Periode und Index eines prinzipal-homogenen Raumes über gewissen Abelschen Varietäten.
Periodic integrals and tautological systems
We study period integrals of CY hypersurfaces in a partial flag variety. We construct a regular holonomic system of differential equations which govern the period integrals. By means of representation theory, a set of generators of the system can be described explicitly. The results are also generalized to CY complete intersections. The construction of these new systems of differential equations has lead us to the notion of a tautological system.
Picard groups of Deligne-Lusztig varieties -- with a view toward higher codimensions.
Pieri Type formula for isotropic Grassmannians; the operator approach.
Pieri's formula for flag manifolds and Schubert polynomials
We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary or a complete symmetric polynomial. Thus, we generalize the classical Pieri’s formula for Schur polynomials (associated to Grassmann varieties) to Schubert polynomials (associated to flag manifolds). Our primary technique is an explicit geometric description of certain...
Pieri-type formulas for maximal isotropic Grassmannians via triple intersections
We give an elementary proof of the Pieri-type formula in the cohomology ring of a Grassmannian of maximal isotropic subspaces of an orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The multiplicities (which are powers of 2) in the Pieri-type formula are seen to arise from the intersection of a collection of quadrics with a linear space.
Pieri-type intersection formulas and primary obstructions for decomposing 2-forms
We study the homological intersection behaviour for the Chern cells of the universal bundle of G(d,Qₙ), the space of [d]-planes in the smooth quadric Qₙ in over the field of complex numbers. For this purpose we define some auxiliary cells in terms of which the intersection properties of the Chern cells can be described. This is then applied to obtain some new necessary conditions for the global decomposability of a 2-form of constant rank.
Pluripolar sets in Grassmann manifolds
Poisson geometry of directed networks in an annulus
As a generalization of Postnikov’s construction [P], we define a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We then show that universal Poisson brackets introduced for the space of edge weights in [GSV3] induce a family of Poisson structures on rational matrix-valued functions and on the space of loops in the Grassmannian. In the former case, this family includes, for a particular kind of networks, the Poisson bracket associated...
Positivity and Kleiman transversality in equivariant -theory of homogeneous spaces
We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth–Ram [GrRa04] concerning the alternation of signs in the structure constants for torus-equivariant -theory of generalized flag varieties . These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant...
Product and puzzle formulae for Belkale-Kumar coefficients.
Projective normality of flag varieties and Schubert varieties.
Puissances extérieures, déterminants et cycles de Schubert
Quantum cohomology of flag manifolds and quantum Toda lattices.
Quelques applications de la cohomologie d'intersection
rc-graphs and Schubert polynomials.
Refined intersection products and limiting linear subspaces of hypersurfaces.
Ropes on a line embedded in a Grassmannian variety.
Schubert functions and the number of reduced words of permutations.
Schubert varieties and Demazure's character formula.