An analytic Grothendieck Riemann Roch theorem
| Autor: | Xiang Tang, Guoliang Yu, Ronald G. Douglas |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Unit sphere
Pure mathematics Conjecture General Mathematics 010102 general mathematics Complete intersection Zero (complex analysis) Grothendieck–Riemann–Roch theorem Quotient space (linear algebra) 01 natural sciences 0103 physical sciences 010307 mathematical physics Ideal (ring theory) 0101 mathematics Atiyah–Singer index theorem Mathematics |
| Zdroj: | Advances in Mathematics. 294:307-331 |
| ISSN: | 0001-8708 |
| DOI: | 10.1016/j.aim.2016.02.031 |
| Popis: | We extend the Boutet de Monvel Toeplitz index theorem to complex manifolds with isolated singularities following the relative K -homology theory of Baum, Douglas, and Taylor for manifolds with boundary. We apply this index theorem to study the Arveson–Douglas conjecture. Let BmBm be the unit ball in CmCm, and I an ideal in the polynomial algebra C[z1,⋯,zm]C[z1,⋯,zm]. We prove that when the zero variety ZIZI is a complete intersection space with only isolated singularities and intersects with the unit sphere S2m−1S2m−1 transversely, the representations of C[z1,⋯,zm]C[z1,⋯,zm] on the closure of I in La2(Bm) and also the corresponding quotient space QIQI are essentially normal. Furthermore, we prove an index theorem for Toeplitz operators on QIQI by showing that the representation of C[z1,⋯,zm]C[z1,⋯,zm] on the quotient space QIQI gives the fundamental class of the boundary ZI∩S2m−1ZI∩S2m−1. In the appendix, we prove with Kai Wang that if f∈La2(Bm) vanishes on ZI∩BmZI∩Bm, then f is contained inside the closure of the ideal I in La2(Bm). |
| Databáze: | OpenAIRE |
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