HB
H.L. Bakker
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This thesis generalises the classical cross-ratio of four projective points in complex projective 1-space to a pairing of two k-dimensional and two (n-k-1)-dimensional linear subspaces of complex projective n-space. We prove that the absolute values of the generalised cross-ratio determine the local height pairings of the corresponding cycles. For the Archimedean height, the proof avoids integrating Green’s forms over high-dimensional subspaces by relating this height, via an incidence correspondence, to an explicit height pairing of points and a principal divisor on a Grassmannian.
We then study degenerating families of these linear subspaces and describe the asymptotic behaviour of all local height pairings as the cycles intersect. We show that, in any degeneration, the asymptotics are governed by the intersection degree of the corresponding families of cycles.
Furthermore, using a global construction, we show that the generalised cross-ratio itself, rather than only its norms, computes a generalised Hodge-theoretic height. This is done by showing that the relative homology group induced by the two k-planes and two (n-k-1)-planes is naturally an extension of mixed Hodge structures of Z(0) by Z(1), and that the height of this extension equals the generalised cross-ratio.
Finally, using local methods, we study regularised limits of local heights and interpret these limits using only the central degenerate geometry, together with small perturbations. This thesis provides a new class of examples associating geometric shapes to limits of local heights. ...
We then study degenerating families of these linear subspaces and describe the asymptotic behaviour of all local height pairings as the cycles intersect. We show that, in any degeneration, the asymptotics are governed by the intersection degree of the corresponding families of cycles.
Furthermore, using a global construction, we show that the generalised cross-ratio itself, rather than only its norms, computes a generalised Hodge-theoretic height. This is done by showing that the relative homology group induced by the two k-planes and two (n-k-1)-planes is naturally an extension of mixed Hodge structures of Z(0) by Z(1), and that the height of this extension equals the generalised cross-ratio.
Finally, using local methods, we study regularised limits of local heights and interpret these limits using only the central degenerate geometry, together with small perturbations. This thesis provides a new class of examples associating geometric shapes to limits of local heights. ...
This thesis generalises the classical cross-ratio of four projective points in complex projective 1-space to a pairing of two k-dimensional and two (n-k-1)-dimensional linear subspaces of complex projective n-space. We prove that the absolute values of the generalised cross-ratio determine the local height pairings of the corresponding cycles. For the Archimedean height, the proof avoids integrating Green’s forms over high-dimensional subspaces by relating this height, via an incidence correspondence, to an explicit height pairing of points and a principal divisor on a Grassmannian.
We then study degenerating families of these linear subspaces and describe the asymptotic behaviour of all local height pairings as the cycles intersect. We show that, in any degeneration, the asymptotics are governed by the intersection degree of the corresponding families of cycles.
Furthermore, using a global construction, we show that the generalised cross-ratio itself, rather than only its norms, computes a generalised Hodge-theoretic height. This is done by showing that the relative homology group induced by the two k-planes and two (n-k-1)-planes is naturally an extension of mixed Hodge structures of Z(0) by Z(1), and that the height of this extension equals the generalised cross-ratio.
Finally, using local methods, we study regularised limits of local heights and interpret these limits using only the central degenerate geometry, together with small perturbations. This thesis provides a new class of examples associating geometric shapes to limits of local heights.
We then study degenerating families of these linear subspaces and describe the asymptotic behaviour of all local height pairings as the cycles intersect. We show that, in any degeneration, the asymptotics are governed by the intersection degree of the corresponding families of cycles.
Furthermore, using a global construction, we show that the generalised cross-ratio itself, rather than only its norms, computes a generalised Hodge-theoretic height. This is done by showing that the relative homology group induced by the two k-planes and two (n-k-1)-planes is naturally an extension of mixed Hodge structures of Z(0) by Z(1), and that the height of this extension equals the generalised cross-ratio.
Finally, using local methods, we study regularised limits of local heights and interpret these limits using only the central degenerate geometry, together with small perturbations. This thesis provides a new class of examples associating geometric shapes to limits of local heights.
This thesis is about homological algebra and singular (co)homology.
In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups.
In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces. ...
In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups.
In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces. ...
This thesis is about homological algebra and singular (co)homology.
In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups.
In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces.
In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups.
In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces.