97 research outputs found
Tautological integrals on Hilbert scheme of points II: Geometric subsets
We develop a formula for tautological integrals over geometric subsets of the
Hilbert scheme of points on complex manifolds. As an illustration of the
theory, we derive a new iterated residue formula for the number of nodal curves
in sufficiently ample linear systems.Comment: 47 pages. Preliminary version, comments are welcome. Survey parts
shared with arXiv:2112.1550
Arithmetic and metric aspects of open de Rham spaces
In this paper we determine the motivic class---in particular, the weight
polynomial and conjecturally the Poincar\'e polynomial---of the open de Rham
space, defined and studied by Boalch, of certain moduli of irregular
meromorphic connections on the trivial bundle on . The
computation is by motivic Fourier transform. We show that the result satisfies
the purity conjecture, that is, it agrees with the pure part of the conjectured
mixed Hodge polynomial of the corresponding wild character variety. We also
identify the open de Rham spaces with quiver varieties with multiplicities of
Yamakawa and Geiss--Leclerc--Schr\"oer. We finish with constructing natural
complete hyperk\"ahler metrics on them, which in the -dimensional cases are
expected to be of type ALF.Comment: 69 page
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Number Theory, Analysis and Geometry: In Memory of Serge Lang
Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future.
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