Holomorphic vector field with one zero on the Grassmannian and cohomology
DOI:
https://doi.org/10.24193/subbmath.2024.4.01Keywords:
Cohomology ring, residues, localization, holomorphic vector fieldAbstract
We consider a holomorphic vector field on the complex Grassmannian constructed from a nilpotent matrix. We show that this vector field vanishes only at a single point. Using the Baum-Bott localization theorem we give a Grothendieck residue formula for the intersection numbers of the Grassmannian. Knowing that Chern classes of the tautological bundle generate the cohomology ring of the Grassmannian we can compute the ideal of relations explicitly from the residue formula. This shows that the cohomology ring of the Grassmannian is determined by holomorphic vector field around its only zero.
Mathematics Subject Classification (2010): 14Cxx, 57Rxx.
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