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We use the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula to calculate the Hirzebruch ?_y genus X_y(S^[n]), where S^[n] is the Hilbert scheme of points of length n of a surface S. Combinatorial interpretation of the weights of the fixed points of the natural torus action on (²)^[n] is used. This is the first step to prove a conjectural formula about the elliptic genus of the Hilbert schemes.

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We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.

We define Stasheff polytopes in the spaces of tropical points of cluster A -varieties. We study the supports of products of elements of canonical bases for cluster A -varieties. We prove that, for the cluster A -variety of type A , such supports are Stasheff polytopes.