We prove that the Leray spectral sequence in rational cohomology for the quotient map U n,d → U n,d /G where U n,d is the affine variety of equations for smooth hypersurfaces of degree d in P n (C) and G is the general linear group, degenerates at E 2 .

We study orbit spaces of generalized gradient vector fields for Morse functions. Typically, these orbit spaces are non-Hausdorff. Nevertheless, they are quite structured topologically and are amenable to study. We show that these orbit spaces are locally contractible. We also show that the quotient map associated to each such orbit space is a weak homotopy equivalence and has the path lifting p...

We prove that the Leray spectral sequence in rational cohomology for the quotient map U n,d → U n,d /G where U n,d is the affine variety of equations for smooth hypersurfaces of degree d in P n (C) and G is the general linear group, degenerates at E 2 .

We prove that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of `-adic étale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive

We prove that the Leray spectral sequence in rational cohomology for the quotient map Un,d → Un,d/G where Un,d is the affine variety of equations for smooth hypersurfaces of degree d in P(C) and G is the general linear group, degenerates at E2. 2000 Math. Subj. Class. 14D20, 14L35, 14J70.

Hyperkähler Analogues of Kähler Quotients by Nicholas James Proudfoot Doctor of Philosophy in Mathematics University of California, Berkeley Professor Allen Knutson, Chair Let X be a Kähler manifold that is presented as a Kähler quotient of Cn by the linear action of a compact group G. We define the hyperkähler analogue M of X as a hyperkähler quotient of the cotangent bundle T ∗Cn by the induc...

Let (M,ω) be a connected, compact symplectic manifold equipped with a Hamiltonian SU(2) or SO(3) action. We prove that, as fundamental groups of topological spaces, π1(M) = π1(Mred), where Mred is the symplectic quotient at any value of the moment map φ.

This paper deals with the geometry of the space (GIT quotient) M8 of 8 points in P, and the Gale-quotient N ′ 8 of the GIT quotient of 8 points in P. The space M8 comes with a natural embedding in P, or more precisely, the projectivization of the S8-representation V4,4. There is a single S8-skew cubic C in P. The fact that M8 lies on the skew cubic C is a consequence of Thomae’s formula for hyp...

Abstract We investigate properties of the Albanese map and fundamental group a complex projective variety with many rational points over some function field prove that every linear quotient such is virtually abelian, as well its surjective, has connected fibres, no multiple fibres in codimension one.