On the power structure over the Grothendieck ring of varieties and its applications

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Google Scholar
Research Projects
Organizational Units
Journal Issue
We discuss the notion of a power structure over a ring and the geometric description of the power structure over the Grothendieck ring of complex quasi-projective varieties and show some examples of applications to generating series of classes of configuration spaces (for example, nested Hilbert schemes of J. Cheah) and wreath product orbifolds.
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 258, pp. 58–69.
V. V. Batyrev, "Non-Archimedean Integrals and Stringy Euler Numbers of Log-Terminal Pairs," J. Eur. Math. Soc. 1, 5–33 (1999). J. Burillo, "The Poincaré-Hodge Polynomial of a Symmetric Product of Compact Kähler Manifolds," Collect. Math. 41, 59–69 (1990). J. Cheah, "On the Cohomology of Hilbert Schemes of Points," J. Algebr. Geom. 5, 479–511 (1996). J. Cheah, "The Virtual Hodge Polynomials of Nested Hilbert Schemes and Related Varieties," Math. Z. 227, 479–504 (1998). J. Cheah, "Cellular Decompositions for Nested Hilbert Schemes of Points," Pac. J. Math. 183, 39–90 (1998). L. Dixon, J. A. Harvey, C. Vafa, and E. Witten, "Strings on Orbifolds. I," Nucl. Phys. B 261, 678–686 (1985). L. Göttsche, "On the Motive of the Hilbert Scheme of Points on a Surface," Math. Res. Lett. 8, 613–627 (2001). S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, "A Power Structure over the Grothendieck Ring of Varieties," Math. Res. Lett. 11, 49–57 (2004). S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, "Power Structure over the Grothendieck Ring of Varieties and Generating Series of Hilbert Schemes of Points," Mich. Math. J. 54 (2), 353–359 (2006); S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, "Integration over Spaces of Nonparametrized Arcs and Motivic Versions of the Monodromy Zeta Function," Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 252, 71–82 (2006) [Proc. Steklov Inst. Math. 252, 63–73 (2006)]. M. Kapranov, "The Elliptic Curve in the S-Duality Theory and Eisenstein Series for Kac-Moody Groups, W.-P. Li and Zh. Qin, "On the Euler Numbers of Certain Moduli Spaces of Curves and Points," I. G. Macdonald, "The Poincaré Polynomial of a Symmetric Product," Proc. Cambridge Philos. Soc. 58, 563–568 (1962). MR0143204 (26 #764) H. Tamanoi, "Generalized Orbifold Euler Characteristic of Symmetric Products and Equivariant Morava K-Theory," Algebr. Geom. Topology 1, 115–141 (2001). W. Wang, "Equivariant K-Theory, Wreath Products, and Heisenberg Algebra," Duke Math. J. 103, 1–23 (2000). W. Wang and J. Zhou, "Orbifold Hodge Numbers of Wreath Product Orbifolds," J. Geom. Phys. 38, 152–169 (2001). E. Zaslow, "Topological Orbifold Models and Quantum Cohomology Rings," Commun. Math. Phys. 156, 301–331 (1993).