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University of Michigan

1. Stevenson, Matthew John. Applications of Canonical Metrics on Berkovich Spaces.

Degree: PhD, Mathematics, 2019, University of Michigan

URL: http://hdl.handle.net/2027.42/151429

This thesis examines the nature of Temkin’s canonical metrics on the sheaves of differentials of Berkovich spaces, and discusses 3 applications thereof. First, we show a comparison theorem between Temkin’s metric on the beth-analytification of a smooth variety over a trivially-valued field of characteristic zero, and a weight metric defined in terms of log discrepancies. This result is the trivially-valued counterpart to a comparison theorem of Temkin between his metric and the weight metric of Mustata–Nicaise in the discretely-valued setting.
These weight metrics are used to define an essential skeleton of a pair over a trivially-valued field; this is done following the approach of Brown–Mazzon in the discretely-valued case, and we show a compatibility result between the essential skeletons of pairs in the two settings. Furthermore, a careful study of the closures of these skeletons enables us to realize the toric skeleton of a toric variety as an essential skeleton.
On the Berkovich unit disc, Temkin’s metric acts a substitute for the Lebesgue measure. Adopting this philosophy, we show a non-Archimedean version of the Ohsawa–Takegoshi extension theorem. As a corollary, we deduce a non-Archimedean analogue of Demailly’s regularization theorem for quasisubharmonic functions on the Berkovich disc.
Finally, we employ Temkin’s metric and essential skeletons to compute the dual boundary complexes of two classes of character varieties that arise in non-abelian Hodge theory. These two results provide the first non-trivial evidence for the geometric P = W conjecture of Katzarkov–Noll–Pandit–Simpson in the compact case. For each result, we give two proofs: one using non-Archimedean geometry over a trivially-valued field, and another in the discretely-valued setting. The latter produces degenerations of compact hyper-Kahler manifolds, which are of independent interest.
*Advisors/Committee Members: Jonsson, Mattias (committee member), Akhoury, Ratindranath (committee member), Canton, Eric (committee member), Mustata, Mircea Immanuel (committee member), Smith, Karen E (committee member).*

Subjects/Keywords: Algebraic geometry; Non-Archimedean geometry; Berkovich spaces; Mathematics; Science

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

APA (6^{th} Edition):

Stevenson, M. J. (2019). Applications of Canonical Metrics on Berkovich Spaces. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/151429

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

Stevenson, Matthew John. “Applications of Canonical Metrics on Berkovich Spaces.” 2019. Doctoral Dissertation, University of Michigan. Accessed August 07, 2020. http://hdl.handle.net/2027.42/151429.

MLA Handbook (7^{th} Edition):

Stevenson, Matthew John. “Applications of Canonical Metrics on Berkovich Spaces.” 2019. Web. 07 Aug 2020.

Vancouver:

Stevenson MJ. Applications of Canonical Metrics on Berkovich Spaces. [Internet] [Doctoral dissertation]. University of Michigan; 2019. [cited 2020 Aug 07]. Available from: http://hdl.handle.net/2027.42/151429.

Council of Science Editors:

Stevenson MJ. Applications of Canonical Metrics on Berkovich Spaces. [Doctoral Dissertation]. University of Michigan; 2019. Available from: http://hdl.handle.net/2027.42/151429

2. Chen, Yuanyuan. Filtration Theorems and Bounding Generators of Symbolic Multi-powers.

Degree: PhD, Mathematics, 2019, University of Michigan

URL: http://hdl.handle.net/2027.42/151674

We prove a very powerful generalization of the theorem on generic freeness that gives countable ascending filtrations, by prime cyclic A-modules A/P, of finitely generated algebras R over a Noetherian ring A and of finitely generated R-modules such that the number of primes P that occur is finite. Moreover, we can control, in a sense that we can make precise, the number of factors of the form A/P that occur.
In the graded case, the number of occurrences of A/P up to a given degree is eventually polynomial. The degree is at most the number of generators of R over A. By multi-powers of a finite sequence of ideals we mean an intersection of powers of the ideals with exponents varying. Symbolic multi-powers are defined analogously using symbolic powers instead of powers. We use our filtration theorems to give new results bounding the number of generators of the multi-powers of a sequence of ideals and of the symbolic multi-powers as well under various conditions. This includes the case of ordinary symbolic powers of one ideal.
Furthermore, we give new results bounding, by polynomials in the exponents, the number of generators of multiple Tor when each input module is the quotient of R by a power of an ideal. The ideals and exponents vary. The bound is given by a polynomial in the exponents. There are similar results for Ext when both of the input modules are quotients of R by a power of an ideal. Typically, the two ideals used are different, and the bound is a polynomial in two exponents.
*Advisors/Committee Members: Hochster, Mel (committee member), Tappenden, James P (committee member), Canton, Eric (committee member), Derksen, Harm (committee member), Smith, Karen E (committee member).*

Subjects/Keywords: symbolic powers; filtration theorems; Mathematics; Science

Record Details Similar Records

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

APA (6^{th} Edition):

Chen, Y. (2019). Filtration Theorems and Bounding Generators of Symbolic Multi-powers. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/151674

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

Chen, Yuanyuan. “Filtration Theorems and Bounding Generators of Symbolic Multi-powers.” 2019. Doctoral Dissertation, University of Michigan. Accessed August 07, 2020. http://hdl.handle.net/2027.42/151674.

MLA Handbook (7^{th} Edition):

Chen, Yuanyuan. “Filtration Theorems and Bounding Generators of Symbolic Multi-powers.” 2019. Web. 07 Aug 2020.

Vancouver:

Chen Y. Filtration Theorems and Bounding Generators of Symbolic Multi-powers. [Internet] [Doctoral dissertation]. University of Michigan; 2019. [cited 2020 Aug 07]. Available from: http://hdl.handle.net/2027.42/151674.

Council of Science Editors:

Chen Y. Filtration Theorems and Bounding Generators of Symbolic Multi-powers. [Doctoral Dissertation]. University of Michigan; 2019. Available from: http://hdl.handle.net/2027.42/151674