Towards studying of the higher rank theory of stable pairs

Calabi-Yau threefold;Stable pairs;Deformation-obstruction theory;Derived categories;Equivariant cohomology;Virtual localization;Wallcrossing

Sheshmani, Artan

关键词: Calabi-Yau threefold;    Stable pairs;    Deformation-obstruction theory;    Derived categories;    Equivariant cohomology;    Virtual localization;    Wallcrossing;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/26229/Sheshmani_Artan.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This thesis is composed of two parts. In the first part we introduce a higher rank analog of the Pandharipande-Thomas theoryof stable pairs on a Calabi-Yau threefold $X$. More precisely, we develop a moduli theory for frozen triples given by the data $\mathcal{O}_X^{\oplus r}(-n)\xrightarrow{\phi} F$ where $F$ is a sheaf of pure dimension $1$. The moduli space of such objects does not naturally determine an enumerative theory: that is, it does not naturally possess a perfect symmetric obstruction theory. Instead, we build a zero-dimensional virtual fundamental class by hand, by truncating a deformation-obstruction theory coming from the moduli of objects in the derivedcategory of $X$. This yields the first deformation-theoretic construction of a higher-rank enumerative theory for Calabi-Yau threefolds. We calculate this enumerative theory for local $\mathbb{P}^1$ using the Graber-Pandharipande virtual localization technique.In the second part of the thesis we compute the Donaldson-Thomas type invariants associated to frozen triples using the wall-crossing formula of Joyce-Song and Kontsevich-Soibelman.