Full Record

Author | Sheshmani, Artan |

Title | Towards studying of the higher rank theory of stable pairs |

URL | http://hdl.handle.net/2142/26229 |

Publication Date | 2011 |

Date Accessioned | 2011-08-25 22:19:38 |

Degree | PhD |

Discipline/Department | 0439 |

Degree Level | doctoral |

University/Publisher | University of Illinois – Urbana-Champaign |

Abstract | This thesis is composed of two parts. In the first part we introduce a higher rank analog of the Pandharipande-Thomas theory of 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 derived category 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. |

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

Contributors | Katz, Sheldon (advisor); Nevins, Thomas A. (advisor); Bradlow, Steven B. (Committee Chair); Katz, Sheldon (committee member); Nevins, Thomas A. (committee member); Schenck, Henry K. (committee member) |

Language | en |

Rights | Copyright 2011 Artan Sheshmani |

Country of Publication | us |

Record ID | handle:2142/26229 |

Repository | uiuc |

Date Retrieved | 2020-03-09 |

Date Indexed | 2020-03-09 |

Grantor | University of Illinois at Urbana-Champaign |

Issued Date | 2011-08-25 22:19:38 |

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…virtual class is a well-behaved deformation-*obstruction* *theory*. The description of the deformation
*obstruction* *theory* differs from case to case depending on the geometric structure of the moduli
space under consideration, hence it is important to study the…

…P ,r,n)
2
2. Hs,FT
(P ,r,n)
2
(τ ) = {(E1 , E2 , φ) ∈ Ms,FT
(τ ) | H1 (E2 (n)) = 0}.
We construct a well-behaved deformation *obstruction* *theory* for DM stack of highly…

…orders
is quasi-isomorphic to the complex given by
⊕r
OX×S
(−n) → F.
As mentioned above, the moduli stack of highly frozen triples has the structure of a DM stack. In
this case, the deformation *obstruction* *theory* is given by a morphism in the…

…and for deformation-*obstruction* *theory*, we require a morphism in the
derived category:
ob : E• → L• (P2 ,r,n)
Hs,FT
(τ )
,
where E• is a perfect complex of amplitude [−1, 1] and moreover h1 (ob) and h0 (…

…ob) are isomorphisms and h−1 (ob) is an epimorphism [25].
(P ,r,n)
2
We construct a deformation *obstruction* *theory* over Hs,FT
(τ ) which has all the nice cohomolog-
ical properties however it is perfect…

…construction in [25] to directly define
(P ,r,n)
2
a virtual fundamental class for Hs,FT
(τ ). On the other hand constructing a well-behaved defor-
(P ,r,n)
2
mation *obstruction* *theory* over Hs,HFT
(τ ) using…

…deformation-*obstruction* *theory* of perfect amplitude [−1, 0]
over the DM stack of stable highly frozen triples:
Theorem (6.12) Consider the 4-term deformation *obstruction* *theory* E•∨ of perfect amplitude
(P ,r,n)
2
[−2, 1…

…x5B;28]. The numerical invariants computed in each *theory* are conjecturally related to each other but the complete understanding of the connection between these
invariants and invariants in Gromov-Witten *theory* has not yet been achieved. During…