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1.
Piyaratne, Hathurusinghege Dulip Bandara.
* Fourier*-

Degree: PhD, 2014, University of Edinburgh

URL: http://hdl.handle.net/1842/9635

Construction of Bridgeland stability conditions on a given Calabi-Yau threefold is an important problem and this thesis realizes the rst known examples of such stability conditions. More precisely, we construct a dense family of stability conditions on the derived category of coherent sheaves on a principally polarized abelian threefold X with Picard rank one. In particular, we show that the conjectural construction proposed by Bayer, Macr and Toda gives rise to Bridgeland stability conditions on X. First we reduce the requirement of the Bogomolov-Gieseker type inequalities to a smaller class of tilt stable objects which are essentially minimal objects of the conjectural stability condition hearts for a given smooth projective threefold. Then we use the Fourier-Mukai theory to prove the strong Bogomolov-Gieseker type inequalities for these minimal objects of X. This is done by showing any Fourier-Mukai transform of X gives an equivalence of abelian categories which are double tilts of coherent sheaves.

Subjects/Keywords: 516.3; derived catagories; stability conditions; abelian threefold; Fourier-Mukai transforms

…construction on threefolds . . . . . . . . . . . . . . . . . . .
23
2.6
*Fourier*-*Mukai* theory… …Similarly the subcategory HNνω,B (I) ⊂ Bω,B is defined.
• For a *Fourier*-*Mukai* functor Υ… …conditions on the derived categories. See [Tod4] for further details.
*Fourier*-*Mukai*… …theory
The notion of *Fourier*-*Mukai* transform (FM transform for short) was introduced… …a *Fourier*-*Mukai* functor (FM
E
functor for short) and when it is an equivalence…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Piyaratne, H. D. B. (2014). Fourier-Mukai transforms and stability conditions on abelian threefolds. (Doctoral Dissertation). University of Edinburgh. Retrieved from http://hdl.handle.net/1842/9635

Chicago Manual of Style (16^{th} Edition):

Piyaratne, Hathurusinghege Dulip Bandara. “Fourier-Mukai transforms and stability conditions on abelian threefolds.” 2014. Doctoral Dissertation, University of Edinburgh. Accessed July 12, 2020. http://hdl.handle.net/1842/9635.

MLA Handbook (7^{th} Edition):

Piyaratne, Hathurusinghege Dulip Bandara. “Fourier-Mukai transforms and stability conditions on abelian threefolds.” 2014. Web. 12 Jul 2020.

Vancouver:

Piyaratne HDB. Fourier-Mukai transforms and stability conditions on abelian threefolds. [Internet] [Doctoral dissertation]. University of Edinburgh; 2014. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/1842/9635.

Council of Science Editors:

Piyaratne HDB. Fourier-Mukai transforms and stability conditions on abelian threefolds. [Doctoral Dissertation]. University of Edinburgh; 2014. Available from: http://hdl.handle.net/1842/9635

2. Pham, Tuan D. On the Picard Varieties of Surfaces with Equivalent Derived Categories.

Degree: 2012, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/9623

It was shown recently by Popa and Schnell that the irregularities of two smooth projective varieties with equivalent bounded derived categories of coherent sheaves are equal. They conjectured that the Picard varieties of smooth projective varieties with equivalent derived categories are derived equivalent. This thesis investigates this conjecture for the case of smooth projective surfaces. More specifically, we showed that the Picard varieties of derived equivalent surfaces are in fact derived equivalent with the possible exception of the case of properly elliptic surfaces with constant j-invariant. In that case, we also provide an analysis of the Picard variety. In addition, we give a statement about the automorphism groups of derived equivalent surfaces.
*Advisors/Committee Members: Popa, Mihnea (advisor), Ein, Lawrence (committee member), Coskun, Izzet (committee member), Schnell, Christian (committee member), Arapura, Donu (committee member).*

Subjects/Keywords: algebraic geometry; derived categories; Picard varieties; automorphism groups; Albanese varieties; Fourier-Mukai transforms

…Proposition 2.2.15 and Proposition 2.2.16).
3
*Fourier*-*Mukai* *transforms* provide important… …For instance, Bridgeland and Maciocia were
able to use *Fourier*-*Mukai* *transforms* to show that… …categories of coherent sheaves and derived functors. We
introduce the *Fourier*-*Mukai* *transforms* and… …surfaces.
CHAPTER 2
*FOURIER*-*MUKAI* *TRANSFORMS*
2.1
Derived categories of coherent sheaves
Here… …*Fourier*-*Mukai* *transforms*
2.2.1
Preliminaries
Let X and Y be smooth projective varieties and…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Pham, T. D. (2012). On the Picard Varieties of Surfaces with Equivalent Derived Categories. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/9623

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Pham, Tuan D. “On the Picard Varieties of Surfaces with Equivalent Derived Categories.” 2012. Thesis, University of Illinois – Chicago. Accessed July 12, 2020. http://hdl.handle.net/10027/9623.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Pham, Tuan D. “On the Picard Varieties of Surfaces with Equivalent Derived Categories.” 2012. Web. 12 Jul 2020.

Vancouver:

Pham TD. On the Picard Varieties of Surfaces with Equivalent Derived Categories. [Internet] [Thesis]. University of Illinois – Chicago; 2012. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/10027/9623.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Pham TD. On the Picard Varieties of Surfaces with Equivalent Derived Categories. [Thesis]. University of Illinois – Chicago; 2012. Available from: http://hdl.handle.net/10027/9623

Not specified: Masters Thesis or Doctoral Dissertation

3.
Grigg, Nathan.
Deformations of Categories of Coherent Sheaves and *Fourier*-*Mukai* * Transforms*.

Degree: PhD, 2013, University of Washington

URL: http://hdl.handle.net/1773/23413

In modern algebraic geometry, an algebraic variety is often studied by way of its category of coherent sheaves or derived category. Recent work by Toda has shown that infinitesimal deformations of the category of coherent sheaves can be described as twisted sheaves on a noncommutative deformation of the variety. This thesis generalizes Toda's work by creating a chain of inclusions from deformations of schemes to commutative deformations to deformations of the category of coherent sheaves. We define projections from coherent deformations to commutative deformations to scheme deformations and show that the fiber of the projection from commutative deformations to schemes is a gerbe. We also prove that for two derived equivalent K3 surfaces in characteristic p and any scheme deformation of one of these, there is a scheme deformation of the other so that the two deformations are also derived equivalent.
*Advisors/Committee Members: Lieblich, Max (advisor).*

Subjects/Keywords: algebraic geometry; coherent sheaves; deformation theory; fourier-mukai transforms; K3 surfaces; Mathematics; mathematics

…x28;X0 ).
4
*Fourier*-*Mukai* *Transforms*
A *Fourier*-*Mukai* transform is a type of functor… …*Fourier*-*Mukai*
transform [Orl1].
Toda also studied the relationship between *Fourier*… …lifting *Fourier*-*Mukai*
partnerships.
2 Deformations of schemes and
categories
For the… …*Mukai* equivalences and deformations of Coh(X0 ) and showed that given an equivalence… …the derived categories of the deformations [Toda].
These *transforms* enrich the…

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Grigg, N. (2013). Deformations of Categories of Coherent Sheaves and Fourier-Mukai Transforms. (Doctoral Dissertation). University of Washington. Retrieved from http://hdl.handle.net/1773/23413

Chicago Manual of Style (16^{th} Edition):

Grigg, Nathan. “Deformations of Categories of Coherent Sheaves and Fourier-Mukai Transforms.” 2013. Doctoral Dissertation, University of Washington. Accessed July 12, 2020. http://hdl.handle.net/1773/23413.

MLA Handbook (7^{th} Edition):

Grigg, Nathan. “Deformations of Categories of Coherent Sheaves and Fourier-Mukai Transforms.” 2013. Web. 12 Jul 2020.

Vancouver:

Grigg N. Deformations of Categories of Coherent Sheaves and Fourier-Mukai Transforms. [Internet] [Doctoral dissertation]. University of Washington; 2013. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/1773/23413.

Council of Science Editors:

Grigg N. Deformations of Categories of Coherent Sheaves and Fourier-Mukai Transforms. [Doctoral Dissertation]. University of Washington; 2013. Available from: http://hdl.handle.net/1773/23413