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Register a new userAlgebraic differential equations of periods integrals
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Authors
Daniel Barlet
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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.
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The existence and uniqueness of formal Puiseux series solutions of non-autonomous algebraic differential equations of the first order at a nonsingular point of the equation is proven. The convergence of those Puiseux series is established. Several new examples are provided. Relationships to the celebrated Painleve theorem and lesser-known Petrovics results are discussed in detail.
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.
A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$. A web of curves on $X$ induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of $X$. We study how the local differential geometry of the web-structure affects the global algebraic geometry of $X$. Under two geometric assumptions on the web-structure, the pairwise non-integrability condition and the bracket-generating condition, we prove that the local differential geometry determines the global algebraic geometry of $X$, up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when $X subset {bf P}^N$ is a Fano submanifold of Picard number 1, and the family of lines covering $X$ becomes a web. In this special case, we have a stronger result that the local differential geometry of the web-structure determines $X$ up to biregular equivalences. As an application, we show that if $X, X subset {bf P}^N, dim X geq 3,$ are two such Fano manifolds of Picard number 1, then any surjective morphism $f: X to X$ is an isomorphism.
In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) in such a way that two generic curves have the same signatures if and only if they are $G$-equivalent. We prove that for any $G$-action, there exists a pair of rational differential invariants, called classifying invariants, that can be used to construct signatures. We derive a formula for the degree of a signature curve in terms of the degree of the original curve, the size of its symmetry group and some quantities depending on a choice of classifying invariants. For the full projective group, as well as for its affine, special affine and special Euclidean subgroups, we give explicit sets of rational classifying invariants and derive a formula for the degree of the signature curve of a generic curve as a quadratic function of the degree of the original curve. We show that this generic degree is the sharp upper bound.
Using Galois theory, we construct explicitly (in all complex dimensions >1) an infinite family of simple complex tori of algebraic dimension 0 with Picard number 0.
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