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Algebraic C*-actions and homotopy continuationEklund, David January 2008 (has links)
<p>Let X be a smooth projective variety over C equipped with a C*-action whose fixed points are isolated. Let Y and Z be subvarieties of complementary dimentions in X which intersect properly. In this thesis we present an algorithm for computing the points of intersection between Y and Z based on homotopy continuation and the Bialynicki-Birula decompositions of X into locally closed invariant subsets. As an application we present a new solution to the inverse kinematic problem of a general six-revolute serial-link manipulator.</p>
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Algebraic C*-actions and homotopy continuationEklund, David January 2008 (has links)
Let X be a smooth projective variety over C equipped with a C*-action whose fixed points are isolated. Let Y and Z be subvarieties of complementary dimentions in X which intersect properly. In this thesis we present an algorithm for computing the points of intersection between Y and Z based on homotopy continuation and the Bialynicki-Birula decompositions of X into locally closed invariant subsets. As an application we present a new solution to the inverse kinematic problem of a general six-revolute serial-link manipulator. / QC 20101108
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C*-actions on rational homogeneous varieties and the associated birational mapsFranceschini, Alberto 20 March 2023 (has links)
Given a birational map among projective varieties, it is known that there exists a variety Z with a one-dimensional torus action such that the birational map is induced from two geometric quotients of Z. We proceed in the opposite direction: given a smooth projective variety X with a one-dimensional torus action, one can define a birational map associated to the action and study the properties of the map via the geometry of X. Rational homogeneous varieties admit natural torus actions, so they are a good class of example to test the general theory. In the thesis, we obtain and discuss some results about the birational maps associated to some one-dimensional torus actions on rational homogeneous varieties.
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Geometric realizations of birational mapsBarban, Lorenzo 29 January 2024 (has links)
In this thesis we study the relation between algebraic torus actions on complex projective varieties and the birational geometry of their geometric quotients. Given a C*-action on a normal projective variety X, there exist two unique connected components of the fixed point locus, called the sink Y− and the source Y+, containing the limit at ∞ and 0 of the general orbit. Let GX− (resp. GX+) be the variety parametrizing the orbits converging to the sink (resp. the source). Since there exists an open subset of points converging to Y±, we obtain a birational map
ψ: GX->GX+. By choosing different linearizations of ample line bundles on X, we obtain a factorization of the birational map ψ among inner geometric quotient, parametrizing different open subsets of stable points.
In this setting, we investigate the local analytic geometry of the birational map ψ. On one hand we link certain birational transformations, called rooftop flips, with varieties with two projective bundles structures. On the other we study when the birational map ψ can be locally described by a toric flip of Atiyah type.
If on one side a C*-action naturally induces a birational map among geometric quotients, it is meaningful to study the opposite direction: more precisely, given a birational map φ: Z+->Z− among normal projective varieties, how can we construct a normal projective variety X, endowed with a C*-action, such that Z− is the sink, Z+ is the source, and the natural birational map ψ constructed above coincide with φ? Such an X is called a geometric realization of the birational map φ. We propose a construction of a geometric realization of φ, whose geometry reflects the factorization of the map as a composition of flips, blow-ups and blow-downs. We describe in particular the case in which φ is a small modification of dream type, namely a birational map which is an isomorphism in codimension 1 associated to a finitely generated multisection ring. Moreover, we show that the cone of divisors associated to such multisection rings admits a chamber decomposition where the models are the geometric quotients of the C*-action. If in addition Z± are assumed to be toric varieties, we construct a function in SageMath to compute the polytope of the associated toric geometric realization.
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Semi-toric integrable systems and moment polytopesWacheux, Christophe 17 June 2013 (has links) (PDF)
Un système intégrable semi-torique sur une variété symplectique de dimension 2n est un système intégrable dont le flot de n − 1 composantes de l'application moment est 2 -périodique. On obtient donc une action hamiltonienne du tore Tn−1. En outre, on demande que tous les points critiques du système soient non-dégénérés et sans composante hyperbolique. En dimension 4, San V˜u Ngo.c et Álvaro Pelayo ont étendu à ces systèmes semi-toriques les résultats célèbres d'Atiyah, Guillemin, Sternberg et Delzant concernant la classification des systèmes toriques. Dans cette thèse nous proposons une extension de ces résultats en dimension quelconque, à commencer par la dimension 6. Les techniques utilisées relèvent de l'analyse comme de la géométrie symplectique, ainsi que de la théorie de Morse dans des espaces différentiels stratifiés. Nous donnons d'abord une description de l'image de l'application moment d'un point de vue local, en étudiant les asymptotiques des coordonnées actionangle au voisinage d'une singularité foyer-foyer, avec le phénomène de monodromie du feuilletage qui en résulte. Nous passons ensuite à une description plus globale dans la veine des polytopes d'Atiyah, Guillemin et Sternberg. Ces résultats sont basés sur une étude systématique de la stratification donnée par les fibres de l'application moment. Avec ces résultats, nous établissons la connexité des fibres des systèmes intégrables semi-toriques de dimension 6 et indiquons comment nous comptons démontrer ce résultat en dimension quelconque.
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Semi-toric integrable systems and moment polytopes / Systèmes intégrables semi-toriques et polytopes momentWacheux, Christophe 17 June 2013 (has links)
Les systèmes intégrables toriques sont des systèmes intégrables dont toutes les composantes de l'application moment sont périodiques de même période. Il s'agit donc de variétés symplectiques munies d'actions Hamiltoniennes de tores. Au début des années 80, Atiyah-Guillemin-Sternberg ont démontré que l'image de l'application moment était un polytope convexe à face rationnelles. Peu de temps après, Delzant a démontré que dans le cas intégrable qui nous intéresse, ce polytope caractérisait entièrement le système : la variété symplectique comme l'action du tore. Le champs d'étude s'est ensuite élargi aux systèmes dits semi-toriques. Ce sont des systèmes intégrables dont toutes les composantes de l'application moment sauf une sont périodiques de même période. En outre, pour simplifier l'étude de ces systèmes, on demande que tous les points critiques du systèmes soient non-dégénérés, et sans composante hyperbolique pour la hessienne. En revanche les points critiques des systèmes semi-toriques peuvent comporter des composantes dites "foyer-foyer". Celles-ci ont une dynamique plus riche que les singularités elliptiques, mais conservent certaines propriétés qui rendent leur analyse plus aisée que les singularités hyperboliques. San Vu-Ngoc et Alvaro Pelayo ont réussi à étendre pour ces systèmes semi-toriques les résultats d'Atiyah-Guillemin-Sternberg et Delzant en dimension 2. L'objectif de cette thèse est de proposer une extension de ces résultats en dimension quelconque, à commencer par la dimension 3. Les techniques utilisées relèvent de l'analyse comme de la géométrie symplectique, ainsi que de la théorie de Morse dans des espaces différentiels stratifiés. / Semi-toric integrable systems are integrable systems whose every component of the moment map are periodic of the same period. They are symplectic manifolds endowed with a Hamiltonian torus actions. At the beginning of the 80's, Atiyah-Guillemin-Sternberg proved that the image of the moment map was a polytope with rational faces. A bit after that, Delzant showed that in the integrable case that matters to us, this polytope characterized entirely the system, that is, the symplectic manifold as well as the torus action. Next, field of study widened to semi-toric systems. They are integrable systems whose all components except one are periodic with the same period. Moreover, to simplify their study, we ask that these systems have only non-degenerate critical points without hyperbolic components. On the other hand, critical points of semi-toric systems can have so-called ''focus-focus'' components. They have a richer dynamic than elliptic singularities, but it retains some properties that makes them easier to study than hyperbolic singularities. San Vu-Ngoc and Alvaro Pelayo have managed to extend to these semi-toric systems the results of Atiyah-Guillemin-Sternberg and Delzant in dimension 2. The objective of this thesis is to propose an extension of these results to any dimension, starting with dimension 3. Techniques involved are analysis as well as symplectic geometry, and Morse theory in stratified differential spaces.
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Poincaré duality in equivariant intersection theory / Poincaré duality in equivariant intersection theoryGonzales Vilcarromero, Richard Paul 25 September 2017 (has links)
We study the Poincaré duality map from equivariant Chow cohomology to equivariant Chow groups in the case of torus actions on complete, possibly singular, varieties with isolated fixed points. Our main results yield criteria for the Poincaré duality map to become an isomorphism in this setting. The methods rely on the localization theorem for equivariant Chow cohomology and the notion of algebraic rational cell. We apply our results to complete spherical varieties and their generalizations. / En este artículo estudiamos el homomorfismo de dualidad de Poincaré, el cual relaciona cohomología de Chow equivariante y grupos de Chow equivariante en aquellos casos donde un toro algebraico actúa sobre una variedad singular compacta y con puntos fijos aislados. Nuestros resultados proporcionan criterios bajo los cuales el homomorfismo de dualidadde Poincaré es un isomorfismo. Para ello, usamos el teorema de localización en cohomología de Chow equivariante y la noción de célula algebraica racional. Aplicamos nuestros resultados a las variedades esféricas compactas y sus generalizaciones.
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