Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space

By Nick Sheridan. 2011

154 citations recorded by the source corpus.

We prove Homological Mirror Symmetry for a smooth $$d$$d-dimensional Calabi–Yau hypersurface in projective space, for any $$d \ge 3$$d≥3 (for example, $$d=3$$d=3 is the quintic threefold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the ‘$$d$$d-dimensional pair of pants’; the introduction of the ‘relative Fukaya category’, and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an ‘orbifold’ Fukaya category); a Morse–Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e238c0679ff60f47b4f290a9199e0190a4b8307

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Citations in the corpus

Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c} (2013, methodology): “tive answer to this question. An important geometric ingredient in the proof of homological mirror symmetry for genus two curves by Seidel [Sei11] and Fermat-type Calabi-Yau hypersurfaces by Sheridan [She15] is the use of specic Lagrangian immersions. Their works motivates our current program: inste”
Mirror symmetry: from categories to curve counts (2015, methodology): “In particular, the mirror map ψ ∗ that appears in Theorem 1.18 was not determined in [She15b], beyond the ﬁrst term (see Remark 1.19 for more on this).”
A Beginner’s Introduction to Fukaya Categories (2013, methodology): “lectic manifolds whose Fukaya categories have been determined in this manner include (but are not limited to) pairs of pants [9], genus 2 curves [44], and Calabi-Yau hypersurfaces in projective space [46]. 4.5. An application to Heegaard-Floer homology. Heegaard-Floer homology associates to a closed”
The wall-crossing formula and Lagrangian mutations (2017, background): “The statement about the critical point is verified by an explicit computation, and the Floer-theoretic implications are well known given the fact that w is a simple eigenvalue, see [31] for details.”
On the homological mirror symmetry conjecture for pairs of pants (2010, background): “omological Mirror Symmetry for hypersurfaces in projective space (not necessarily Calabi-Yau). The author has made considerable progress in this direction, which will appear in a forthcoming preprint [50]. In this section we give a brief outline of our anticipated results in the Calabi-Yau case, bor”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “Associated to that is the relative Fukaya category, which is a formal deformation of F(M) (inverting the deformation parameter then yields a full subcategory of the Fukaya category F(M̄), see [38, 33, 49]).”
Khovanov homology from Floer cohomology (2015, methodology): “The case in which some of the Lagrangians are equal is handled by counting configurations of holomorphic discs and (perturbed) gradient flow lines, as in [10, 46, 52].”
Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow (2014, background): “Seidel [63] proved it forM a quartic surface in CP3 (a Calabi–Yau 2-fold), and Sheridan [65] proved it for M a smooth Calabi–Yau m-fold hypersurface in CPm+1 for m > 3.”
Homological mirror symmetry for punctured spheres (2011, background): “mensional). Seidel’s argument was subsequently extended to higher genus curves [12], to pairs of pants and their higher-dimensional analogues [42], and to Calabi-Yau hypersurfaces in projective space [43]. Unfortunately, the ordinary Fukaya category consisting of closed Lagrangians is insuﬃ- cient i”
Generating the Fukaya categories of Hamiltonian 𝐺-manifolds (2015, background): “li split-generates then K is not displaceable from any Lagrangian submanifold L with HF(L,L) 6= 0. Finding a split-generator is also the ﬁrst step in many modern proofs of homological mirror symmetry [6, 51, 52, 54, 55, 56]. The aim of this paperis todeterminewhen a free Lagrangian orbit of a Hamilt”
On the Fukaya category of a Fano hypersurface in projective space (2013, methodology): “Again we follow [36] closely: we define a category of matrix factorizations mirror to the relative Fukaya category, compute a full subcategory B̃, apply the Homological Perturbation Lemma and the classification results from Section 2 to prove it is quasi-equivalent to Ã, and prove that it split-gene”
The family Floer functor is faithful (2014, background): “Such descriptions are possible in a limited class of examples, which include Calabi-Yau hypersurfaces in projective space [19, 21] and toric varieties [12, 5].”