This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott’s original approach for semistable holomorphic bundles. This leads to a natural proof that the hyperk¨ahler Kirwan map is surjective for the non-fixed determinant case.

We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A1 structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given embeddings with isomorphic normal bundles of a closed manifold B into manifolds Q1 and Q2, we construct a differential graded category from the singular cochains of these spaces, and prove that it is equivalent to the A1 category obtained by considering exact Lagrangian embeddings of Q1 and Q2 which intersect cleanly along B.

In a series of works, one of the authors has developed with J.- M. Hwang a geometric theory of uniruled projective manifolds, especially those of Picard number 1, basing on the study of varieties of minimal rational tangents. A fundamental result in this theory is a principle of analytic continuation under very mild assumptions,called Cartan-Fubini extension, of biholomorphisms between connected open subsets of two Fano manifolds of Picard number 1 which preserve varieties of minimal rational tangents. In this article we develop a generalization of Cartan-Fubini extension for non-equidimensional holomorphic immersions from a connected open subset of a Fano manifold of Picard number 1 into a uniruled projective manifold, under the assumptions that the map sends varieties of minimal rational tangents onto linear sections of varieties of minimal rational tangents and that it satisfies a mild geometric condition formulated in terms of second fundamental forms on varieties of minimal rational tangents. Formerly such a result was known only in the very special case of irreducible Hermitian symmetric manifolds of rank at least two, and the proof relied on the existence of flattening coordinates, viz.,Harish-Chandra coordinates, with respect to which the varieties of minimal rational tangents form a constant family. The proof of the main result, which is based on the deformation theory of rational curves, is differential-geometric in nature and is applicable to the general situation of uniruled projective manifolds without any assumption on the existence of special coordinate systems. As an application, we give a characterization of standard embeddings for certain pairs of rational homogeneous manifolds in terms of embeddings of varieties of minimal rational tangents.

In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a spectral invariant. In the present paper we describe a group theoretic condition under which Sunada’s method gives manifolds with identical covering spectra. When the group theoretic condition of our method is not met, we are able to construct Sunada isospectral manifolds with distinct covering spectra in dimension 3 and higher. Hence, the covering spectrum is not a spectral invariant. The main geometric ingredient of the proof has an interpretation as the minimum-marked-length-spectrum analogue of Colin de Verdi`ere’s classical result on constructing metrics where the first k eigenvalues of the Laplace spectrum have been prescribed.

Let (M, !) be a compact symplectic manifold, and L a compact embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono [8] construct Lagrangian Floer cohomology, yielding groups HF(L, b;nov) for one or HF􀀀 (L1, b1), (L2, b2);nov
for two Lagrangians, where b, b1, b2 are choices of bounding cochains, and exist if and only if L,L1,L2 have unobstructed Floer cohomology. These are independent of choices up to isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M, !), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich. The goal of this paper is to extend [8] to immersed Lagrangians  : L ! M, with transverse self-intersections. In the embedded case, Floer cohomology HF(L, b;nov) is a modified, ‘quantized’ version of singular homology Hn−(L;nov) over the Novikov ring nov. In our immersed case, HF(L, b;nov) turns out to be a quantized version of Hn−(L;nov)  L (p−,p+)2R nov · (p−, p+), where R =  (p−, p+) : p−, p+ 2 L, p− 6= p+, (p−) = (p+) is a set of two extra generators for each self-intersection point of L, and (p−, p+) has degree (p−,p+) 2 Z, an index depending on how L intersects itself at (p−) = (p+). The theory becomes simpler and more powerful for graded Lagrangians in Calabi–Yau manifolds, when we can work over a smaller Novikov ring CY. The proofs involve associating a gapped filtered A1 algebra over 0 nov or 0 CY to  : L ! M, which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called AN,0 algebras for N = 0, 1, 2, . . ..