Anand Dessai

Rigidity theorems for SpinC-manifolds

Abstract We prove a rigidity and vanishing theorem for Spin C -manifolds. Special cases include the rigidity of the elliptic genus and the elliptic genera of higher level. We state some applications to Spin C -manifolds with nice Pin (2)-action.

Characteristic numbers of positively curved spin-manifolds with symmetry

Let M be a spin-manifold of positive sectional curvature and dimension > 8. Suppose a compact connected Lie group G acts smoothly on M. We show that the characteristic number A(M, TM) vanishes if G contains two commuting involutions acting isometrically on M.

On the rigidity theorem for elliptic genera

We give a detailed proof of the rigidity theorem for elliptic gen- era. Using the Lefschetz fixed point formula we carefully analyze the relation between the characteristic power series defining the elliptic genera and the equivariant elliptic genera. We show that equivariant elliptic genera converge to Jacobi functions which are holomorphic. This implies the rigidity of elliptic genera. Our approach can be easily modified to give a proof of the rigidity theorem for the elliptic genera of level N.

Nonconnected moduli spaces of nonnegative sectional curvature metrics on simply connected manifolds

We show that in each dimension 4n+3, n⩾1, there exist infinite sequences of closed smooth simply connected manifolds M of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension 7, and our result also holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, in conjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such M the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold M×R also has infinitely many path components.

Homotopy complex projective spaces with Pin(2)-action

Let M be a manifold homotopy equivalent to the complex projective space CPm. Petrie conjectured that M has standard total Pontrjagin class if M admits a non-trivial action by S1. We prove the conjecture for m<12 under the assumption that the action extends to a nice Pin(2)-action with fixed point. The proof involves equivariant index theory for Spinc-manifolds and Jacobi functions as well as classical results from the theory of transformation groups.

COMPLETE INTERSECTIONS WITH S 1 -ACTION

We give the diffeomorphism classification of complete intersections with S1-symmetry in dimension ≤ 6. In particular, we show that a 6-dimensional complete intersection admits a smooth non-trivial S1-action if and only if it is diffeomorphic to the complex projective space or the quadric. We also prove that in any odd complex dimension only finitely many complete intersections can carry a smooth effective action by a torus of rank > 1.

Bordism-Finiteness and Semi-simple Group Actions

We give bordism-finiteness results for smooth S3-manifolds. Consider the class of oriented manifolds which admit an S1-action with isolated fixed points such that the action extends to an S3-action with fixed point. We exhibit various subclasses, characterized by an upper bound for the Euler characteristic and properties of the first Pontryagin class p1, for example p1 = 0, which contain only finitely many oriented bordism types in any given dimension. Also we show finiteness results for homotopy complex projective spaces and complete intersections with S3-action as above.