Complex Analysis, Harmonic Analysis, & Complex Geometry Seminar

Abstract: The concept of sparse domination was first introduced for singular integral operators and then extended to many other operators in harmonic analysis. I will discuss a general sparse domination principle that applies to large classes of operators beyond the Calderón--Zygmund theory, but usually does not cover endpoint cases.  There are several new results in this direction, covering for example classical oscillatory multipliers and Bochner-Riesz means at the critical index.

This is joint work with David Beltran and Joris Roos.

Abstract: The Hartogs triangle in the complex Euclidean space is an important example in several complex variables. It is a bounded pseudoconvex domain with non-Lipschitz boundary. In this talk, we discuss the extendability of Sobolev spaces on the Hartogs triangle and show that the weak and strong maximal extensions of the Cauchy-Riemann operator agree (joint work with A. Burchard, J. Flynn and G. Lu). These results are related to the Dolbeault cohomology groups with Sobolev coefficients on the complement of the Hartogs triangle. We will also discuss some recent progress for the Cauchy-Riemann equations on Hartogs triangles in the complex projective space (joint work with C. Laurent- Thiébaut).

This talk is about subtle properties of averages on spheres in two and higher-dimensional Euclidean space.

Maximal spherical averages are a classic topic in harmonic analysis originating in questions on differentiability properties of functions. We consider maximal spherical averages with a supremum taken over a given dilation set. It turns out that the sharp Lp improving properties of such operators are closely related to fractal dimensions of the dilation set such as the Minkowski and Assouad dimensions. This leads to a surprising characterization of the closed convex sets which can occur as closure of the sharp Lp improving region of such a maximal operator. 

This is joint work with Andreas Seeger. If time allows, we will also mention recent work on the Heisenberg group and work in progress on the variational setting.

Abstract: https://math.rutgers.edu/news-events/list-all-events/icalrepeat.detail/2023/04/07/11959/-/estimate-for-global-newlander-nirenberg-theorem-on-strongly-pseudoconvex-domains

Abstract: It is a famous result  of M. Riesz that the Szego projection operator, initially defined as the orthogonal projection from the space $L^2(\mathbb{T})$ of square integrable functions on the circle to the Hardy space $H^2(\mathbb{D)$, extends 

continuously as a projection operator from $L^p(\mathbb{T})$ onto $H^p(\mathbb{D})$. There is a long history of similar results in the setting of Bergman spaces, and a long list of domains where an analogous statement does not hold in the Bergman setting. We try to understand the geometric distinction between the Hardy and the Bergman situations in $L^p$, and propose a new projection operator on Reinhardt domains which is expected to have better mapping properties. We verify that the new operator satisfies $L^p$ estimates in some situations where the Bergman projection operator does not satisfy such estimates.  This is joint work with Luke Edholm of the University of Vienna.

Abstract: In this talk, we will study the residual Monge-Ampere mass of a plurisubharmonic function with isolated singularity at the origin in C^2. We proved that the residual mass is zero if its Lelong number is zero at the origin, provided that it is S1-invariant and radially regular. This result answers the zero mass conjecture raised by Guedj and Rashkovskii in this special case. 

Abstract: ALG gravitational instantons i.e. non-compact complete hyper-K”ahler surfaces asymptotic to a twisted product of the complex plane and an elliptic curve, are intensively studied. Following the classical work of Tian-Yau and Hein etc on Monge-Ampere methods for Ricci flat K”ahler metrics on quasi-projective varieties, we provide an geometric existence theorem for generalized ALG Ricci flat K”ahler 3-folds on isotrivial K3 fibrations/crepant resolutions. These metrics decay to the ALG model in any poly-nomial rate, and topogical numbers/data can be calculated. 

Abstract: We will discuss some classes of PDEs (on Hermitian manifolds) factoring through complex hessian but not its eigenvalues. In particular, this will include the class of PDEs which, if solvable, would confirm the quaternionic Calabi conjecture.

Abstract: In the study of K-stability, Fujita and Li proposed the valuative criterion of K-stability on Fano varieties, which has played an essential role of the algebraic theory of K-stability. Recently, Dervan-Legendre considered the valuative criterion of polarized varieties, which is a generalization of Fujita-Li criterion on Fano varieties. We will show that uniformly valuative stability is an open condition. We would like to study the valuative criterion for the Donaldson's J-equation. Motivated by the beta-invariant of Dervan-Legendre, we introduce a notion, the so-called valuative J-stability and prove that J-stability implies valuative J-stability. 

Abstract: In Euclidean spaces, convolution with a smoothing kernel gives regularization for subharmonic functions, plurisubharmonic functions, and more generally, functions whose complex Hessian is in a convex cone. We wish to explore how to generalize these results on K"ahler manifolds. Very few results are known beyond the plurisubharmonic case, and the usual integration approach runs into serious difficulties. Instead, we use a sup-convolution to obtain regularization results for a general convex cone on compact Kahler manifolds with nonnegative bisectional curvature. This is based on joint work with Yulun Xu. 

Abstract: Title: K¨ahler-Einstein metrics and obstruction flatness of circle bundles Abstract: Obstruction flatness of a strongly pseudoconvex hypersurface ? in a complex manifold refers to the property that any (local) K¨ahler-Einstein metric on the pseudoconvex side of ?, complete up to ?, has a potential ? log u such that u is C?-smooth up to ?. In general, u has only a finite degree of smoothness up to ?. In this talk, we are interested in obstruction flatness of hypersurfaces ? that arise as unit circle bundles S(L) of negative Hermitian line bundles (L, h) over a complex manifold M, whose dual line bundle induces a K¨ahler metric g on M. The main result we will discuss can be summarized as follows: If (M, g) has constant Ricci eigenvalues, then S(L) is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and (M, g) is complete, then the corresponding disk bundle admits a complete K¨ahler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of S(L) in terms of the K¨ahler geometry of (M, g) in some special cases. The talk is based on a recent joint paper with Ebenfelt and Xu

Abstract: In this talk, I will introduce the Hörmander's weighted L² estimates for the Cauchy-Riemann operator and then present some applications which include sharp pointwise and uniform estimates for the canonical solution for the Cauchy-Riemann equation u = f on classical bounded symmetric domains and product domains in Cn. The second application is my recent work on applying the weighted L2 estimates to study the Corona problem in several complex variables. This seminar is held at Rutgers-Camden, as part of a joint seminar with the Complex Analysis and Geometry seminar at Rutgers-New Brunswick. 

Abstract: Uniform estimates for complex Monge-Ampere equations have been extensively studied, ever since Yau’s resolution of the Calabi conjecture. Subsequent developments have led to many geometric applications to many other fields, but all relied on the pluripotential theory from complex analysis. In this talk, we will discuss a new PDE-based method of obtaining sharp uniform C^0 estimates for complex Monge-Ampere (MA) and other fully nonlinear PDEs, without the pluripotential theory. This new method extends more generally to other interesting geometric estimates for MA and Hessian equations. This is based on the joint works with D.H. Phong, F. Tong.

Abstract: Motivated by holomorphic sectional curvature and Ricci curvature, in 2018, Lei Ni introduced the definition of k-Ricci curvature. In this talk, I will show that the canonical bundle of a compact Kahler manifold with quasi-negative k-Ricci curvature is nef and big. This is a joint work with Man-Chun Lee and Luen-Fai Tam.

Abstract: In the study of complex algebraic varieties, the combinatorial methods play an important role. On a toric variety, it is well-known that toric invariant geometric objects can be described by objects in the Euclidean space. For example, one can naturally assign a convex body (Newton polytope) to each toric invariant big line bundle. A few years ago, Lazarsfeld—Mustaţă and Kaveh—Khovanskii introduced a generalisation of this assignment to the non-toric setting. They assign convex bodies known as Okounkov bodies to each big line bundle on a projective variety. It turns out that Okounkov bodies provide tremendous information about the geometry of the variety.

In this talk, I will talk about a generalisation of the Okounkov body construction to line bundles equipped with pluri-subharmonic metrics. I will explain various applications of this construction in pluripotential theory and in non-Archimedean geometry.

Abstract: It is well known that the De Rham cohomology of a compact Lie group is isomorphic to the Chevalley-Eilenberg complex. While the former is a topological invariant of the Lie group, the latter can be computed by using simple linear algebra methods. In this talk, we discuss how to obtain an injective homomorphism between the cohomology spaces associated with left-invariant involutive structures and the cohomology of a generalized Chevalley-Eilenberg complex.

We discuss some cases in which the homomorphism is surjective, such as the Dolbeault cohomology and certain elliptic and CR structures. The results provide new insights regarding the general theory of involutive structures as, for example, they reveal algebraic obstructions for solvability for the associated differential complexes.

Abstract: Given a bounded Lipschitz domain $\Omega\subset\mathbb R^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $\Omega$ which is bounded in Besov and Triebel-Lizorkin spaces. In this talk we take a look on two new properties the extension operator $\mathcal E$ and give some applications. We prove the equivalent norms $\|f\|_{\mathscr A_{pq}^s(\Omega)}\approx\sum_{|\alpha|\le m}\|\partial^\alpha f\|_{\mathscr A_{pq}^{s-m}(\Omega)}$ for general Besov and Triebel-Lizorkin spaces, which appears to be a well-known result. We also derive some quantitative smoothing estimates of the extended function in $\overline{\Omega}^c$ up to boundary. This is joint work with Ziming Shi. 

Abstract: While the equivalence between ampleness and positivity holds for vector bundles of rank one, its higher rank counterpart known as Griffiths' conjecture is still open. There is also a similar but weaker conjecture by Kobayashi who proposed to use Finsler rather than Hermitian metrics to study the equivalence. We will review these two conjectures and state our progress. One of our results is that we can construct a positively curved Finsler metric on $E$ if the symmetric power of the dual $S^kE^*$ has a negatively curved $L^2$ Hermitian metric.

Abstract: The classical Abel-Jacobi map induces the Torelli map from a moduli space of curves of genus $g\geq 2$ into the corresponding Siegel modular variety. The goal of the talk is to explain some geometric problems related to the mapping, focusing on a conjecture of Oort on scarcity of totally geodesic subvarieties in the Torelli image.  We will also explain its relation to a rigidity problem of representation of a lattice of semi-simple Lie groups in a mapping class group.  Some new approaches will be explained.

Abstract: The asymptotic behavior of Bergman kernel on polarized Kähler manifold has been studied by Tian, Ruan, Zelditch, Catlin and many others since 1990. The works related to Bergman kernel play important roles in complex geometry. In this talk, I will discuss some uniform asymptotic estimates (L^p and C^0) of Bergman kernels on various collections of polarized Kähler manifolds.

Abstract: I will explain how local identities for almost Kähler manifolds lead to various unexpected symmetries on certain subspaces of the cohomology of a compact almost Kähler manifold. This allows to deduce several geometric and topological consequences for these manifolds. In particular, we obtain new obstructions to the existence of a symplectic form compatible with a given almost complex structure. This is joint work with Scott Wilson.

Abstract: I will present several examples of compact almost complex manifolds with a 1-parameter family of almost complex structures having arbitrarily small Nijenhuis tensors in the C^0-norm. The 4 dimensional examples possess no complex structure, while the 6 dimensional examples do not possess left invariant complex structures, and whether they possess complex structures appears to be unknown. This is joint work with Luis Fernandez and Tobias Shin.

Abstract: In the last decades geometric flows have been proved to be a powerful tool in the classification and uniformization problems in geometry and topology. Despite the wide range of applicability of the existing analytical methods, we are still lacking efficient tools adapted to the study of general (non-Kähler) complex manifolds. In my talk I will discuss the pluriclosed flow - a modification of the Ricci flow - which was introduced by Streets and Tian, and shares many nice features of the Ricci flow. The important open questions driving the ongoing research in complex geometry are the classification of compact non-Kähler surfaces, and the Global Spherical Shell conjecture. Our hope is that understanding the long-time behaviour and singularities of the pluriclosed flow well enough, we can use it to approach these open questions.

To apply an analytic flow to any geometric problem, we need to make the first necessary step - classify the stationary points of the flow, and, more generally, its solitons (stationary points modulo diffeomorphisms). For the pluriclosed flow, this question reduces to a non-linear elliptic PDE for an Hermitian metric on a given complex manifold. We will discuss this problem on compact/complete complex surfaces, and provide exhaustive classification under natural extra geometric assumptions. In the course of our classification we will discover a natural extension of the famous Gibbons-Hawking ansatz for hyperKähler manifolds.

Jones (1980) identified extension domains for BMO with uniform domains.  In joint work with Almaz Butaev, we show the analogous result for classes of functions of vanishing mean oscillation, and for the nonhomogeneous space bmo defined by Goldberg.  In the latter case we identify the extension domains with the epsilon-delta domains used in Jones' extension theorem for Sobolev spaces, which can be considered as local versions of uniform domains.

Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor.  The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths – and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled. Another outcome of the approach is a new concept of volume for vector bundles.

ABSTRACT:  Flat directions are obstacles and at the same time also essential tools for a number of fundamental problems in several complex variables involving rigidity and regularity.  Among them are the following examples.

(i)  The global non-deformability of irreducible compact Hermitian symmetric manifolds.

(ii) The strong rigidity and super rigidity problem of holomorphic maps with curvature condition on the target manifold.

(iii) The regularity question of the complex Neumann problem for weakly pseudoconvex domains.

(iv) Rigidity and strong rigidity problems of holomorphic vector bundles.

(v) Rigidity and strong rigidity problems of CR manifolds.

For global nondeformability and regularity problems for pseudoconvexity domains flat directions are obstacles.  For rigidity of metrics and CR manifolds with the possibility of small perturbations, flat directions are essential tools. The talk starts with the historic motivations of the problems and does not assume any background more than basic complex analysis.  After discussing the general techniques involving flat directions, we will focus on the global non-deformability problem and some recent methods in this area.

Abstract: It was a long standing question in pluripotential theory asking whether one can recover higher Lelong numbers of a plurisubharmonic function by using its analytic data. Chi Li recently gave an explicit counter-example to this question. Motivated by the theory of non-pluripolar products in the compact setting, he conjectured that his example can be generalized to a much larger natural class of psh functions. We present in the talk an affirmative answer to this question. This is a joint-work with Do Duc Thai.

Abstract

Abstract: Here we develop a theory for oscillatory integrals with complex phases. Basic scale-invariant bounds for these oscillatory integrals do not hold in the generality that they do in the real setting. In fact they fail in the category of complex analytic phases but we develop a perspective and arguments to establish scale-invariant bounds for complex polynomial phases.

Abstract: We study regularity of $\dbar$ solutions on a relatively compact $C^2$ domain $D$ in a complex manifold. Suppose that the boundary of the domain has everywhere either $(q+1)$ negative or $(n-q)$ positive Levi eigenvalues. Under a necessary condition on the existence of a locally $L^2$ solution on the domain, we show the existence of the solutions on the closure of the domain that gain $1/2$ derivative when $q=1$ and the given $(0,q)$ form in the $\dbar$ equation is in the H\"older-Zygmund space $\Lambda^r(\overline D)$ with $r>1$. For $q>1$, the same regularity for the solutions is achieved when the boundary is either sufficiently smooth or of $(n-q)$ positive Levi eigenvalues everywhere.