Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the quantum cohomology of homogeneous varieties
84
0
0.0
(
0
)
Authors
William Fulton
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
We give a complete description of the equivariant quantum cohomology ring of any smooth hypertoric variety, and find a mirror formula for the quantum differential equation.
Given a smooth quasiprojective variety $Y$ over $mathbb C$ that is not projective, consider its unordered configuration spaces $mathrm{Conf}^n(Y)$ for $ngeq 0$. Remove a point $P$ of $Y$ and obtain a one-puncture $Y-P$ of $Y$. We give a decomposition formula that computes the singular cohomology groups of $mathrm{Conf}^n(Y-P)$ in terms of those of $mathrm{Conf}^m(Y); (0leq mleq n)$, and prove it for several families of examples of $Y$, including the case where $Y$ is obtained from a smooth projective variety by puncturing one or more points. This formula keeps track of the mixed Hodge structures of the cohomology groups as well. This result simultaneously implies a result of Kallel involving Betti numbers and a consequence of a combinatorial property of configuration spaces due to Vakil and Wood. We also obtain intermediate results involving ordered configuration spaces that potentially work for more examples of $Y$.
In this paper we compute the cohomology of the Fano varieties of $k$-planes in the smooth complete intersection of two quadrics in $mathbb{P}^{2g+1}$, using Springer theory for symmetric spaces.
We show that the infinitesimal deformations of the Brill--Noether locus $W_d$ attached to a smooth non-hyperelliptic curve $C$ are in one-to-one correspondence with the deformations of $C$. As an application, we prove that if a Jacobian $J$ deforms together with a minimal cohomology class out the Jacobian locus, then $J$ is hyperelliptic. In particular, this provides an evidence to a conjecture of Debarre on the classification of ppavs carrying a minimal cohomology class. Finally, we also study simultaneous deformations of Fano surfaces of lines and intermediate Jacobians.
We prove a structure theorem for the Albanese maps of varieties with Q-linearly trivial log canonical divisors. Our start point is the action of a nonlinear algebraic group on a projective variety.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
