Online Talk

The singular cohomology ring of a smooth compact toric variety is given by a theorem of Danilov and Jurkiewicz. However, even the group structure of the integral cohomology is vastly unknown for singular compact toric varieties. Consider the Leray-Serre spectral sequence of the filtration by orbit dimensions of the compact torus in a compact toric variety. It collapses at the second page for cellular cohomology with complex coefficients (by Danilov) and for rational Borel-Moore homology (by Totaro). In the talk, a collapse result of this spectral sequence will be given for singular cohomology with integral coefficients (with trivial extension in abelian groups).m Explicit formulas for ordinary and bigraded Betti numbers for skeletons of compact nonsingular toric varieties will be presented.Based on arXiv:2401.14146.