Related papers: Gromov-Witten invariants of a class of toric varie…

Combining geometric group theory techniques with geometric topology tools, we show how primitive cohomologies provide useful insights towards unifying the mathematical formulation of Gromov-Witten invariants. In particular, we emphasise the…

We give a description of the (small) quantum cohomology ring of the flag variety as a certain commutative subalgebra in the tensor product of the Nichols algebras. Our main result can be considered as a quantum analog of a result by Y.…

In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even when it is singular) in terms of combinatorial data coming from the fan of the toric variety.

The present article is a continuation of QA/1303.4046, where we discussed the classification of quantum groups with quasi-classical limit $\mathfrak{g}$ and introduced a theory of Belavin-Drinfeld cohomology associated to any…

Let G be a virtually cyclic of the form (Z_a x Z_b) x Z or [Z_a x (Z b x Q_{2^i})] x Z. We compute the integral cohomology ring of G, and then obtain the periodicity of the Farell cohomology of these groups.

Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups…

We give elementary geometric proofs of the main theorems about the (small) quantum cohomology of partial flag varieties SL(n)/P, including the quantum Pieri and quantum Giambelli formulas and the presentation.

We describe explicitly the cohomology of the total complex of certain diagrams of invertible sheaves on normal toric varieties. These diagrams, called wheels, arise in the study of toric singularities associated to dimer models. Our main…

Let $X$ be a complete $\Q$-factorial toric variety of dimension $n$ and $\del$ the fan in a lattice $N$ associated to $X$. For each cone $\sigma$ of $\del$ there corresponds an orbit closure $V(\sigma)$ of the action of complex torus on…

A permutohedral variety is a remarkable object in various areas of mathematics, and its topological invariants are widely recognized. However, only little is known about a real permutohedral variety, that is, the real locus of a…

The main result of the paper is a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subvariety in the torus equivariant integral cohomology ring of the Grassmannian. As a corollary, we…

The main result of this note gives an explicit presentation of the $S^1$-equivariant cohomology ring of the $(n-k,k)$ Springer variety (in type $A$) as a quotient of a polynomial ring by an ideal $I$, in the spirit of the well-known Borel…

We build the abstract theory of Gromov-Witten invariants of genus 0 for quantum minimal Fano varieties (a minimal natural (with respect to Gromov-Witten theory) class of varieties). In particular, we consider ``the minimal Gromov-Witten…

We study Givental's Lagrangian cone for the quantum orbifold cohomology of toric stack bundles and prove that the I-function gives points in the Lagrangian cone, namely we construct an explicit slice of the Lagrangian cone defined by the…

We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On the level of Grothendieck groups this…

Let $G$ be a classical complex Lie group, $P$ any parabolic subgroup of $G$, and $G/P$ the corresponding partial flag variety. We prove an explicit combinatorial Giambelli formula which expresses an arbitrary Schubert class in the…

We present a recursive algorithm computing all the genus-zero Gromov-Witten invariants from a finite number of initial ones, for Fano varieties with generically tame semi-simple quantum (and small quantum) (p, p)- type cohomology, whose…

We generalize classical results about the topology of toric varieties to the case of projective Q-factorial T-varieties of complexity one using the language of divisorial fans. We describe the Hodge-Deligne polynomial in the smooth case,…

We explicitly describe cohomology of the sheaf of differential forms with poles along a semiample divisor on a complete simplicial toric variety. As an application, we obtain a new vanishing theorem which is an analogue of the…

This is an expository lecture, for the Abel bicentennial (Oslo, 2002), describing some recent work on the (small) quantum cohomology ring of Grassmannians and other homogeneous varieties.