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University of Illinois – Chicago

1. Lombardi, Luigi. Derived Equivalences of Irregular Varieties and Constraints on Hodge Numbers.

Degree: 2013, University of Illinois – Chicago

We study derived equivalences of smooth projective irregular varieties. More specifically, as suggested by a conjecture of Popa, we investigate the behavior of cohomological support loci associated to the canonical bundle (around the origin) under derived equivalence. We approach this problem in two ways. In the first approach we establish and apply the derived invariance of a ``twisted'' version of Hochschild homology taking into account an isomorphism due to Rouquier and related to autoequivalences of derived categories. In the second approach we relate the derived invariance of cohomological support loci to the derived invariance of Hodge numbers. As a result, we obtain the derived invariance of the first two and the last two cohomological support loci, leading to interesting geometric applications. For instance, we deduce the derived invariance of a few numerical quantities attached to irregular varieties, and furthermore we describe the geometry of Fourier-Mukai partners of Fano fibrations, and hence of Mori fiber spaces, fibered over curves of genus at least two. Finally, we also study constraints on Hodge numbers of special classes of irregular compact Kaehler manifolds. More specifically, we write down nequalities for all the Hodge numbers by studying the exactness of BGG complexes associated to bundles of holomorphic p-forms and by using classical results in the theory of vector bundles on projective spaces. As an application of our techniques, we bound the regularity of cohomology modules in terms of the defect of semismallness of the Albanese map. Advisors/Committee Members: Popa, Mihnea (advisor), Budur, Nero (committee member), Coskun, Izzet (committee member), Ein, Lawrence (committee member), Libgober, Anatoly (committee member).

Subjects/Keywords: Derived Categories; Equivalences; Non-vanishing Loci; Irregular Varieties; Picard Variety; Hodge Numbers; Derivative Complex; Hochschild homology

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

APA (6th Edition):

Lombardi, L. (2013). Derived Equivalences of Irregular Varieties and Constraints on Hodge Numbers. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/10294

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Lombardi, Luigi. “Derived Equivalences of Irregular Varieties and Constraints on Hodge Numbers.” 2013. Thesis, University of Illinois – Chicago. Accessed July 12, 2020. http://hdl.handle.net/10027/10294.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Lombardi, Luigi. “Derived Equivalences of Irregular Varieties and Constraints on Hodge Numbers.” 2013. Web. 12 Jul 2020.

Vancouver:

Lombardi L. Derived Equivalences of Irregular Varieties and Constraints on Hodge Numbers. [Internet] [Thesis]. University of Illinois – Chicago; 2013. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/10027/10294.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Lombardi L. Derived Equivalences of Irregular Varieties and Constraints on Hodge Numbers. [Thesis]. University of Illinois – Chicago; 2013. Available from: http://hdl.handle.net/10027/10294

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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