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Degree of the generalized Plücker embedding of a Quot scheme and quantum cohomology


Ravi, M; Rosenthal, J; Wang, X (1998). Degree of the generalized Plücker embedding of a Quot scheme and quantum cohomology. Mathematische Annalen, 311(1):11-26.

Abstract

We compute the degree of the generalized Plücker embedding $\kappa$ of a Quot scheme $X$ over $\PP^1$. The space $X$ can also be considered as a compactification of the space of algebraic maps of a fixed degree from $\PP^1$ to the Grassmanian $\rm{Grass}(m,n)$. Then the degree of the embedded variety $\kappa (X)$ can be interpreted as an intersection product of pullbacks of cohomology classes from $\rm{Grass}(m,n)$ through the map $\psi$ that evaluates a map from $\PP^1$ at a point $x\in \PP^1$. We show that our formula for the degree verifies the formula for these intersection products predicted by physicists through Quantum cohomology~\cite{va92}~\cite{in91}~\cite{wi94}. We arrive at the degree by proving a version of the classical Pieri's formula on the variety $X$, using a cell decomposition of a space that lies in between $X$ and $\kappa (X)$.

Abstract

We compute the degree of the generalized Plücker embedding $\kappa$ of a Quot scheme $X$ over $\PP^1$. The space $X$ can also be considered as a compactification of the space of algebraic maps of a fixed degree from $\PP^1$ to the Grassmanian $\rm{Grass}(m,n)$. Then the degree of the embedded variety $\kappa (X)$ can be interpreted as an intersection product of pullbacks of cohomology classes from $\rm{Grass}(m,n)$ through the map $\psi$ that evaluates a map from $\PP^1$ at a point $x\in \PP^1$. We show that our formula for the degree verifies the formula for these intersection products predicted by physicists through Quantum cohomology~\cite{va92}~\cite{in91}~\cite{wi94}. We arrive at the degree by proving a version of the classical Pieri's formula on the variety $X$, using a cell decomposition of a space that lies in between $X$ and $\kappa (X)$.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1998
Deposited On:12 Mar 2010 08:40
Last Modified:20 Feb 2018 08:05
Publisher:Springer
ISSN:0025-5831
Additional Information:The original publication is available at www.springerlink.com
OA Status:Green
Publisher DOI:https://doi.org/10.1007/s002080050173

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