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 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…