An analytic Grothendieck Riemann Roch theorem
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© 2016 Elsevier Inc. We extend the Boutet de Monvel Toeplitz index theorem to complex manifolds with isolated singularities following the relative Khomology theory of Baum, Douglas, and Taylor for manifolds with boundary. We apply this index theorem to study the ArvesonDouglas conjecture. Let B m be the unit ball in C m , and I an ideal in the polynomial algebra C[z 1 ,...,z m ]. We prove that when the zero variety Z I is a complete intersection space with only isolated singularities and intersects with the unit sphere S2m1 transversely, the representations of C[z 1 ,...,z m ] on the closure of I in L a2 (B m ) and also the corresponding quotient space Q I are essentially normal. Furthermore, we prove an index theorem for Toeplitz operators on Q I by showing that the representation of C[z 1 ,...,z m ] on the quotient space Q I gives the fundamental class of the boundary ZI∩S 2m1 . In the appendix, we prove with Kai Wang that if f∈La2(B m ) vanishes on ZI∩B m , then f is contained inside the closure of the ideal I in La2(B m ).
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Douglas, R. G., Tang, X., & Yu, G.
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Douglas, Ronald GTang, XiangYu, Guoliang
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Algebraic Variety With Singularities

Grothendieckriemannroch Theorem

Index Theorem

The Arvesondouglas Conjecture

Toeplitz Operator
Identity
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