Commuting partial differential operators and vector bundles over abelian varieties. (English) Zbl 0809.14016

Nakayashiki, Atsushi

Am. J. Math. 116, No. 1, 65-100 (1994).

The aim of this paper is to construct commuting partial differential operators with matrix coefficients from vector bundles on abelian varieties. For this the author introduces and studies in detail BA- modules (Baker-Akhiezer modules) defined as follows. Let \(X\), \(\widehat X\) denote respectively an abelian variety and its dual. Let \(P\) denote the normalised Poincaré bundle on \(X \times \widehat X\). Let \(D\) be an ample irreducible divisor on \(X\) and \(F\) a coherent sheaf which is an \({\mathcal O}_ X\)-module. There is a natural affine bundle \(X_ 0\) on \(X\) with fibres isomorphic to \(H^ 0 (\widehat X, \Omega^ 1_{\widehat X})\). Let \(s\) be a section of this affine bundle over \(X-D\) with poles of order at most one on \(D\). Let \(p_ 1\) and \(p_ 2\) denote the projections from \(X \times \widehat X\) to \(X\) and \(\widehat X\). Define the BA-module \({\mathcal F}\) associated to the quartet \((X,D,F,s)\) as \({\mathcal F} = {\mathcal F} (X,D,F,s) : = p_{2*} (p^*_ 1F(*D) \bigotimes P)\), where \(F(*D) = \varinjlim F \otimes {\mathcal O}_ X (nD)\). The author shows that \({\mathcal F}\) is a coherent \({\mathcal D}_{\widehat X}\)-module (the module structure being determined by \(s)\) and gives a set \((C)\) of (sufficient) conditions for \({\mathcal F}_{\widehat x}\) to be a free \({\mathcal D}_{\widehat X, \widehat x}\)-module. He gives a construction of quartets satisfying conditions \((C)\) using the method of Fourier transform of S. Mukai [Nagoya Math. J. 81, 153-175 (1981; Zbl 0417.14036)]. He also studies the characteristic variety of \({\mathcal F}\), relation of its dimension to the dimension of the support of \({\mathcal F}\) and rank reduction. The BA-module is, in addition, a module over the affine commutative ring \(A = H^ 0 (X-D, {\mathcal O}_ X)\). This is exploited to give an embedding of \(A\) into the ring of partial differential operators with matrix coefficients. This gives a ring of commuting differential operators.

Reviewer: U.N.Bhosle (Bombay)

MSC:

14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14K20	Analytic theory of abelian varieties; abelian integrals and differentials
58J40	Pseudodifferential and Fourier integral operators on manifolds

Keywords:

Fourier transform; \({\mathcal D}\)-module; Baker-Akhiezer modules; commuting partial differential operators; vector bundles on abelian varieties; BA- modules; abelian variety