Volume 2013, Issue 684

Abstract. We show how the Atiyah–Singer family index theorem for both usual and self-adjoint elliptic operators fits naturally into the framework of the Madsen–Tillmann–Weiss spectra. Our main theorem concerns bundles of odd-dimensional manifolds. Using completely functional-analytic methods, we show that for any smooth proper oriented fibre bundle with odd-dimensional fibres, the family index of the odd signature operator is trivial. The Atiyah–Singer theorem allows us to draw a topological conclusion: the generalized Madsen–Tillmann–Weiss map kills the Hirzebruch -class in rational cohomology. If , this means that induces the zero map in rational cohomology. In particular, the three-dimensional analogue of the Madsen–Weiss theorem is wrong. For 3-manifolds M , we also prove the triviality of in mod p cohomology in many cases. We show an appropriate version of these results for manifold bundles with boundary.

Abstract. This paper studies stable sheaves on abelian surfaces of Picard number one. Our main tools are semi-homogeneous sheaves and Fourier–Mukai transforms. We introduce the notion of semi-homogeneous presentation and investigate the behavior of stable sheaves under Fourier–Mukai transforms. As a consequence, an affirmative answer is given to the conjecture proposed by Mukai in the 1980s. This paper also includes an explicit description of the birational correspondence between the moduli spaces of stable sheaves and the Hilbert schemes.

Abstract. We construct an equivariant algebraic cobordism theory for schemes with an action by a linear algebraic group over a field of characteristic zero.

Abstract. We establish a close connection between intersection multiplicities of special cycles on arithmetic models of the Shimura variety for and Fourier coefficients of derivatives of certain incoherent Eisenstein series, confirming a conjecture of Kudla and Rapoport.

Abstract. We develop a theory of holomorphic differentials on a certain class of non-compact Riemann surfaces obtained by opening infinitely many nodes.

Abstract. In this paper we provide sufficient conditions for the graphs of holomorphic mappings on compact sets in complex manifolds to have Stein neighborhoods. We show that under these conditions the mappings have properties analogous to properties of holomorphic functions on compact sets in ℂ n .

Abstract. Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over and formulate conjectures about their rationality properties. Moreover, if K is a real quadratic field, E is an elliptic curve over without complex multiplication and is a ring class character such that , we prove a Gross–Zagier type formula relating Darmon points to a suitably defined algebraic part of ; this generalizes results of Bertolini, Darmon and Dasgupta to the case of division quaternion algebras and arbitrary characters. Finally, as an application of this formula, assuming the rationality conjectures for Darmon points we obtain vanishing results for Selmer groups of E over extensions of K contained in narrow ring class fields when the analytic rank of E is zero, as predicted by the Birch and Swinnerton-Dyer conjecture.