Jorge Vitorio Pereira, Carlo Perrone
We study the subvariety of integrable 1-forms in a finite dimensional vector space $W \subset Ω^1(\mathbb C^n,0)$. We prove that the irreducible components with dimension comparable with the rank of $W$ are of minimal degree.
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This paper was inspired by Guan and Zhou's recent proof of the so-called strong openness conjecture for plurisubharmonic functions. We give a proof shorter than theirs and extend the result to possibly singular hermitian metrics on vector bundles.
We introduce a certain index of a collection of germs of 1-forms on a germ of a singular variety which is a generalization of the local Euler obstruction corresponding to Chern numbers different from the top one.
Bi-Lipschitz triviality of function-germs on singular varieties
本文研究奇异簇上解析函数芽形变的双李普希茨平凡性，引入强有理双李普希茨平凡族概念并给出无穷小判别准则，进而得到齐次与加权齐次情形下的平凡性和刚性结果。
For a lattice $Γ$ of a simply connected solvable Lie group $G$, we describe the analytic germ in the variety of representations of $Γ$ at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by $G$. By this description, under certain assumption, we study the singularity of the analytic germ in the variety of representations of $Γ$ at the trivial representation by using the Kuranishi space construction. By a similar technique, we also study deformations of holomorphic structures of trivial vector bund…
We associate to any germ of an analytic variety a Lie algebra of tangent vector fields, the {\it tangent algebra}. Conversely, to any Lie algebra of vector fields an analytic germ can be associated, the {\it integral variety}. The paper investigates properties of this correspondence: The set of all tangent algebras is characterized in purely Lie algebra theoretic terms. And it is shown that the tangent algebra determines the analytic type of the variety.
We study analytic integrable deformations of the germ of a holomorphic foliation given by $df=0$ at the origin $0 \in \mathbb C^n, n \geq 3$. We consider the case where $f$ is a germ of an irreducible and reduced holomorphic function. Our central hypotheses is that, {\em outside of a dimension $\leq n-3$ analytic subset $Y\subset X$, the analytic hypersurface $X_f : (f=0)$ has only normal crossings singularities}. We then prove that, as germs, such deformations also exhibit a holomorphic first integral, depending analytically on the parameter of the deformation. This applies to the study of in…
It is shown that a germ of a holomorphic mapping sending a real-analytic generic submanifold of finite type into another is determined by its projection on the Segre variety of the target manifold. A necessary and sufficient condition is given for a germ of a mapping into the Segre variety of the target manifold to be the projection of a holomorphic mapping sending the source manifold into the target. An application to the biholomorphic equivalence problem is also given.
Given a germ of an analytic variety $X$ and a germ of a holomorphic function $f$ with a stratified isolated singularity with respect to the logarithmic stratification of $X$, we show that under certain conditions on the singularity type of the pair $(f,X)$, the following relative analog of the well known K. Saito's theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce-Roberts numbers) is equivalent to the relative quasihomogeneity of the pair $(f,X)$, i.e. to the existence of a coordinate system such that both $f$ and $X$ are quasih…
For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs we consider general types of multivariate Bernstein-Sato ideals associated to arbitrary factorizations of our germ. We show the zero loci of these ideals are purely codimension one and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we show the Bernstein-Sato ideals attached to a factorization into linear forms are principal. As an application, we independently verify and improve an estimate of Maisonobe's regarding standar…
The systematic study of neighborhoods of analytic varieties was started by H. Grauert in his celebrated article 1962. In that article he considers a manifold X and a negatively embedded submanifold A subset X. He introduces n-neighborhood, n in N, of A and studies when an isomorphism of two n-neighborhoods can be extended to an isomorphism of (n+1)-neighborhoods. He observes that obstructions to this extension problem lie in the first cohomology group of certain sheaves involving the normal bundle of A in X. Using a version of Kodaira vanishing theorem he shows that for a large n these cohomol…
The Bruce-Roberts Tjurina number of holomorphic 1-forms along complex analytic varieties
本文定义了全纯1-形式沿复解析簇对的Bruce-Roberts Tjurina数，证明其与多个已知不变量之间的关系，并应用于复二维全纯叶层芽的拟齐性问题。
We show that if the variety of minimal rational tangents (VMRT) of a uniruled projective manifold at a general point is projectively equivalent to that of a symplectic or an odd-symplectic Grassmannian, the germ of a general minimal rational curve is biholomorphic to the germ of a general line in a presymplectic Grassmannian. As an application, we characterize symplectic and odd-symplectic Grassmannians, among Fano manifolds of Picard number 1, by their VMRT at a general point and prove their rigidity under global Kähler deformation. Analogous results for $G/P$ associated with a long root were…
We study algebraic isomonodromic deformations of flat logarithmic connections on the Riemann sphere with $n\geq 4$ poles, for arbitrary rank. We introduce a natural property of algebraizability for the germ of universal deformation of such a connection. We relate this property to a peculiarity of the corresponding monodromy representation: to yield a finite braid group orbit on the appropriate character variety. Under reasonable assumptions on the deformed connection, we may actually establish an equivalence between both properties. We apply this result in the rank two case to relate finite br…
By attaching a Lie algebra of germs of analytic vector fields to every point of a (real or complex) analytic variety V we construct the Nagano foliation of the variety. We prove that the Nagano foliation of V is a stratification. The treatment of the subject is totally coordinate free but relies on the Oka-Cartan-Serre theory of coherent analytic sheaves.
Given a complex germ $f$ near the point $\mathfrak{x}$ of the complex manifold $X$, equipped with a factorization $f = f_{1} \cdots f_{r}$, we consider the $\mathscr{D}_{X,\mathfrak{x}}[s_{1}, \dots, s_{r}]$-module generated by $ F^{S} := f_{1}^{s_{1}} \cdots f_{r}^{s_{r}}$. We show for a large class of germs that the annihilator of $F^{S}$ is generated by derivations and this property does not depend on the chosen factorization of $f$. We further study the relationship between the Bernstein-Sato variety attached to $F$ and the cohomology support loci of $f$, via the $\mathscr{D}_{X,\mathfrak{…
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