Elliptic classes of Schubert varieties
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Mathematische Annalen
Elliptic classes of Schubert varieties Shrawan Kumar1 · Richárd Rimányi1
· Andrzej Weber2
Received: 10 October 2019 / Revised: 9 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov–Libgober classes of Schubert varieties in general homogeneous spaces G/P. While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi–Tarasov–Varchenko we prove that weight functions represent elliptic classes of Schubert varieties.
1 Introduction Schubert calculus is usually considered in ordinary cohomology or in K -theory. Generalized cohomology theories correspond to formal group laws. Under this correspondence ordinary cohomology and K -theory correspond to the one-dimensional algebraic groups C and C∗ respectively. There is another one-dimensional complex algebraic group, the elliptic curve E = C∗ /q Z , (|q| < 1 fixed). The corresponding cohomology theory is called elliptic. In this paper we study the thus obtained (equivariant) elliptic Schubert calculus. A key step in any Schubert calculus is assigning a characteristic class to a Schubert variety. Traditionally this characteristic class is the fundamental class notion of the
Communicated by Jean-Yves Welschinger.
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Richárd Rimányi [email protected] Shrawan Kumar [email protected] Andrzej Weber [email protected]
1
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA
2
Institute of Mathematics, University of Warsaw, Warsaw, Poland
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S. Kumar et al.
given cohomology theory. However, it is known that in elliptic cohomology the notion of fundamental class is not well defined [8], or in other words, the notion depends on choices. There are important works (e.g. [15,25] and references therein) on elliptic fundamental classes based on making some natural choices—the choice can be geometric (a resolution) or algebraic (a basis in a Hecke algebra). In this paper we are suggesting a notion which does not depend on choices. Our class is not the elliptic fundamental class (as just discussed, it does not exist); we regard our class as an analogue of the cohomological Chern–Schwartz–MacPherson (CSM) class, and the K-theoretic motivic Chern (MC) class. In fact, certain limits of our elliptic class recovers the CSM and the MC classes. The CSM and MC characteristic classes are one-parameter deformations of the fundamental classes in their respective cohomology theories. The parameter is usually denoted by1 h. At “h = ∞” and h = 1 the CSM and MC classes specialize to the fundamental class of the theory. Our elliptic class also depends on the extra h parame
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