Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2306.13589

About Homological Mirror Symmetry

Alessandro Imparato
View original source ↗Read source document ↗Export metadata ↓

Abstract

The B-side of Kontsevich's Homological Mirror Symmetry Conjecture is discussed. We give first a self-contained study of derived categories and their homological algebra, and later restrict to the bounded derived category of schemes and ultimately Calabi--Yau manifolds, with particular emphasis on the basics of the underlying sheaf theory, and the algebraic features therein. Finally, we loosely discuss the lowest dimensional manifestations of homological mirror symmetry, namely for elliptic curves and $K3$ surfaces. The present work is a sequel to the author's survey "Towards Homological Mirror Symmetry" on the A-side of homological mirror symmetry.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alessandro Imparato. 2023-06-27. About Homological Mirror Symmetry. https://arxiv.org/abs/2306.13589

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

KEEP EXPLORING

Related papers

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗
arXiv2026

Nodal degeneration of chiral algebras I: Global structure and gluing formula

We define a natural extension of a universal factorization algebra $\mathcal{A}$ to families of stable punctured curves, by integrating over all semistable modifications. We prove that the resulting sheaf of factorization homology satisfies a natural gluing formula, by tensoring over a certain derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, generalizing the Verlinde formula for gluing of conformal blocks.

math.AG↗