A vanishing theorem on arithmetic surfaces.
Let be a complex algebraic group, simple and simply connected, a maximal torus and the Weyl group. One shows that the coarse moduli space parametrizing -equivalence classes of semistable -bundles over an elliptic curve is isomorphic to . By a result of Looijenga, this shows that is a weighted projective space.
We express the Euler-Poincaré characteristic of a semi-algebraic set, which is the intersection of a non-singular complete intersection with two polynomial inequalities, in terms of the signatures of appropriate bilinear symmetric forms.
In this paper we show that on a general hypersurface of degree r = 3,4,5,6 in P5 a rank 2 vector bundle ε splits if and only if h1ε(n) = h2ε(n) = 0 for all n ∈ Z. Similar results for r = 1,2 were obtained in [15], [16] and [2].
We review some facts about rank two arithmetically Cohen-Macaulay bundles on quintic threefolds. In particular, we separate them into seventeen natural classes, only fourteen of which can appear on a general quintic. We discuss some enumerative problems arising from these.
In this paper we construct and study an action of the affine braid group associated with a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a “categorical version” of Kazhdan-Lusztig-Ginzburg’s construction of the affine Hecke algebra, and is used in particular by the first author and I. Mirković in the course...
Let be a proper smooth variety over a field of characteristic and an effective divisor on with multiplicity. We introduce a generalized Albanese variety Alb of of modulus , as higher dimensional analogue of the generalized Jacobian with modulus of Rosenlicht-Serre. Our construction is algebraic. For we give a Hodge theoretic description.
For a complex projective manifold Gromov-Witten invariants can be constructed either algebraically or symplectically. Using the versions of Gromov-Witten theory by Behrend and Fantechi on the algebraic side and by the author on the symplectic side, we prove that both points of view are equivalent