Math 2.0

Differential geometry Symplectic geometry Conjecture

Homological mirror symmetry

Builds on Fukaya category

Statement

Let XX be a compact Calabi–Yau manifold, viewed as a symplectic manifold. Let Xˇ\check{X} be a Calabi–Yau manifold that is mirror to XX in the sense of mirror symmetry. Then the derived category of coherent sheaves DbCoh(Xˇ)D^b\mathrm{Coh}(\check{X}) is equivalent, as a triangulated category, to the split-closed derived Fukaya category Dπℱ(X)D^\pi \mathcal{F}(X). The same holds with the roles of XX and Xˇ\check{X} exchanged.

Remarks

Kontsevich proposed the conjecture in his address to the International Congress of Mathematicians in Zürich in 1994, as a categorical explanation of mirror symmetry. The conjecture is not a single precise statement, because "mirror" is not defined in general. It is proved in many cases where a mirror pair is given by an explicit construction. Polishchuk and Zaslow proved it for elliptic curves, Seidel for the quartic surface, Abouzaid and Smith for the four-dimensional torus, and Sheridan for Calabi–Yau hypersurfaces in projective space. Versions for Fano manifolds, whose mirrors are Landau–Ginzburg models, replace the derived category on one side by a category of matrix factorizations or a Fukaya–Seidel category.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph
Fukaya category, concept, no evidence levelFukaya categoryconceptHomological mirror symmetry, conjecture, no evidence levelHomological mirror sy…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Fukaya category concept
  2. Homological mirror symmetry conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Homological_mirror_symmetry
About this page
TagB5N9WJH5F0
URIm2://B5N9WJH5F0
Kindconjecture
WritingW0 stub
MSC53D37, 14J33, 18G80
StatusThis copy of the site has no build data. Run m2 status B5N9WJH5F0.
AboutFukaya category
UsesFukaya category
Same as
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.