Floer cohomology in the mirror of the projective plane and a binodal cubic curve
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.
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| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2011
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| Online Access: | http://hdl.handle.net/1721.1/67812 |
| _version_ | 1811070154135568384 |
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| author | Pascaleff, James Thomas |
| author2 | Dennis Auroux. |
| author_facet | Dennis Auroux. Pascaleff, James Thomas |
| author_sort | Pascaleff, James Thomas |
| collection | MIT |
| description | Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. |
| first_indexed | 2024-09-23T08:28:28Z |
| format | Thesis |
| id | mit-1721.1/67812 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T08:28:28Z |
| publishDate | 2011 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/678122019-04-09T18:17:46Z Floer cohomology in the mirror of the projective plane and a binodal cubic curve Pascaleff, James Thomas Dennis Auroux. Massachusetts Institute of Technology. Dept. of Mathematics. Massachusetts Institute of Technology. Dept. of Mathematics. Mathematics. Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. Cataloged from PDF version of thesis. Includes bibliographical references (p. 115-117). We construct a family of Lagrangian submanifolds in the Landau-Ginzburg mirror to the projective plane equipped with a binodal cubic curve as anticanonical divisor. These objects correspond under mirror symmetry to the powers of the twisting sheaf 0(1), and hence their Floer cohomology groups form an algebra isomorphic to the homogeneous coordinate ring. An interesting feature is the presence of a singular torus fibration on the mirror, of which the Lagrangians are sections. This gives rise to a distinguished basis of the Floer cohomology and the homogeneous coordinate ring parameterized by fractional integral points in the singular affine structure on the base of the torus fibration. The algebra structure on the Floer cohomology is computed using the symplectic techniques of Lefschetz fibrations and the TQFT counting sections of such fibrations. We also show that our results agree with the tropical analog proposed by Abouzaid-Gross-Siebert. Extensions to a restricted class of singular affine manifolds and to mirrors of the complements of components of the anticanonical divisor are discussed. by James Thomas Pascaleff. Ph.D. 2011-12-19T19:00:46Z 2011-12-19T19:00:46Z 2011 2011 Thesis http://hdl.handle.net/1721.1/67812 767908243 eng M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 117 p. application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Pascaleff, James Thomas Floer cohomology in the mirror of the projective plane and a binodal cubic curve |
| title | Floer cohomology in the mirror of the projective plane and a binodal cubic curve |
| title_full | Floer cohomology in the mirror of the projective plane and a binodal cubic curve |
| title_fullStr | Floer cohomology in the mirror of the projective plane and a binodal cubic curve |
| title_full_unstemmed | Floer cohomology in the mirror of the projective plane and a binodal cubic curve |
| title_short | Floer cohomology in the mirror of the projective plane and a binodal cubic curve |
| title_sort | floer cohomology in the mirror of the projective plane and a binodal cubic curve |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/67812 |
| work_keys_str_mv | AT pascaleffjamesthomas floercohomologyinthemirroroftheprojectiveplaneandabinodalcubiccurve |