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**Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables.**
*(English)*
Zbl 0722.11026

In a series of papers the authors have studied the relationship between two types of cohomology classes for arithmetic quotients of the symmetric spaces attached to orthogonal and unitary groups. On one hand these cohomology classes come up in a geometric way as the Poincaré dual classes to natural cycles on the arithmetic quotients. These cycles are themselves unions of arithmetic quotients of totally geodesic subsymmetric spaces associated to smaller orthogonal or unitary groups. On the other hand there are cohomology classes given in terms of automorphic forms being constructed by means of the Weil representation (or theta correspondence).

The main new twist in the paper under review is the use of the Cauchy- Riemann equations in the general theory of the theta correspondence. This method is based on a study of the double complex of relative Lie cohomology with values in the oscillator representation associated to a dual reductive pair. The authors construct liftings from the cohomology with compact supports of locally symmetric spaces associated to O(p,q) (resp. U(p,q)) of degree nq (resp. Hodge type nq,nq) to the space of classical holomorphic Siegel modular forms of weight \((p+q)/2\) and genus n (resp. holomorphic Hermitian modular forms of weight \(p+q\) and genus n). The Fourier coefficients of the lift of a given \(\phi\) can be expressed in terms of periods of \(\phi\) over certain totally geodesic cycles. For a specific choice of \(\phi\) (choose \(\phi\) to be the Poincaré dual of a finite cycle) a collection of formulas analogous to those of F. Hirzebruch and D. Zagier [Invent. Math. 36, 57- 113 (1976; Zbl 0332.14009)] is obtained.

The main new twist in the paper under review is the use of the Cauchy- Riemann equations in the general theory of the theta correspondence. This method is based on a study of the double complex of relative Lie cohomology with values in the oscillator representation associated to a dual reductive pair. The authors construct liftings from the cohomology with compact supports of locally symmetric spaces associated to O(p,q) (resp. U(p,q)) of degree nq (resp. Hodge type nq,nq) to the space of classical holomorphic Siegel modular forms of weight \((p+q)/2\) and genus n (resp. holomorphic Hermitian modular forms of weight \(p+q\) and genus n). The Fourier coefficients of the lift of a given \(\phi\) can be expressed in terms of periods of \(\phi\) over certain totally geodesic cycles. For a specific choice of \(\phi\) (choose \(\phi\) to be the Poincaré dual of a finite cycle) a collection of formulas analogous to those of F. Hirzebruch and D. Zagier [Invent. Math. 36, 57- 113 (1976; Zbl 0332.14009)] is obtained.

Reviewer: J.Schwermer (Eichstätt)

### MSC:

11F30 | Fourier coefficients of automorphic forms |

11F27 | Theta series; Weil representation; theta correspondences |

11F75 | Cohomology of arithmetic groups |

### Keywords:

theta correspondence; holomorphic Siegel modular forms; holomorphic Hermitian modular forms; Fourier coefficients; lift### Citations:

Zbl 0332.14009
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\textit{S. S. Kudla} and \textit{J. J. Millson}, Publ. Math., Inst. Hautes Étud. Sci. 71, 121--172 (1990; Zbl 0722.11026)

### References:

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