Distributions and regular holonomic systems

Summary

The first section treats analytic D-module theory on real analytic manifolds and some basic results concerned with extendible distributions is presented in section 2 as a preparation to section 3. There we prove that every regular holonomic D _X -module on a complex manifold is locally a cyclic module generated by a distribution on the underlying real manifold. The main result is Theorem 7.3.5 which gives an exact functor from RH(D _X ) into the category of regular holonomic modules on the conjugate complex manifold defined by

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOUdS2aaS % baaSqaaiaadIfaaeqaaOGaaiikaiaad2eacaGGPaGaeyypa0Jaamis % aiaad+gacaWGTbWaaSbaaSqaaiaadseadaWgaaadbaGaamiwaaqaba % aaleqaaOGaaiikaiaad2eacaGGSaGaamiraiaadkgadaWgaaWcbaGa % amiwamaaBaaameaacaWGsbaabeaaaSqabaGccaGGPaGaaiOlaaaa!4811!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\kappa _X}(M) = Ho{m_{{D_X}}}(M,D{b_{{X_R}}}).$$

We refer to К _X as Kashiwara’s conjugation functor. Reversing the roles between X and EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiwayaara % aaaa!36EA!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\bar X$$ there exists the conjugation functor EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa % aaleaaceWGybGbaebaaeqaaaaa!37E6!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${K_{\bar X}}$$ from EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOuaiaabI % eacaqGOaGaamiramaaBaaaleaaceWGybGbaebaaeqaaOGaaiykaiab % gkziUkaabkfacaqGibGaaeikaiaadseadaWgaaWcbaGaamiwaaqaba % GccaGGPaaaaa!41A1!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\text{RH(}}{D_{\bar X}}) \to {\text{RH(}}{D_X})$$. We prove that the compsed functor EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa % aaleaaceWGybGbaebaaeqaaOGaeSigI8Maam4samaaBaaaleaacaWG % ybaabeaaaaa!3B03!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${K_{\bar X}} \circ {K_X}$$ is the identity on RH(D _X ).

Distributions whose cyclic D _X -modules are regular holonomic will be called regular holonomic distributions. Various examples of regular holonomic distributions are given in subsequent sections. In particular we mention the principal value distribu?tions defined by

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacq % aH8oqBcaGGSaGaeuiQdKfacaGLPmIaayPkJaGaeyypa0ZaaCbeaeaa % caWGmbGaamyAaiaad2gaaSqaaiabew7aLjabgkziUkaaicdaaeqaaO % Waa8quaeaacaWGMbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaeuiQ % dKfaleaacaGG8bGaamOzaiaacYhaaeqaniabgUIiYdaaaa!4D8F!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\left\langle {\mu ,\Psi } \right\rangle = \mathop {Lim}\limits_{\varepsilon \to 0} \int\limits_{|f|} {{f^{ - 1}}\Psi } $$

where Ψ is any test-form on X _R and f ∈ O(X). Meromorphic continuations of distributions are also discussed.

In the final sections we use the conjugation functor to exhibit an inverse functor to the de Rham functor in the Riemann-Hilbert correspondence. The inverse functor is obtained from a temperate Hom-functor composed with the 6-complex. This leads to properties of regular holonomic modules which go beyond those in Chapter V. The main results occur at the end of section 9. The last section contains a discussion about D-module theory related to Hodge theory.