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    syllabus [16.02.2023 17:42]
    timashev создано     		syllabus [08.04.2025 16:43] (текущий)
    Строка 1: Строка 1:
     ====== Spherical varieties ====== ====== Spherical varieties ======
      
    -=== Introduction of the course ===+=== Introduction to the course ===
      
     Spherical homogeneous spaces are both classical and modern objects of study in algebra and geometry. Particular examples were studied by geometers since the XIX-th century, starting from spheres and projective spaces and passing to Grassmannians, flag varieties, spaces of quadrics and, more generally, symmetric spaces. However, the unity of properties and approaches to the study of spherical spaces was well understood not too long ago, which led to an active development of the theory during last 40 years. Spherical homogeneous spaces lie at the crossroads of algebraic geometry, theory of algebraic groups, enumerative geometry, harmonic analysis, and representation theory. We shall consider various properties of spherical spaces from viewpoints of algebraic transformation groups, harmonic analysis, and equivariant symplectic geometry. Spherical homogeneous spaces are both classical and modern objects of study in algebra and geometry. Particular examples were studied by geometers since the XIX-th century, starting from spheres and projective spaces and passing to Grassmannians, flag varieties, spaces of quadrics and, more generally, symmetric spaces. However, the unity of properties and approaches to the study of spherical spaces was well understood not too long ago, which led to an active development of the theory during last 40 years. Spherical homogeneous spaces lie at the crossroads of algebraic geometry, theory of algebraic groups, enumerative geometry, harmonic analysis, and representation theory. We shall consider various properties of spherical spaces from viewpoints of algebraic transformation groups, harmonic analysis, and equivariant symplectic geometry.
    Строка 14: Строка 14:
      
       - Two motivating problems: geometric proof of the Clebsch-Gordan formula and Steiner's conic problem.   - Two motivating problems: geometric proof of the Clebsch-Gordan formula and Steiner's conic problem.
    - 
       - Equivariant line bundles over homogeneous spaces and geometric models for representations of algebraic groups. Frobenius reciprocity.   - Equivariant line bundles over homogeneous spaces and geometric models for representations of algebraic groups. Frobenius reciprocity.
    - 
       - Spherical homogeneous spaces, equivalent defining properties: existence of an open orbit for a Borel subgroup, the only Borel-invariant rational functions are constant, multiplicity free representation in global sections of line bundles.   - Spherical homogeneous spaces, equivalent defining properties: existence of an open orbit for a Borel subgroup, the only Borel-invariant rational functions are constant, multiplicity free representation in global sections of line bundles.
    - 
       - Examples of spherical homogeneous spaces. Symmetric spaces are spherical.   - Examples of spherical homogeneous spaces. Symmetric spaces are spherical.
    - 
       - Spherical homogeneous spaces from the symplectic viewpoint: coisotropic property for the Hamiltonian action in the cotangent bundle, commutativity of the Poisson algebra of invariant functions and of the algebra of invariant differential operators.   - Spherical homogeneous spaces from the symplectic viewpoint: coisotropic property for the Hamiltonian action in the cotangent bundle, commutativity of the Poisson algebra of invariant functions and of the algebra of invariant differential operators.
    - 
       - Spherical varieties as equivariant (partial) compactifications of spherical homogeneous spaces: examples and the problem of classification.   - Spherical varieties as equivariant (partial) compactifications of spherical homogeneous spaces: examples and the problem of classification.
    - 
       - Birational invariants of a spherical variety: weight lattice, valuation cone, colors.   - Birational invariants of a spherical variety: weight lattice, valuation cone, colors.
    - 
       - Finiteness of the orbit set of a spherical variety.   - Finiteness of the orbit set of a spherical variety.
    - 
       - Simple spherical varieties and colored cones.   - Simple spherical varieties and colored cones.
    - 
       - Combinatorial classification of spherical varieties via colored fans.   - Combinatorial classification of spherical varieties via colored fans.
    - 
       - Geometry of spherical varieties via colored fans: criteria of compactness and affinity, configuration and adherence order on orbits.   - Geometry of spherical varieties via colored fans: criteria of compactness and affinity, configuration and adherence order on orbits.
    - 
       - Toric varieties as a particular case of spherical varieties.   - Toric varieties as a particular case of spherical varieties.
    - 
       - Reductive algebraic monoids and group embeddings.   - Reductive algebraic monoids and group embeddings.
    - 
       - Wonderful completions of symmetric spaces and groups.   - Wonderful completions of symmetric spaces and groups.
    - 
       - Divisors and line bundles on a spherical variety, a description of Picard group and divisor class group.   - Divisors and line bundles on a spherical variety, a description of Picard group and divisor class group.
    - 
       - Globally generated, nef and ample line bundles on spherical varieties.   - Globally generated, nef and ample line bundles on spherical varieties.
    - 
       - Spherical double flag varieties and tensor product decompositions.   - Spherical double flag varieties and tensor product decompositions.
    - 
       - Cellular decomposition and cohomology of a smooth projective spherical variety.   - Cellular decomposition and cohomology of a smooth projective spherical variety.
    - 
       - Enumerative geometry on a spherical homogeneous space, Halphen ring (ring of conditions).   - Enumerative geometry on a spherical homogeneous space, Halphen ring (ring of conditions).
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       - Intersection index of divisors on a projective spherical variety and on a spherical homogeneous space. Generalized Bezout theorem on a reductive group.   - Intersection index of divisors on a projective spherical variety and on a spherical homogeneous space. Generalized Bezout theorem on a reductive group.
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       - Solution of Steiner's conic problem via spherical varieties.   - Solution of Steiner's conic problem via spherical varieties.
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       - Frobenius splitting and its properties.   - Frobenius splitting and its properties.
    - 
       - Frobenius splitting of spherical varieties reduced to positive characteristic and its geometric consequences: rationality of singularities, vanishing of higher cohomology of nef line bundles, projective normality.   - Frobenius splitting of spherical varieties reduced to positive characteristic and its geometric consequences: rationality of singularities, vanishing of higher cohomology of nef line bundles, projective normality.
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       - Wonderful varieties and spherical systems, classification of spherical homogeneous spaces.   - Wonderful varieties and spherical systems, classification of spherical homogeneous spaces.
      
     +=== References ===
     +  - M. Brion. [[http://www-fourier.univ-grenoble-alpes.fr/~mbrion/spheriques.pdf|Variétés sphériques]]. Notes de la session de la S. M. F. «Opérations hamiltoniennes et opérations de groupes algébriques» (Grenoble, 1997).
     +  - D.A. Timashev. Homogeneous spaces and equivariant embeddings. [[https://doi.org/10.1007/978-3-642-18399-7|DOI link]]
     +  - N. Perrin. On the geometry of spherical varieties. [[https://doi.org/10.1007/s00031-014-9254-0|DOI link]]
     +  - J. Gandini. Embeddings of spherical homogeneous spaces. [[https://doi.org/10.1007/s10114-018-7162-2|DOI link]]
    syllabus.1676558559.txt.gz · Последнее изменение: 08.04.2025 16:42 (внешнее изменение)

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