Spin polynomial invariants, as denned by Pidstrigatch and Tyurin, are used to define differentiable invariants for Dolgachev surfaces. In order to define and compute these invariants, some machinery is developed to study the wall structure for Spin polynomials, to deal with the fact that the moduli spaces have the wrong dimension and to handle nonreduced structures. The invariants are then computed.

We study the set of connected components of certain unions of affine Deligne-Lusztig varieties arising from the study of Shimura varieties. We determine the set of connected components for basic $\s $-conjugacy classes. As an application, we verify the Axioms in\cite {HR} for certain PEL type Shimura varieties. We also show that for any nonbasic $\s $-conjugacy class in a residually split group, the set of connected components is" controlled" by the set of straight elements associated to the $\s $-conjugacy class, together with the obstruction from the corresponding Levi subgroup. Combined with\cite {Zhou}, this allows one to verify in the residually split case, the description of the mod- p isogeny classes on Shimura varieties conjectured by Langland and Rapoport. Along the way, we determine the Picard group of the Witt vector affine Grassmannian of\cite {BS} and\cite {Zhu} which is of independent interest.

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.

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.

A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: Given two mildly singular projective varieties with some of the first variety9s pseudonormed spaces being isometric to the corresponding ones of the second variety9s, can one construct a birational map between them that induces these isometries? In this work, a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence.