Complex Analytic Set Research Articles

On Lipschitz Geometry at infinity of complex analytic sets

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein’s problem. Moser’s Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex parametric version of Moser’s Bernstein Theorem. More precisely, we prove that any entire complex analytic set in $$\mathbb {C}^n$$ which is Lipschitz regular at infinity must be an affine linear subspace of $$\mathbb {C}^n$$ . In particular, a complex analytic set which is a graph of an entire Lipschitz function must be affine linear subspace. That result comes as a consequence of the following characterization of algebraic sets, which is also proved here: if X and Y are entire complex analytic sets which are bi-Lipschitz homeomorphic at infinity then X is a complex algebraic set if and only if Y is a complex algebraic set too. Thus, an entire complex analytic set is a complex algebraic set if and only if it is bi-Lipschitz homeomorphic at infinity to a complex algebraic set. No restrictions on the singular set, dimension nor codimension are required in the results proved here.

On characterization of smoothness of complex analytic sets

On characterization of smoothness of complex analytic sets

Quantum computing for fusion energy science applications

This is a review of recent research exploring and extending present-day quantum computing capabilities for fusion energy science applications. We begin with a brief tutorial on both ideal and open quantum dynamics, universal quantum computation, and quantum algorithms. Then, we explore the topic of using quantum computers to simulate both linear and nonlinear dynamics in greater detail. Because quantum computers can only efficiently perform linear operations on the quantum state, it is challenging to perform nonlinear operations that are generically required to describe the nonlinear differential equations of interest. In this work, we extend previous results on embedding nonlinear systems within linear systems by explicitly deriving the connection between the Koopman evolution operator, the Perron–Frobenius evolution operator, and the Koopman–von Neumann evolution (KvN) operator. We also explicitly derive the connection between the Koopman and Carleman approaches to embedding. Extension of the KvN framework to the complex-analytic setting relevant to Carleman embedding, and the proof that different choices of complex analytic reproducing kernel Hilbert spaces depend on the choice of Hilbert space metric are covered in the appendixes. Finally, we conclude with a review of recent quantum hardware implementations of algorithms on present-day quantum hardware platforms that may one day be accelerated through Hamiltonian simulation. We discuss the simulation of toy models of wave–particle interactions through the simulation of quantum maps and of wave–wave interactions important in nonlinear plasma dynamics.

Non-negative divisors and the Grauert metric

Grauert showed that it is possible to construct complete Kähler metrics on the complement of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics on the complement of a principal divisor in \(\mathbb {C}^n\), \(n \ge 1\). In addition, we also study how this metric and its holomorphic sectional curvature behave when the corresponding principal divisors vary continuously.

On the Fukui–Kurdyka–Paunescu conjecture

In this paper, we prove the Fukui–Kurdyka–Paunescu conjecture, which says that sub-analytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (respectively, degree) of real and complex analytic (respectively, algebraic) sets. For instance, still in the real case, we prove a global version of the Fukui–Kurdyka–Paunescu conjecture. In the complex case, one of the results that we prove is the following: if$(X,0)\subset (\mathbb {C}^{n},0)$,$(Y,0)\subset (\mathbb {C}^{m},0)$are germs of analytic sets and$h\colon (X,0)\to (Y,0)$is a semi-bi-Lipschitz homeomorphism whose graph is a complex analytic set, then the germs$(X,0)$and$(Y,0)$have the same multiplicity. One of the results that we prove in the global case is the following: if$X\subset \mathbb {C}^{n}$,$Y\subset \mathbb {C}^{m}$are algebraic sets and$\phi \colon X\to Y$is a semi-algebraic semi-bi-Lipschitz homeomorphism such that the closure of its graph in$\mathbb {P}^{n+m}(\mathbb {C})$is an orientable homological cycle, then${\rm deg}(X)={\rm deg}(Y)$.

On Grauert’s examples of complete Kähler metrics

Grauert showed that the existence of a complete Kähler metric does not characterize domains of holomorphy by constructing such metrics on the complements of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics in two prototype cases namely, C n ∖ { 0 } , n ≥ 2 \mathbb {C}^n \setminus \{0\}, n \ge 2 and B N ∖ A \mathbb {B}^N \setminus A , N ≥ 2 N \ge 2 and A ⊂ B N A \subset \mathbb {B}^N is a hyperplane of codimension at least two. This is done by computing the Gaussian curvature of the restriction of these metrics to the leaves of a suitable holomorphic foliation in these two examples. We also examine this metric on the punctured plane C ∗ \mathbb {C}^{\ast } and show that it behaves very differently in this case.

Differential invariance of the multiplicity of real and complex analytic sets

This paper is devoted to proving the differential invariance of the multiplicity of real and complex analytic sets. In particular, we prove the real version of Gau-Lipman's Theorem, i.e., it is proved that the multiplicity mod 2 of real analytic sets is a differential invariant. We prove also a generalization of Gau-Lipman's Theorem.

Moderately Discontinuous Homology

AbstractWe introduce a new metric homology theory, which we call Moderately Discontinuous Homology, designed to capture Lipschitz properties of metric singular subanalytic germs. The main novelty of our approach is to allow “moderately discontinuous” chains, which are specially advantageous for capturing the subtleties of the outer metric phenomena. Our invariant is a finitely generated graded abelian group for any and homomorphisms for any . Here is a “discontinuity rate”. The homology groups of a subanalytic germ with the inner or outer metric are proved to be finitely generated and only finitely many homomorphisms are essential. For Moderately Discontinuous Homology recovers the homology of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for ‐Homology recovers the homology of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating from the germ to its tangent cone. Our homology theory is a bi‐Lipschitz subanalytic invariant, is invariant by suitable metric homotopies, and satisfies versions of the relative and Mayer‐Vietoris long exact sequences. Moreover, fixed a discontinuity rate b we show that it is functorial for a class of discontinuous Lipschitz maps, whose discontinuities are b‐moderated; this makes the theory quite flexible. In the complex analytic setting we introduce an enhancement called Framed MD homology, which takes into account information from fundamental classes. As applications we prove that Moderately Discontinuous Homology characterizes smooth germs among all complex analytic germs, and recovers the number of irreducible components of complex analytic germs and the embedded topological type of plane branches. Framed MD homology recovers the topological type of any plane curve singularity and relative multiplicities of complex analytic germs. © 2020 Wiley Periodicals LLC.

On the infinitesimal automorphisms of principal bundles

We review some basic facts on vector fields, in the complex-analytic setting, thus, obtaining a rationality result and an extension of the Birkhoff-Grothendieck theorem, as follows: (1) Let $Z$ be a compact complex manifold endowed with a very ample line bundle $L$. Denote by $\mathfrak{g}_L$ the extended Lie algebra of infinitesimal automorphisms of $L$. If the representation of $\mathfrak{g}_L$ on the space of holomorphic sections of $L$ is irreducible then $Z$ is rational; (2) Let $P$ be a holomorphic principal bundle over the Riemann sphere, with structural group $G$ whose Lie algebra is not equal to its nilpotent radical. Then there exists a Lie subgroup $H$ of $G$ which is a quotient of a Borel subgroup of ${\rm SL}(2)$ and such that $P$ admits a reduction to $H$.

Algebraic elliptic cohomology and flops II: SL-cobordism

In this paper, we study the algebraic Thom spectrum MSL in Voevodsky's motivic stable homotopy category over an arbitrary perfect field k. Using the motivic Adams spectral sequence, we compute the geometric part of the η-completion of MSL. As an application, we study Krichever's elliptic genus with integral coefficients, restricted to MSL. We determine its image, and identify its kernel as the ideal generated by differences of SL-flops. This was proved by B. Totaro in the complex analytic setting. In the appendix, we prove some convergence properties of the motivic Adams spectral sequence.

On unitary invariants of quotient Hilbert modules along smooth complex analytic sets

Let Ω⊂Cm be an open, connected and bounded set and A(Ω) be a function algebra of holomorphic functions on Ω. Suppose that M is a reproducing kernel Hilbert module over A(Ω). In this article, we first obtain a model for the quotient Hilbert modules obtained from submodules of functions in M vanishing to order k along a smooth irreducible complex analytic set Z⊂Ω of codimension at least 2, assuming that the Hilbert module M is in the Cowen-Douglas class over Ω. This model is used to show that such a quotient module happens to be in the Cowen-Douglas class over Z∩Ω which then enables us to determine unitary equivalence classes of the aforementioned quotient modules in terms of the geometric invariants of hermitian holomorphic vector bundles. As an application, we obtain that unitary equivalence classes of a large family of these Hilbert modules are completely determined by those of certain quotient modules of the above kind.

Multiplicity, regularity and blow-spherical equivalence of complex analytic sets

The author was also partially supported by CNPq-Brazil grant 303811/2018-8 and Gobierno Vasco Grant IT1094-16.

Algebraic elliptic cohomology theory and flops I

We define the algebraic elliptic cohomology theory coming from Krichever’s elliptic genus as an oriented cohomology theory on smooth varieties over an arbitrary perfect field. We show that in the algebraic cobordism ring with rational coefficients, the ideal generated by differences of classical flops coincides with the kernel of Krichever’s elliptic genus. This generalizes a theorem of B. Totaro in the complex analytic setting.

On Lipschitz Rigidity of Complex Analytic Sets

We prove that any complex analytic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and has tangent cone at infinity that is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$ itself. No restrictions on the singular set, dimension nor codimension are required. In particular, a complex algebraic set in $\mathbb{C}^n$ which is Lipschitz regular at infinity is an affine linear subspace.

A transcendental approach to injectivity theorem for log canonical pairs

In this paper, we study transcendental aspects of the cohomology groups of adjoint bundles of log canonical pairs, aiming to establish an analytic theory for log canonical singularities. As a result, in the case of purely log terminal pairs, we give an analytic proof of the injectivity theorem originally proved by the Hodge theory. Our method is based on the theory of harmonic integrals and the L2-method for the dbar-equation, and it enables us to generalize the injectivity theorem to the complex analytic setting.

Obstructed thickenings and supermanifolds

Associated to any supermanifold is a filtration by spaces, referred to as thickenings. It is the objective of this article to study them up to a certain equivalence and then up to isomorphism in the complex-analytic setting. We study them from two points of view: (1) as structures embedded in supermanifolds and (2) abstractly. Throughout, we will be guided by the goal to clarify and address the question: when does a given thickening embed in a supermanifold? Such a question was, in essence, first studied by Eastwood and LeBrun. In this article we begin with a pedagogical account of their study, after which we further study thickenings in supergeometry and present a classification of thickenings of a given order. As a complement to our study, we comment on the moduli problem for complex supermanifolds and consider the analogous problem for thickenings. Finally, to illustrate the ideas in this article, we conclude by describing some obstructed thickenings of the complex projective plane.

Local topological algebraicity with algebraic coefficients of analytic sets or functions

We prove that any complex or real analytic set or function germ is topologically equivalent to a germ defined by polynomial equations whose coefficients are algebraic numbers.

A cup product lemma for continuous plurisubharmonic functions

A version of Gromov’s cup product lemma in which one factor is the (1, 0)-part of the differential of a continuous plurisubharmonic function is obtained. As an application, it is shown that a connected noncompact complete Kähler manifold that has exactly one end and admits a continuous plurisubharmonic function that is strictly plurisubharmonic along some germ of a [Formula: see text]-dimensional complex analytic set at some point has the Bochner–Hartogs property; that is, the first compactly supported cohomology with values in the structure sheaf vanishes.

A proof of the differentiable invariance of the multiplicity using spherical blowing-up

In this paper we use some properties of spherical blowing-up to give an alternative and more geometric proof of Gau–Lipman Theorem about the differentiable invariance of the multiplicity of complex analytic sets. Moreover, we also provide a generalization of the Ephraim–Trotman Theorem.

Tangent Cones of Lipschitz Normally Embedded Sets Are Lipschitz Normally Embedded. Appendix by Anne Pichon and Walter D. Neumann

AbstractWe prove that tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded. We also extend to real subanalytic sets the notion of reduced tangent cone and we show that subanalytic Lipschitz normally embedded sets have reduced tangent cones. In particular, we get that Lipschitz normally embedded complex analytic sets have reduced tangent cones.