Equivariant Lefschetz number of differential operators

Felder, Giovanni; Tang, Xiang

doi:10.1007/s00209-009-0579-7

Cited by 5 publications

(1 citation statement)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where i in the first line denotes the connected components, the first isomorphism follows directly from Propositions 3 and 4 in [DE05]. From [FT10] we know that the homology H…”

mentioning

confidence: 96%

Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

Vitanov¹

2020

Preprint

View full text Add to dashboard Cite

A. Etingof introduces a sheaf of Cherednik algebras ℋ 1,c,X,G , attached to a complex algebraic variety X with an action by a finite group G, by means of generators and relations. In a previous paper the author showed how to realize the sheaf of Cherednik algebras on a general global quotient orbifold X/G by gluing sheaves of flat sections of flat holomorphic vector bundles on orbit type strata in X. In the current note we use this realization of the sheaf of global Cherednik algebras to define trace density maps for ℋ 1,c,X,G attached to the various orbit type strata. The maps are not entirely explicit. By means of the trace density morphisms we obtain a morphism between the hypercohomology of the sheaf of Hochschild chain complexes of ℋ 1,c,X,G for a Coxeter group G and the Chen-Ruan orbifold cohomology of X/G. We subsequently show that for any complex reflection group G, when c is a formal parameter, the hyercohomology of the sheaf of Hochschild chain complexes of ℋ 1,c,X,G is isomorphic to the Chen-Ruan cohomology of X/G with values in the Laurent field. Finally, we present an algebraic index theorem for the sheaf of formal global Cherendik algebras ℋ 1,(( )),X,G localized at . C H,(( )) n−l,l 14 4.2. Construction of the trace density maps 15 5. Algebraic index theorem 20 Acknowledgemnts 24 References 24

show abstract

“…where i in the first line denotes the connected components, the first isomorphism follows directly from Propositions 3 and 4 in [DE05]. From [FT10] we know that the homology H…”

mentioning

confidence: 96%

Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

Vitanov¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

Hochschild Lefschetz class for $$\mathcal{D }$$ -modules

2012

Self Cite

View full text Add to dashboard Cite

We introduce a notion of Hochschild Lefschetz class for a good coherent D-module on a compact complex manifold, and prove that this class is compatible with the direct image functor. We prove an orbifold Riemann-Roch formula for a D-module on a compact complex orbifold. This is actually a slight abuse of terminology. In fact, HH( EX , EX) is an object in the derived category of sheaves of C-vector spaces on X.2 See Eq.(1) for the definition of HH( EX , E γ X ).

show abstract

Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

Vitanov

2022

International Mathematics Research Notices

View full text Add to dashboard Cite

We show how a novel construction of the sheaf of Cherednik algebras $\mathscr {H}_{1, c, X, G}$ on a quotient orbifold $Y:=X/G$ in author’s prior work leads to results for $\mathscr {H}_{1, c, X, G}$, which until recently were viewed as intractable. First, for every orbit type stratum in $X$, we define a trace density map for the Hochschild chain complex of $\mathscr {H}_{1, c, X, G}$, which generalizes the standard Engeli–Felder’s trace density construction for the sheaf of differential operators $\mathscr {D}_X$. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the derived category of complexes of $\mathbb {C}_{Y}\llbracket \hbar \rrbracket $-modules, which computes the hypercohomology of the Hochschild chain complex of the sheaf of formal Cherednik algebras $\mathscr {H}_{1, \hbar , X, G}$. We show that this hypercohomology is isomorphic to the Chen–Ruan cohomology of the orbifold $Y$ with values in the ring of formal power series $\mathbb {C}\llbracket \hbar \rrbracket $. We infer that the Hochschild chain complex of the sheaf of skew group algebras $\mathscr {H}_{1, 0, X, G}$ has a well-defined Euler characteristic that is equal to the orbifold Euler characteristic of $Y$. Finally, we prove an algebraic index theorem.

show abstract

Equivariant Lefschetz number of differential operators

Abstract: Let G be a compact Lie group acting on a compact complex manifold M by holomorphic transformations. We prove a trace density formula for the G-Lefschetz number of a holomorphic differential operator on M. We generalize the recent results of Engeli and the first author to orbifolds.

Cited by 5 publications

References 12 publications

Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

Hochschild Lefschetz class for $$\mathcal{D }$$ -modules

Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

Contact Info

Product

Resources

About