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SUMMARY:Eivind Schneider
DTSTART:20240515T162000Z
DTEND:20240515T180000Z
DTSTAMP:20260423T010250Z
UID:GDEq/112
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GDEq/112/">I
 nvariant divisors and equivariant line bundles</a>\nby Eivind Schneider as
  part of Geometry of differential equations seminar\n\n\nAbstract\nScalar 
 relative invariants play an important role in the theory of group actions 
 on a manifold as their zero sets are invariant hypersurfaces. Relative inv
 ariants are central in many applications\, where they often are treated lo
 cally\, since an invariant hypersurface is not necessarily the locus of a 
 single function. Our aim is to outline a global theory of relative invaria
 nts in the complex analytic setting. For a Lie algebra $\\mathfrak{g}$ of 
 holomorphic vector fields on a complex manifold $M$\, any holomorphic $\\m
 athfrak{g}$-invariant hypersurface is given in terms of a $\\mathfrak{g}$-
 invariant divisor. This generalizes the classical notion of scalar relativ
 e $\\mathfrak{g}$-invariant. Since any $\\mathfrak{g}$-invariant divisor g
 ives rise to a $\\mathfrak{g}$-equivariant line bundle\, we investigate th
 e group $\\mathrm{Pic}_{\\mathfrak{g}}(M)$ of $\\mathfrak{g}$-equivariant 
 line bundles. A cohomological description of $\\mathrm{Pic}_{\\mathfrak{g}
 }(M)$ is given in terms of a double complex interpolating the Chevalley-Ei
 lenberg complex for $\\mathfrak{g}$ with the Čech complex of the sheaf of
  holomorphic functions on $M$. In the end we will discuss applications of 
 the theory to jet spaces and differential invariants.\n\nThe talk is based
  on joint work with Boris Kruglikov (<a href="https://arxiv.org/abs/2404.1
 9439">arXiv:2404.19439</a>).\n
LOCATION:https://researchseminars.org/talk/GDEq/112/
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