LetMbe a compact Riemannian manifold of dimension n > 2. The k-curvature, for k = 1, 2, . . . , n, is defined as the k-th ele- mentary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a con- formal metric whose k-curvature is a constant. When k = 1, it reduces to the well-known Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions is known for the case k = 2, n = 4, for locally conformally flat manifolds and for the cases k > n/2. In this paper we prove the solvability of the k-Yamabe problem in the remaining cases k ≤ n/2, under the hypothesis that the problem is variational. This includes all of the cases k = 2 as well as the locally conformally flat case.

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As a continuation of\lianyaufour, we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties. In Part A of this paper, motivated by the recent work of Kachru-Vafa, we degenerate a three-moduli family of Calabi-Yau toric varieties along a codimension one subfamily which can be described by the vanishing of certain Mori coordinate, corresponding to going to the``large volume limit''in a certain direction. Then we see that the deformation space of the subfamily is the same as a certain family of K3 toric surfaces. This family can in turn be studied by further degeneration along a subfamily which in the end is described by a family of elliptic curves. The periods of the K3 family (and hence the original Calabi-Yau family) can be described by the squares of the periods of the elliptic curves. The consequences include:(1) proofs of various conjectural formulas of physicists\vk\lkm involving mirror maps and modular functions;(2) new identities involving multi-variable hypergeometric series and modular functions--generalizing\lianyaufour. In Part B, we study for two-moduli families the perturbation series of the mirror map and the type A Yukawa couplings near certain large volume limits. Our main tool is a new class of polynomial PDEs associated with Fuchsian PDE systems. We derive the first few terms in the perturbation series. For the case of degree 12 hypersurfaces in\P^ 4 [6, 2, 2, 1, 1], in one limit the series of the couplings are expressed in terms of the j function. In another limit, they are expressed in terms of rational functions. The latter give explicit formulas for infinite sequences of``instanton

Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of Q-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincar´e transform, which has several consequences. In many concrete geometric contexts these functions provide useful invariants. We illustrate this with two different applications, the first one to GVsubschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.

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