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Complex Rank 3 Vector Bundles on Complex Projective 5-Space.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Complex Rank 3 Vector Bundles on Complex Projective 5-Space./
Author:	Opie, Morgan Peck.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2021,
Description:	162 p.
Notes:	Source: Dissertations Abstracts International, Volume: 83-02, Section: B.
Contained By:	Dissertations Abstracts International83-02B.
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28499268
ISBN:	9798534671032

Complex Rank 3 Vector Bundles on Complex Projective 5-Space.
Opie, Morgan Peck.

Complex Rank 3 Vector Bundles on Complex Projective 5-Space. - Ann Arbor : ProQuest Dissertations & Theses, 2021 - 162 p.

Source: Dissertations Abstracts International, Volume: 83-02, Section: B.

Thesis (Ph.D.)--Harvard University, 2021.

This item must not be sold to any third party vendors.

This work concerns two aspects of the study of complex rank 3 topological vector bundles on complex projective five-space. The first aim is to classify such bundles: to give complete, computable algebraic invariants. This is a nontrivial project because classical invariants like Chern classes do not uniquely determine such bundles. Our classification strategy is motivated by the classical results of Atiyah and Rees for complex rank 2 topological bundles on ℂP3: Atiyah and Rees show that these bundles are determined by their Chern classes, together with an additional ℤ/2-invariant which should be understood as arising from a twisted version of real topological K-theory.We re-examine Atiyah and Rees' approach and excise specialized geometry which does not generalize to higher rank bundles or higher-dimensional spaces. From this perspective, certain algebraic analogies emerge that allow for a similar approach to complex rank 3 topological bundles on ℂP5. Indeed, we show that complex rank 3 topological vector bundles on ℂP5 are determined by their Chern classes together with a ℤ/3-valued invariant, which arises from a universal invariant valued in twisted, 3-local tmf-cohomology (where tmf is the spectrum of topological modular forms).The second aim of this thesis is to concretely construct interesting topological bundles of complex rank 3 on ℂP5. While our classification results predict the number of complex rank 3 topological bundles on ℂP5 with prescribed Chern classes, and offer an invariant to distinguish bundles with the same Chern classes, they are inexplicit. Ideally, one would like a procedure to realize all topological equivalence classes by concrete bundles. A prototypical construction is given by results of Horrocks, which produce algebraic representatives for all topological equivalence classes of complex rank 2 bundles on ℂP3. Due to the complexity of higher-codimension subschemes of complex projective spaces, this algebraic construction does not admit an obvious generalization.However, we show that it can be reinterpreted through a homotopical lens, where it admits a broadly applicable generalization. In a particular application of this machinery, we produce a homotopical method for constructing new rank 3 bundles on ℂP5, using an explicit construction on classifying spaces. We show that iteratively applying this construction to rank 3 sums of line bundles produces new rank 3 bundles that are not themselves sums of line bundles, demonstrating that our methods allow for the construction of interesting bundles from simple ones.

ISBN: 9798534671032Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Algebraicity

Complex Rank 3 Vector Bundles on Complex Projective 5-Space.
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500 $a Source: Dissertations Abstracts International, Volume: 83-02, Section: B.
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506 $a This item must not be sold to any third party vendors.
520 $a This work concerns two aspects of the study of complex rank 3 topological vector bundles on complex projective five-space. The first aim is to classify such bundles: to give complete, computable algebraic invariants. This is a nontrivial project because classical invariants like Chern classes do not uniquely determine such bundles. Our classification strategy is motivated by the classical results of Atiyah and Rees for complex rank 2 topological bundles on ℂP3: Atiyah and Rees show that these bundles are determined by their Chern classes, together with an additional ℤ/2-invariant which should be understood as arising from a twisted version of real topological K-theory.We re-examine Atiyah and Rees' approach and excise specialized geometry which does not generalize to higher rank bundles or higher-dimensional spaces. From this perspective, certain algebraic analogies emerge that allow for a similar approach to complex rank 3 topological bundles on ℂP5. Indeed, we show that complex rank 3 topological vector bundles on ℂP5 are determined by their Chern classes together with a ℤ/3-valued invariant, which arises from a universal invariant valued in twisted, 3-local tmf-cohomology (where tmf is the spectrum of topological modular forms).The second aim of this thesis is to concretely construct interesting topological bundles of complex rank 3 on ℂP5. While our classification results predict the number of complex rank 3 topological bundles on ℂP5 with prescribed Chern classes, and offer an invariant to distinguish bundles with the same Chern classes, they are inexplicit. Ideally, one would like a procedure to realize all topological equivalence classes by concrete bundles. A prototypical construction is given by results of Horrocks, which produce algebraic representatives for all topological equivalence classes of complex rank 2 bundles on ℂP3. Due to the complexity of higher-codimension subschemes of complex projective spaces, this algebraic construction does not admit an obvious generalization.However, we show that it can be reinterpreted through a homotopical lens, where it admits a broadly applicable generalization. In a particular application of this machinery, we produce a homotopical method for constructing new rank 3 bundles on ℂP5, using an explicit construction on classifying spaces. We show that iteratively applying this construction to rank 3 sums of line bundles produces new rank 3 bundles that are not themselves sums of line bundles, demonstrating that our methods allow for the construction of interesting bundles from simple ones.
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