can be smoothed by rational…
Several properties of a hyepergeometric series related to Gromov-Witten theory of some Calabi-Yau geometries was studied in [8]. These properties play basic role in the study of higher genus Gromov-Witten theories. We extend the results of…
We prove several formulas related to Hodge theory and the Kodaira-Spencer-Kuranishi deformation theory of K\"ahler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami…
Slicing a module into semisimple ones is useful to study modules. Loewy structures provide a means of doing so. To establish the Loewy structures of projective modules over a finite dimensional symmetric algebra over a field $F$, the…
We consider a generalization of Calabi-Yau structures in the context of $\alpha$-Sasakian manifolds. We study deformations of a special class of Legendrian submanifolds and classify invariant contact Calabi-Yau structures on 5-dimensional…
We give a mathematically precise statement of the SYZ conjecture between mirror space pairs and prove it for any toric Calabi-Yau manifold with the Gross Lagrangian fibration. To date, it is the first time we realize the SYZ proposal with…
We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and…
We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the…
To paraphrase, part I constructs a bundle of $A _{\infty}$ categories given the input of a Hamiltonian fibration over a smooth manifold. Here we show that this bundle is generally non-trivial by a sample computation. One principal…
We study aspects related to Kontsevich's homological mirror symmetry conjecture in the case of Calabi-Yau complete intersections in toric varieties. In a 1996 lecture at Rutgers University, Kontsevich indicated how his proposal implies that…
Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been…
A twin Lagrangian fibration, originally introduced by Yau and the first author, is roughly a geometric structure consisting of two Lagrangian fibrations whose fibers intersect with each other cleanly. In this paper, we show the existence of…