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Register a new userA Characterization of Symplectic Grassmannians
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We provide a characterization of Symplectic Grassmannians in terms of their Varieties of Minimal Rational Tangents.
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In this paper we discuss physical derivations of the quantum K theory rings of symplectic Grassmannians. We compare to standard presentations in terms of Schubert cycles, but most of our work revolves around a proposed description in terms of two other bases, involving shifted Wilson lines and lambda_y classes, which are motivated by and amenable to physics, and which we also provide for ordinary Grassmannians.
Let $X$ be an integral scheme of finite presentation over a field. Let $q$ be a singular closed point of $X$. We prove that there exists an open subset $V$ of $X$ containing $q$ such that $V$ admits a resolution, that is, there exists a smooth scheme $widetilde V$ and a proper birational morphism from $widetilde V$ onto $V$.
We complement our previous computation of the Chow-Witt rings of classifying spaces of special linear groups by an analogous computation for the general linear groups. This case involves discussion of non-trivial dualities. The computation proceeds along the lines of the classical computation of the integral cohomology of ${rm BO}(n)$ with local coefficients, as done by Cadek. The computations of Chow-Witt rings of classifying spaces of ${rm GL}_n$ are then used to compute the Chow-Witt rings of the finite Grassmannians. As before, the formulas are close parallels of the formulas describing integral cohomology rings of real Grassmannians.
Let k be a field. Denote by Spc(k)_* the unstable, pointed motivic homotopy category and by Omega_Gm: Spc(k)_* to Spc(k)_* the Gm-loops functor. For a k-group G, denote by Gr_G the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence Omega_Gm G = Gr_G. If k is perfect, we use this to compute the motive M(Omega_Gm G) in DM(k, Z).
Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 - the symplectically primitive but complex imprimitive groups - and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving 39+9=48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution.
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