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56
Schubert cells and cohomology of the space G/P
 31 Fomin S.V. and Kirillov A.N., Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in Geometry, 147182, Progress in Math. 172
, 1973
"... We study the homological properties of the factor space G/P, where G is a complex semisimple Lie group and Ρ a parabolic subgroup of G. To this end we compare two descriptions of the cohomology of such spaces. One of these makes use of the partition of G/P into cells (Schubert cells), while the oth ..."
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We study the homological properties of the factor space G/P, where G is a complex semisimple Lie group and Ρ a parabolic subgroup of G. To this end we compare two descriptions of the cohomology of such spaces. One of these makes use of the partition of G/P into cells (Schubert cells), while the other consists in identifying the cohomology of G/P with certain polynomials on the Lie algebra of the Cartan subgroup Η of G. The results obtained are used to describe the algebraic action of the Weyl group W of G on the cohomology of G/P. Contents
Complex and Kähler structures on compact solvmanifolds
 J. Symplectic Geom
"... Abstract. We discuss our recent results on the existence and classification problem of complex and Kähler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact solvmanifolds (and compact homogeneous manifolds in gen ..."
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Cited by 30 (4 self)
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Abstract. We discuss our recent results on the existence and classification problem of complex and Kähler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact solvmanifolds (and compact homogeneous manifolds in general). 1.
Hofer metric and geometry of conjugacy classes in Lie groups
 Invent. Math
"... Given a closed symplectic manifold (M, ω) we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group Ham (M, ω) by means of the Hofer metric on Ham (M, ω). We use pseudoholomorphic curves involved in the definition of the multiplicative structure ..."
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Cited by 18 (0 self)
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Given a closed symplectic manifold (M, ω) we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group Ham (M, ω) by means of the Hofer metric on Ham (M, ω). We use pseudoholomorphic curves involved in the definition of the multiplicative structure on the Floer cohomology of a symplectic manifold (M, ω) to estimate this quantity in terms of actions of some periodic orbits of related Hamiltonian flows. As a corollary we get a new way to obtain AgnihotriBelkaleWoodward inequalities for eigenvalues of products of unitary matrices. As another corollary we get a new proof of the geodesic property (with respect to the Hofer metric) of Hamiltonian flows generated by certain autonomous Hamiltonians. Our main technical tool is Karea defined for Hamiltonian fibrations over a surface with boundary in the spirit of L.Polterovich’s work on Hamiltonian fibrations over S 2.
Arithmetic Intersection Theory on Flag Varieties
 Dept. Math., Massachusetts Institute of Technology
, 1996
"... Let F be the complete flag variety over SpecZwith the tautological filtration 0 ae E 1 ae E 2 ae \Delta \Delta \Delta ae En = E of the trivial bundle E over F . The trivial hermitian metric on E(C ) induces metrics on the quotient line bundles L i (C ). Let bc 1 (L i ) be the first Chern class of ..."
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Cited by 13 (5 self)
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Let F be the complete flag variety over SpecZwith the tautological filtration 0 ae E 1 ae E 2 ae \Delta \Delta \Delta ae En = E of the trivial bundle E over F . The trivial hermitian metric on E(C ) induces metrics on the quotient line bundles L i (C ). Let bc 1 (L i ) be the first Chern class of L i in the arithmetic Chow ring d CH(F ) and b x i = \Gammabc 1 (L i ). Let h2Z[X 1 ; : : : ; Xn ] be a polynomial in the ideal he 1 ; : : : ; e n i generated by the elementary symmetric polynomials e i . We give an effective algorithm for computing the arithmetic intersection h(bx 1 ; : : : ; b xn ) in d CH(F ), as the class of a SU (n)invariant differential form on F (C ). In particular we show that all the arithmetic Chern numbers one obtains are rational numbers.
Characteristic polynomials of random Hermitian matrices and DuistermaatHeckman localisation on noncompact Kahler manifolds
 Nucl. Phys. B
"... We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N × N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard ”supersymmetry ” approach, we int ..."
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Cited by 11 (6 self)
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We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N × N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard ”supersymmetry ” approach, we integrate out Grassmann variables at the early stage and circumvent the use of the HubbardStratonovich transformation in the ”bosonic ” sector. The method, suggested recently by one of us [19], is shown to be capable of calculation when reinforced with a generalization of the ItzyksonZuber integral to a noncompact integration manifold. We arrive to such a generalisation by discussing the DuistermaatHeckman localization principle for integrals over noncompact homogeneous Kähler manifolds. In the limit of large N the asymptotic expression for the correlation function reproduces the result outlined earlier by Andreev and Simons [14]. 1
QUANTUM CHARACTERISTIC CLASSES AND THE HOFER METRIC
, 709
"... Abstract. Given a closed monotone symplectic manifold M, we define certain characteristic cohomology classes of the free loop space LHam(M, ω) with values in QH∗(M), and their S 1 equivariant version. These classes generalize the Seidel representation and satisfy versions of the axioms for Chern cla ..."
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Cited by 9 (3 self)
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Abstract. Given a closed monotone symplectic manifold M, we define certain characteristic cohomology classes of the free loop space LHam(M, ω) with values in QH∗(M), and their S 1 equivariant version. These classes generalize the Seidel representation and satisfy versions of the axioms for Chern classes. In particular there is a Whitney sum formula, which gives rise to a graded ring homomorphism from the ring H∗(LHam(M, ω), Q), with its Pontryagin product to QH2n+∗(M) with its quantum product. As an application we prove an extension of a theorem of McDuff and Slimowitz on minimality in the Hofer metric of a semifree Hamiltonian circle action, to higher dimensional geometry of the loop space LHam(M, ω).
Berezin quantization and unitary representations of Lie groups
, 1994
"... In 1974, Berezin proposed a quantum theory for dynamical systems having a Kähler manifold as their phase space. The system states were represented by holomorphic functions on the manifold. For any homogeneous Kähler manifold, the Lie algebra of its group of motions may be represented either by holo ..."
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Cited by 8 (1 self)
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In 1974, Berezin proposed a quantum theory for dynamical systems having a Kähler manifold as their phase space. The system states were represented by holomorphic functions on the manifold. For any homogeneous Kähler manifold, the Lie algebra of its group of motions may be represented either by holomorphic differential operators (“quantum theory”), or by functions on the manifold with Poisson brackets, generated by the Kähler structure (“classical theory”). The Kähler potentials and the corresponding Lie algebras are constructed now explicitly for all unitary representations of any compact simple Lie group. The quantum dynamics can be represented in terms of a phasespace path integral, and the action principle appears in the semiclassical approximation.
Plane curves with a big fundamental group of the complement
 MATHÉMATIQUES, 354, GRENOBLE 1996, 26P.; EPRINT ALGGEOM/9607006
, 1997
"... Let C ⊂ IP² be an irreducible plane curve whose dual C ∗ ⊂ IP² ∗ is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincaré group G = π1(IP 2 \ C) contains a free group with two generators. If the geometric genus g of C is at least 2, then a subgroup o ..."
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Cited by 8 (0 self)
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Let C ⊂ IP² be an irreducible plane curve whose dual C ∗ ⊂ IP² ∗ is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincaré group G = π1(IP 2 \ C) contains a free group with two generators. If the geometric genus g of C is at least 2, then a subgroup of G can be mapped epimorphically onto the fundamental group of the normalization of C, and the result follows. To handle the cases g = 0, 1, we construct universal families of immersed plane curves and their Picard bundles. This allows us to reduce the consideration to the case of Plücker curves. Such a curve C can be regarded as a plane section of the corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying Zariski–Lefschetz type arguments we deduce the result from ‘the bigness’ of the braid group Bd, g, that is, of the group of d–string braids of a compact genus g Riemann surface.