More Like this

1. Operator realizations of non-commutative analytic functions

   https://doi.org/10.1017/fms.2025.10038  

   Augat, Méric L; Martin, Robert T_W; Shamovich, Eli (January 2025, Forum of Mathematics, Sigma)

   A realization is a triple, (A,b,c), consisting of a d−tuple, A=(A1,⋯,Ad), d∈N, of bounded linear operators on a separable, complex Hilbert space, H, and vectors b,c∈H. Any such realization defines an analytic non-commutative (NC) function in an open neighbourhood of the origin, 0:=(0,⋯,0), of the NC universe of d−tuples of square matrices of any fixed size. For example, a univariate realization, i.e., where A is a single bounded linear operator, defines a holomorphic function of a single complex variable, z, in an open neighbourhood of the origin via the realization formula b∗(I−zA)−1c . It is well known that an NC function has a finite-dimensional realization if and only if it is a non-commutative rational function that is defined at 0 . Such finite realizations contain valuable information about the NC rational functions they generate. By extending to infinite-dimensional realizations, we construct, study and characterize more general classes of analytic NC functions. In particular, we show that an NC function is (uniformly) entire if and only if it has a jointly compact and quasinilpotent realization. Restricting our results to one variable shows that a formal Taylor series extends globally to an entire or meromorphic function in the complex plane, C, if and only if it has a realization whose component operator is compact and quasinilpotent, or compact, respectively. This motivates our definition of the field of global (uniformly) meromorphic NC functions as the field of fractions generated by NC rational expressions in the ring of NC functions with jointly compact realizations. This definition recovers the field of meromorphic functions in C when restricted to one variable. 

   more » « less
2. The symplectic isotopy problem for rational cuspidal curves

   https://doi.org/10.1112/s0010437x2200762x  

   Golla, Marco; Starkston, Laura (July 2022, Compositio Mathematica)

   We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology. 

   more » « less
3. On finite energy monopoles on $$C x \Sigma$$.

   Wang, Donghao (July 2019, ArXiv.org)

   Let $$X=\mathbb{C}\times\Sigma$$ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on $$X$$ with finite analytic energy. The spin bundle $$S^+\to X$$ splits as $$L^+\oplus L^-$$. When $$2-2g\leq c_1(S^+)[\Sigma]<0$$, the moduli space is in bijection with the moduli space of pairs $$((L^+,\bar{\partial}), f)$$ where $$(L^+,\bar{\partial})$$ is a holomorphic structure on $L^+$ and $$f: \mathbb{C}\to H^0(\Sigma, L^+,\bar{\partial})$$ is a polynomial map. Moreover, the solution has analytic energy $$-4\pi^2d\cdot c_1(S^+)[\Sigma]$$ if $$f$$ has degree $$d$$. When $$c_1(S^+)=0$$, all solutions are reducible and the moduli space is the space of flat connections on $$\bigwedge^2 S^+$$. We also estimate the decay rate at infinity for these solutions. 

   more » « less
4. Rationality of Real Conic Bundles With Quartic Discriminant Curve

   https://doi.org/10.1093/imrn/rnad003  

   Ji, Lena; Ji, Mattie (February 2023, International Mathematics Research Notices)

   We study real double covers of $$\mathbb P^{1}\times \mathbb P^{2}$$ branched over a $(2,2)$-divisor, which are conic bundles with smooth quartic discriminant curve by the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space is $$\mathbb R$$-rational. For five of the six isotopy classes, we construct $$\mathbb C$$-rational examples with obstructions to rationality over $$\mathbb R$$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with irrational members, we characterize rationality using the real locus and the intermediate Jacobian torsor obstruction of Hassett–Tschinkel and Benoist–Wittenberg. These double cover models were introduced by Frei, Sankar, Viray, Vogt, and the first author, who determined explicit descriptions for their intermediate Jacobian torsors. 

   more » « less
5. Equivalence of Neighborhoods of Embedded Compact Complex Manifolds and Higher Codimension Foliations

   https://doi.org/10.1007/s40598-021-00192-w  

   Gong, Xianghong; Stolovitch, Laurent (March 2022, Arnold mathematical journal)

   We consider an embedded n-dimensional compact complex manifold in n+d dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert’s formal principle program. We will give conditions ensuring that a neighborhood of C in M is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold’s result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in Mn+d having C as a compact leaf, extending Ueda’s theory to the high codimension case. Both problems appear as a kind of linearization problems involving small divisors conditions arising from solutions to their cohomological equations. 

   more » « less