For a simplicial complex ∆, the graded Betti number βi,j(k[∆]) of the StanleyReisner ring k[∆] over a field k has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if ∆ is the boundary complex of a d-dimensional stacked polytope with n vertices for d ≥ 3, then βk−1,k(k[∆]) = (k − 1) ` n−d k ́ . We prove this combinatorially.

We prove a tight lower bound on the algebraic Betti numbers of tree and forest ideals and an upper bound on certain graded Betti numbers of squarefree monomial ideals.

For any moduli space of stable representations of quivers, certain smooth varieties, compactifying projective space fibrations over the moduli space, are constructed. The boundary of this compactification is analyzed. Explicit formulas for the Betti numbers of the smooth models are derived. In the case of moduli of simple representations, explicit cell decompositions of the smooth models are co...

We present TopMesh, a tool for extracting topological information from non-manifold three-dimensional objects with parts of non-uniform dimensions. The boundary of such objects is discretized as a mesh of triangles and of dangling edges, representing one-dimensional parts of the object. The geometrical and topological information extracted include the number of elements in the mesh, the number ...

We study the rational cohomology groups of the real De Concini–Procesi model corresponding to a finite Coxeter group, generalizing the type-A case of the moduli space of stable genus 0 curves with marked points. We compute the Betti numbers in the exceptional types, and give formulae for them in types B and D. We give a generating-function formula for the characters of the representations of a ...

Classical Hodge theory gives a decomposition of the complex cohomology of a compact Kähler manifold M , which carries the standard Hodge structure {H(M), p + q = k} of weight k. Deformations of M then lead to variations of the Hodge structure. This is best understood when reformulating the Hodge decomposition in an abstract manner. Let HC = HR ⊕ C be a complex vector space with a real structure...

A degeneration of compact Kähler manifolds gives rise to a monodromy action on Betti moduli space H (X,G) = Hom(π1(X), G)/G over smooth fibres with a complex algebraic structure group G being either abelian or reductive. Assume that the singularities of the central fibre is of normal crossing. When G = C, the invariant cohomology classes arise from the global classes. This is no longer true in ...

We study some symplectic geometric aspects of rationally connected 4-folds. As a corollary, we prove that any smooth projective 4-fold symplectic deformation equivalent to a Fano 4-fold of pseudo-index at least 2 or a rationally connected 4-fold whose second Betti number is 2 is rationally connected.

The notion of a characteristic fibration is introduced. This fibration consists of a base space M and a set of fibres which are dimension groups associated to a noncommutative ring R. Every dimension group of the fibration is isomorphic to the first Betti group of M with a ‘positive cone’ depending continuously on the fibre. The characteristic fibrations are linked to the codimension–1 regular ...

We introduce Fuß–Catalan complexes as d-dimensional generalisations of triangulations of a convex polygon. These complexes are used to refine Catalan numbers and Fuß–Catalan numbers, by introducing colour statistics for triangulations and Fuß–Catalan complexes. Our refinements consist in showing that the number of triangulations, respectively of Fuß–Catalan complexes, with a given colour distri...