We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work Quantum cohomology of flag varieties (Internat. Math. Res. Notices, no. 6 (1995), 263–277). We also give a geometric proof of the quantum Monk formula.

If an algebraic torus T acts on a complex projective algebraic variety X then the affine scheme Spec H∗ T (X;C) associated to the equivariant cohomology is often an arrangement of linear subspaces of the vector space H 2 (X;C). In many situations the ordinary cohomology ring of X can be described in terms of this arrangement.

States in the absolute (semi-relative) cohomology but not in the relative cohomology are examined through the component decomposition of the string field theory action for the 2-D string. It is found that they are auxiliary fields without kinetic terms, but are important for instance in the master equation for the Ward-Takahashi identities. The ghost structure is analyzed in the Siegel gauge, b...

We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree d rational curves in P. We deduce as special cases algebro-geometric and arithmetic refinements of topological computations of Segal, Cohen–Cohen–Mann–Milgr...

Let f1, . . . , fr ∈ K[x], K a field, be homogeneous polynomials and put F = ∑r i=1 yifi ∈ K[x, y]. The quotient J = K[x, y]/I, where I is the ideal generated by the ∂F/∂xi and ∂F/∂yj , is the Jacobian ring of F . We describe J by computing the cohomology of a certain complex whose top cohomology group is J .

We give an example of a compact connected Lie group the lowest rank such that mod 2 cohomology ring its classifying space has nonzero nilpotent element.

Torus manifolds are topological generalization of smooth projective toric manifolds. We compute the rational cohomology ring a class locally standard torus whose orbit space may have proper non-acyclic faces.

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of C2. The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. A relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves is pr...