Analytic Subvariety Research Articles

Complete norm-preserving extensions of holomorphic functions

We show that for every connected analytic subvariety V there is a pseudoconvex set Ω such that every bounded matrix-valued holomorphic function on V extends isometrically to Ω. We prove that if V is two analytic discs intersecting at one point, if every bounded scalar valued holomorphic function extends isometrically to Ω, then so does every matrix-valued function. In the special case that Ω is the symmetrized bidisc, we show that this cannot be done by finding a linear isometric extension from the functions that vanish at one point.

On the Extension of Quasiplurisubharmonic Functions

Let (V, ω) be a compact Kähler manifold such that V admits a cover by Zariski-open Stein sets with the property that ω has a strictly plurisubharmonic exhaustive potential on each element of the cover. If X ⊂ V is an analytic subvariety, we prove that any ω∣x-plurisubharmonic function on X extends to a ω-plurisubharmonic function on V.This result generalizes a previous result of ours on the extension of singular metrics of ample line bundles. It allows one to show that any transcendental Kähler class in the real Neron—Severi space NSℝ(V) has this extension property.

Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below

We show that non-collapsed Gromov–Hausdorff limits of polarized Kähler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson–Sun, in which 2-sided Ricci curvature bounds were assumed. As a basic ingredient we show that, under lower Ricci curvature bounds, almost Euclidean balls in Kähler manifolds admit good holomorphic coordinates. Further applications are integral bounds for the scalar curvature on balls, and a rigidity theorem for Kähler manifolds with almost Euclidean volume growth.

Extension of cohomology classes and holomorphic sections defined on subvarieties

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general L 2 L^2 extension theorem obtained by Demailly.

Courants résidus et opérateurs de Monge–Ampère

We extend to the case when the support is a possibly singular analytic subvariety the Federer theorem on the structure of the residue current. On another hand, we determine the general law of transformation of the distribution γf associated with a holomorphic map f=(f1,…,fN). In such a way, we arrive at the cohomological interpretation of the fundamental class of an effective analytic cycle, which is not necessarily a local complete intersection. By this same law, we obtain a characterization of the pure dimensional algebraic subsets of Cn, which are complete intersections. We also characterize the complete intersections of codimension q in Pn in terms of the solutions of the singular Monge–Ampère equation in Pq. Lastly, we express the condition on the dimension of the poles of the plurisubharmonic function u, so that the Monge–Ampère operator Q∧dcu has measure coefficients, for all closed positive current Q of bidimension (k,k).

Norm preserving extensionsof bounded holomorphic functions

Let $V$ be an analytic subvariety of a domain $\Omega$ in $\mathbb{C}^{n}$. When does $V$ have the property that every bounded holomorphic function $f$ on $V$ has an extension to a bounded holomorphic function on $\Omega$ with the same norm? An obvious sufficient condition is if $V$ is a holomorphic retract of $\Omega$. We shall discuss for what domains $\Omega$ this is also necessary. This is joint work with Łukasz Kosinski.

Algebraic separatrices for non-dicritical foliations on projective spaces of dimension at least four

Non-dicritical codimension one foliations on projective spaces of dimension four or higher always have an invariant algebraic hypersurface. The proof relies on a strengthening of a result by Rossi on the algebraization/continuation of analytic subvarieties of projective spaces.

Holomorphic Foliations Tangent to Levi-Flat Subsets

An irreducible real analytic subvariety H of real dimension $$2n +1$$ in a complex manifold M is a Levi-flat subset if its regular part carries a complex foliation of dimension n. Locally, a germ of real analytic Levi-flat subset is contained in a germ of irreducible complex variety $$H^{\imath }$$ of dimension $$n+1$$ , called intrinsic complexification, which can be globalized to a neighborhood of H in M provided H is a coherent analytic subvariety. In this case, a singular holomorphic foliation $$\mathcal {F}$$ of dimension n in M that is tangent to H is also tangent to $$H^{\imath }$$ . In this paper, we prove integration results of local and global nature for the restriction to $$H^{\imath }$$ of a singular holomorphic foliation $$\mathcal {F}$$ tangent to a real analytic Levi-flat subset H. From a local viewpoint, if $$n=1$$ and $$H^{\imath }$$ has an isolated singularity, then $$\mathcal {F}|_{H^{\imath }}$$ has a meromorphic first integral. From a global perspective, when $$M = \mathbb {P}^N$$ and H is coherent and of low codimension, $$H^{\imath }$$ extends to an algebraic variety. In this case, $$\mathcal {F}|_{H^{\imath }}$$ has a rational first integral provided infinitely many leaves of $$\mathcal {F}$$ in H are algebraic.

A Note on Curvature Estimate of the Hermitian–Yang–Mills Flow

In this paper, we study the curvature estimate of the Hermitian-Yang-Mills flow on holomorphic vector bundles. In one simple case, we show that the curvature of the evolved Hermitian metric is uniformly bounded away from the analytic subvariety determined by the Harder-Narasimhan-Seshadri filtration of the holomorphic vector bundle.

On the singular sets of solutions to the Kapustin\u2013Witten equations and the Vafa\u2013Witten ones on compact K\xe4hler surfaces

This article finds a structure of singular sets on compact Kähler surfaces, which Taubes introduced in the studies of the asymptotic analysis of solutions to the Kapustin–Witten equations and the Vafa–Witten ones originally on smooth four-manifolds. These equations can be seen as real four-dimensional analogues of the Hitchin equations on Riemann surfaces, and one of common obstacles to be overcome is a certain unboundedness of solutions to these equations, especially of the “Higgs fields”. The singular sets by Taubes describe part of the limiting behaviour of a sequence of solutions with this unboundedness property, and Taubes proved that the real two-dimensional Haussdorff measures of these singular sets are finite. In this article, we look into the singular sets, when the underlying manifold is a compact Kähler surface, and find out that they have the structure of an analytic subvariety in this case.

Closed Groups of Automorphisms of Products of Hyperbolic Riemann Surfaces

In this paper, we provide the complete list of all closed groups G of automorphisms of a product R of hyperbolic Riemann surfaces such that the order of any element in $$\textit{G/G}_e$$ , where $$G_e$$ is the identity component of G, is finite. In particular, if X is an analytic subvariety of R then the identity component of the stabilizer of X in $${\text {Aut}}R$$ is on this list. In its turn, it allows us to state that the identity component of the group $${\text {Aut}}X$$ must contain a group from this list.

The codimension-three conjecture for holonomic DQ-modules

On a complex symplectic manifold, we prove that any holonomic DQ-module endowed with a lattice and defined outside of an analytic subvariety of codimension 3 of its support extends as an holonomic DQ-module. This is an analogue for DQ-modules of the codimension-three conjecture for microdifferential modules recently proved by Kashiwara and Vilonen.

Dimension of the minimum set for the real and complex Monge–Ampère equations in critical Sobolev spaces

We prove that the zero set of a nonnegative plurisubharmonic function that solves $\det (\partial \overline{\partial} u) \geq 1$ in $\mathbb{C}^n$ and is in $W^{2, \frac{n(n-k)}{k}}$ contains no analytic sub-variety of dimension $k$ or larger. Along the way we prove an analogous result for the real Monge-Amp\`ere equation, which is also new. These results are sharp in view of well-known examples of Pogorelov and B{\l}ocki. As an application, in the real case we extend interior regularity results to the case that $u$ lies in a critical Sobolev space (or more generally, certain Sobolev-Orlicz spaces).

Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

Let [Formula: see text] be a connected open Riemann surface. Let [Formula: see text] be an Oka domain in the smooth locus of an analytic subvariety of [Formula: see text], [Formula: see text], such that the convex hull of [Formula: see text] is all of [Formula: see text]. Let [Formula: see text] be the space of nondegenerate holomorphic maps [Formula: see text]. Take a holomorphic 1-form [Formula: see text] on [Formula: see text], not identically zero, and let [Formula: see text] send a map [Formula: see text] to the cohomology class of [Formula: see text]. Our main theorem states that [Formula: see text] is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on [Formula: see text] can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstnerič and Lárusson in 2016.

On the Volumes of Analytic Varieties Near the Diagonal for a Compact Quotient of the Polydisk

We give sharp lower bounds for the volumes of one-dimensional complex analytic subvarieties in factorwise geodesic tubular neighborhoods of the Cartesian self-product of a compact quotient of the polydisk. As an application, we obtain a sufficient condition for the pluri-canonical bundles of a compact quotient of the polydisk to satisfy the property $$(N_p)$$ on linear syzygies.

The minimum sets and free boundaries of strictly plurisubharmonic functions

We study the minimum sets of plurisubharmonic functions with strictly positive Monge–Ampere densities. We investigate the relationship between their Hausdorff dimension and the regularity of the function. Under suitable assumptions we prove that the minimum set cannot contain analytic subvarieties of large dimension. In the planar case we analyze the influence on the regularity of the right hand side and consider the corresponding free boundary problem with irregular data. We provide sharp examples for the Hausdorff dimension of the minimum set and the related free boundary. We also draw several analogues with the corresponding real results.

The infinite topology of the hyperelliptic locus in Torelli space

Genus g Torelli space is the moduli space of genus g curves of compact type equipped with a homology framing. The hyperelliptic locus is a closed analytic subvariety consisting of finitely many mutually isomorphic components. We use properties of the hyperelliptic Torelli group to show that when \(g\ge 3\) these components do not have the homotopy type of a finite CW complex. Specifically, we show that the second rational homology of each component is infinite-dimensional. We give a more detailed description of the topological features of these components when \(g=3\) using properties of genus 3 theta functions.

Sard property for the endpoint map on some Carnot groups

In Carnot–Carathéodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizations of the abnormal set in the case of Lie groups.

Kähler currents and null loci

We prove that the non-Kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kahler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein-Lazarsfeld-Mustata-Nakamaye-Popa. As an application, we show that finite time non-collapsing singularities of the Kahler-Ricci flow on compact Kahler manifolds always form along analytic subvarieties, thus answering a question of Feldman-Ilmanen-Knopf and Campana. We also extend the second author's results about noncollapsing degenerations of Ricci-flat Kahler metrics on Calabi-Yau manifolds to the nonalgebraic case.

Observations on the Hofer distance between closed subsets

We prove the elementary but surprising fact that the Hofer distance between two closed subsets of a symplectic manifold can be expressed in terms of the restrictions of Hamiltonians to one of the subsets; this helps explain certain energy-capacity inequalities that appeared recently in work of Borman-McLean and Humiliere-Leclercq-Seyfaddini. We also build on arXiv:1201.2926 to obtain new vanishing results for the Hofer distance between subsets, applicable for instance to singular analytic subvarieties of Kahler manifolds.