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    syllabus [16.02.2023 17:43]
    timashev     		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 37: Строка 37:
       - 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.
       - 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.1676558615.txt.gz · Последнее изменение: 08.04.2025 16:42 (внешнее изменение)

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