Algebraic differential equations of periods integrals

We explain that in the study of the asymptotic expansion at the origin of a period integral like $gamma$z $omega$/df or of a hermitian period like f =s $rho$.$omega$/df $land$ $omega$ /df the computation of the Bernstein polynomial of the fresco (filtered differential equation) associated to the pair of germs (f, $omega$) gives a better control than the computation of the Bernstein polynomial of the full Brieskorn module of the germ of f at the origin. Moreover, it is easier to compute as it has a better functoriality and smaller degree. We illustrate this in the case where f $in$ C[x 0 ,. .. , x n ] has n + 2 monomials and is not quasi-homogeneous, by giving an explicite simple algorithm to produce a multiple of the Bernstein polynomial when $omega$ is a monomial holomorphic volume form. Several concrete examples are given.

On partial differential operators which annihilate the roots of the universal equation of degree k

The aim of this paper is to study in details the regular holonomic $D-$module introduced in cite{[B.19]} whose local solutions outside the polar hyper-surface ${Delta(sigma).sigma_k = 0 }$ are given by the local system generated by the local branches of the multivalued function which is the root of the universal degree $k$ equation $z^k + sum_{h=1}^k (-1)^h.sigma_h.z^{k-h} = 0 $. Note that it is surprising that this regular holonomic $D-$module is given by the quotient of $D$ by a left ideal which has very simple explicit generators despite the fact it necessary encodes the analogous systems for any root of the universal degree $l$ equation for each $l leq k$. Our main result is to relate this $D-$module with the minimal extension of the irreducible local system associated to the difference of two branches of the multivalued function defined above. Then we obtain again a very simple explicit description of this minimal extension in term of the generators of its left ideal in the Weyl algebra. As an application we show how these results allow to compute the Taylor expansion of the root near $-1$ of the equation $z^k + sum_{h=-1}^k (-1)^h.sigma_h.z^{k-h} - (-1)^k = 0 $.

Note on Lisbon integrals and their associated D--modules

The aim of this Note is to specify the links between the three kinds of Lisbon integrals, trace functions and trace forms with the corresponding D--modules.

New singularity invariants : the sheaf $beta_X^bullet$

The graded coherent sheaf $alpha_X^bullet$ constructed in [B.18] for any reduced pure dimensional complex space $X$ is stable by exterior product but not by the de Rham differential. We construct here a new graded coherent sheaf $alpha_X^bullet$ containing $alpha_X^bullet$ and stable both by exterior product and by the de Rham differential. We show that it has again the ``pull-back property for holomorphic maps $f : X to Y$ between irreducible complex spaces such that $f(X)$ is not contained in the singular set of $Y$. Moreover, this graded coherent sheaf $alpha_X^bullet$ comes with a natural coherent exhaustive filtration and this filtration is also compatible with the pull-back by such holomorphic maps. These sheaves define new invariants on singular complex spaces. We show on some simple examples that these invariants are new.

On symmetric partial differential operators

Let s 1 ,. .. , s k be the elementary symmetric functions of the complex variables x 1 ,. .. , x k. We say that F $in$ C[s 1 ,. .. , s k ] is a trace function if their exists f $in$ C[z] such that F (s 1 ,. .. , s k ] = k j=1 f (x j) for all s $in$ C k. We give an explicit finite family of second order differential operators in the Weyl algebra W 2 := C[s 1 ,. .. , s k ] $partial$ $partial$s 1 ,. .. , $partial$ $partial$s k which generates the left ideal in W 2 of partial differential operators killing all trace functions. The proof uses a theorem for symmetric differential operators analogous to the usual symmetric functions theorem and the corresponding map for symbols. As a corollary, we obtain for each integer k a holonomic system which is a quotient of W 2 by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree k: z k + k h=1 (--1) h .s h .z k--h = 0.

On Lisbon integrals

We introduce new complex analytic integral transforms, the Lisbon Integrals, which naturally arise in the study of the affine space $mathbb{C}^k$ of unitary polynomials $P_s(z)$ where $sinmathbb{C}^k$ and $zin mathbb{C}$, $s_i$ identified to the $i-$th symmetric function of the roots of $P_s(z)$. We completely determine the $mathcal{D}$-modules (or systems of partial differential equations) the Lisbon Integrals satisfy and prove that they are their unique global solutions. If we specify a holomorphic function $f$ in the $z$-variable, our construction induces an integral transform which associates a regular holonomic module quotient of the sub-holonomic module we computed. We illustrate this correspondence in the case of a $1$-parameter family of exponentials $f_t(z) = exp(t z)$ with $t$ a complex parameter.

On the nearly smooth complex spaces

We introduce a class of normal complex spaces having only mild sin-gularities (close to quotient singularities) for which we generalize the notion of a (analytic) fundamental class for an analytic cycle and also the notion of a relative fundamental class for an analytic family of cycles. We also generalize to these spaces the geometric intersection theory for analytic cycles with rational positive coefficients and show that it behaves well with respect to analytic families of cycles. We prove that this intersection theory has most of the usual properties of the standard geometric intersection theory on complex manifolds, but with the exception that the intersection cycle of two cycles with positive integral coefficients that intersect properly may have rational coefficients. AMS classification. 32 C 20-32 C 25-32 C 36.

The sheaf $alpha$ $bullet$ X

We introduce in a reduced complex space, a new coherent sub-sheaf of the sheaf $omega_{X}^{bullet}$ which has the universal pull-back property for any holomorphic map, and which is in general bigger than the usual sheaf of holomorphic differential forms $Omega_{X}^{bullet}/torsion$. We show that the meromorphic differential forms which are sections of this sheaf satisfy integral dependence equations over the symmetric algebra of the sheaf $Omega_{X}^{bullet}/torsion$. This sheaf $alpha_{X}^{bullet}$ is also closely related to the normalized Nash transform. We also show that these $q-$meromorphic differential forms are locally square-integrable on any $q-$dimensional cycle in $X$ and that the corresponding functions obtained by integration on an analytic family of $q-$cycles are locally bounded and locally continuous on the complement of closed analytic subset.

Gauduchons form and compactness of the space of divisors

We show that in a holomorphic family of compact complex connected manifolds parametrized by an irreducible complex space $S$, assuming that on a dense Zariski open set $S^{*}$ in $S$ the fibres satisfy the $partialbarpartial-$lemma, the algebraic dimension of each fibre in this family is at least equal to the minimal algebraic dimension of the fibres in $S^{*}$. For instance, if each fibre in $S^{*}$ are Moishezon, then all fibres are Moishezon.

Note on the semi-continuity of the algebraic dimension

In this short Note we show that the direct image sheaf R 1 $pi$ * (O X) associated to an analytic family of compact complex manifolds $pi$ : X $rightarrow$ S parametrized by a reduced complex space S is a locally free (coherent) sheaf of O S --modules. This result allows to improve a semi-continuity type result for the algebraic dimension of compact complex manifolds in an analytic family given in [B.15]. AMS Classification 2010. 32G05-32A20-32J10.