We investigate the problem how to describe the equivalence classes of formal deformations of a symplectic manifold $M$ in the case when we have several deformation parameters $\ve_1, \ve_2, \dots, \ve_g$ of non-positive degrees. We define formal deformations of $M$ over the base ring $\bbC[[\ve, \ve_1, \dots, \ve_g]]$ as Maurer-Cartan elements of the differential graded Lie algebra $(\ve, \ve_1, \dots, \ve_g) \sPD^\bullet(M)[[\ve, \ve_1, \dots, \ve_g]]$ where $\sPD^\bullet(M)$ denotes the algebra of polydifferential operators on $M$. The interesting feature of such deformations is that, if at least one formal parameter carries a non-zero degree, then the resulting Maurer-Cartan element corresponds to a $\bbC[[\ve, \ve_1, \dots, \ve_g]]$-multilinear $A_\infty$-structure on the graded vector space $\cO(M)[[\ve, \ve_1, \dots, \ve_g]]$ with the zero differential, where $\cO(M)$ is the algebra of smooth complex-valued functions $M$. This dissertation focuses on formal deformations of $\cO(M)$ with the base ring $\bbC[[\ve, \ve_1, \dots, \ve_g]]$ such that corresponding MC elements $\mu$ satisfy these two conditions: The Kodaira-Spencer class of $\mu$ is $\ve \al$ and $\mu$ satisfies the equation $\mu \rvert_{\ve=0} =0$. The main result of this study gives us a bijection between the set of isomorphism classes of such deformations and the set of all degree 2 vectors of the graded vector space \, $\bigoplus_{q \geq 0} \, (\ve, \ve_1, \dots, \ve_g) \, H^q(M, \bbC)[[\ve, \ve_1, \dots, \ve_g]]$ where $H^\bullet(M, \bbC)$ is the singular cohomology of $M$ with coefficients in $\bbC$ and every vector of $H^q(M, \bbC)$ carries degree $q$. / Mathematics
| Identifer | oai:union.ndltd.org:TEMPLE/oai:scholarshare.temple.edu:20.500.12613/679 |
| Date | January 2017 |
| Creators | Altinay-Ozaslan, Elif |
| Contributors | Dolgushev, Vasily, Letzter, E. S. (Edward S.), 1958-, Lorenz, Martin, 1951-, Stienon, Mathieu |
| Publisher | Temple University. Libraries |
| Source Sets | Temple University |
| Language | English |
| Detected Language | English |
| Type | Thesis/Dissertation, Text |
| Format | 67 pages |
| Rights | IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available., http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | http://dx.doi.org/10.34944/dspace/661, Theses and Dissertations |
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