Titles and abstracts - Several Complex Variables and Complex Geometry Conference

Speaker	Title	Abstract
J. Cao Université Côte d’Azur, France	Hodge Theory for Local Systems on Quasi-Compact Kähler Manifolds and Applications	In this talk, we will present some extension theorems for local systems on quasi-compact Kähler manifolds. We will also discuss several geometric applications, including results on cohomology jumping loci and properties of local systems on special varieties. This talk is based on two joint works with Ya Deng, Christopher Hacon, and Mihai Păun: arXiv:2511.06773 and arXiv:2603.14539.
S. Chanillo Rutgers, USA	To be announced.	To be announced.
T. Collins Toronto University, Canada	To be announced.	To be announced.
D. Coman Syracuse University, USA	Distribution of random degeneracy sets for Grassmannian embeddings	Let (X, ω) be a compact Kähler manifold, (L, h^L) be a positive line bundle, and (E, h^E) be a Hermitian holomorphic vector bundle of rank r on X. We show that the pullback by the Kodaira embedding associated to L^p ⊗ E of the k-th Chern form of the dual universal bundle over the Grassmannian converges as p → ∞ to the k-th power of the Chern form c₁(L, h^L), for 0 ≤ k ≤ r. The degeneracy set of a k-tuple of holomorphic sections of L^p ⊗ E is the locus of points in X where they are linearly dependent. We compute the expectation of the currents of integration along degeneracy sets of random k-tuples of holomorphic sections of L^p ⊗ E. Using these results and a sequence of suitable meromorphic transforms associated to the degeneracy sets, we prove the almost sure convergence of these currents as p → ∞. This talk is based on joint work with Turgay Bayraktar, Bingxiao Liu and George Marinescu.
T. Darvas Univ of Maryland, USA	Sharp C^1,1̄ Estimates in Kähler Quantization	I will discuss sharp C^1,1̄ estimates for Bergman potentials in Kähler quantization. For a plurisubharmonic weight φ, with weighted Bergman kernel K_φ, we prove upper and positive lower bounds for i∂∂̄ log K_φ in terms of the corresponding bounds for i∂∂̄φ. These estimates hold both locally and on compact Kähler manifolds. I will explain the analytic ideas behind the estimates and how they lead to optimal C^1,α-convergence of Bergman approximations for Kähler currents with bounded coefficients. Time permitting, I will also discuss applications to the quantization of non-pluripolar Radon measures. The talk is based on joint work with Zbigniew Blocki.
P. Ebenfelt University of California at San Diego, USA	To be announced.	To be announced.
X. Gong University of Wisconsin at Madison, USA	Integrability of Koszul connections on complex vector bundles over domains in ℂⁿ	We study invertible matrix solutions A to the equation A⁻¹∂̄A = ω^(0,1) on a small open subset U of the closure M of a domain M ⊂ ℂⁿ, where ω^(0,1) is a matrix of (0,1) forms on M satisfying the formal integrable condition ∂̄ω^(0,1) = ω^(0,1) ∧ ω^(0,1). For a C² domain M that is either strongly pseudoconvex or has at least 3 negative Levi eigenvalues at a boundary point contained in U, we obtain existence and sharp regularity of the solutions.
V. Guedj L’Université Paul Sabatier, France	Geometric smoothing by the Kähler-Ricci flow	We study the geometric regularization of a positive closed current by the (twisted) Kähler-Ricci flow on a compact Kähler manifold. We conjecture that the local Arnold multiplicities linearly decrease to zero, while the flow produces complete Kähler metrics in the Zariski open subset of points that have zero Lelong numbers. We prove this conjecture in complex dimension 1 and provide several partial results in higher dimension. This is joint work with E. Di Nezza and H. C. Lu.
B. Guo Rutgers University at Newark, USA	To be announced.	To be announced.
S. Y. Kim Institute of Basic Sciences, South Korea	Proper holomorphic maps between bounded symmetric domains of rank p and 2p − 1	By a recent result of Kim–Mok–Seo, it is known that any proper holomorphic map between two irreducible bounded symmetric domains of the same type is semi-standard if rank(Ω′) ≤ 2 rank(Ω) − 2. On the other hand, Seo constructed examples for type-I bounded symmetric domains with rank(Ω′) = 2 rank(Ω) − 1, which are not semi-standard. In this talk, we classify the proper holomorphic maps between two type-I bounded symmetric domains Ω, Ω′ with 5 ≤ rank(Ω) and rank(Ω′) = 2 rank(Ω) − 1, under the assumption that the proper holomorphic maps respect the maximal invariantly geodesic subspaces. We first classify the rational proper holomorphic maps between generalized balls ℬ_p,q and ℬ_2p−1,q′ with 5 ≤ p ≤ q and 2p − 1 ≤ q′, that are the associated moduli maps of those between the corresponding type-I domains. Then we give an explicit expression of the corresponding proper holomorphic maps between Ω and Ω′. This is joint work with Y. Gao and S.-C. Ng.
T. Koike Osaka Metropolitan University, Japan	Recent progress on semi-positivity criteria for holomorphic line bundles	I will discuss geometric criteria for the semi-positivity of holomorphic line bundles on compact Kähler manifolds. Let D be an effective nef divisor on a compact Kähler manifold X, and assume that the numerical dimension of the line bundle [D] associated with D is one. I will explain a characterization of the semi-positivity of [D] in terms of the unitary flatness of [D] on a neighborhood of the support of D, which leads naturally to a Levi-flat neighborhood geometry and a holomorphic foliation associated with the divisor. I will also explain the relation with Ueda theory and some related formal-principle-type questions for line bundles.
Lukasz Kosinski Jagiellonian University, Poland	To be announced.	To be announced.
Y. Kusakabe Kyushu University, Japan	On pseudoconvexity of Gromov elliptic manifolds	In contrast to Kobayashi hyperbolic manifolds, Gromov elliptic manifolds are defined as complex manifolds admitting a dominating holomorphic spray, that is, a holomorphic family of dominating entire maps ℂⁿ → X parametrized by X. By Gromov’s Oka principle, every Gromov elliptic manifold is an Oka manifold, and therefore admits sufficiently many holomorphic maps from Stein manifolds to satisfy approximation and extension properties. Geometrically, while complete Kobayashi hyperbolic manifolds are Hartogs pseudoconvex, recent developments suggest that Oka manifolds are closely related to pseudoconcavity. In this talk, however, we show that Gromov elliptic manifolds also enjoy a certain form of pseudoconvexity. As an application, we construct compact Oka manifolds that are not Gromov elliptic, thereby providing compact counterexamples to Gromov’s question.
B. Lamel University of Vienna, Austria	To be announced.	To be announced.
S. Y. Li University of California at Irvine, USA	Stein manifolds with Bergman metrics of constant holomorphic sectional curvature	Let M be a complex manifold of dimension n. Assume that the Bergman metric of M has constant holomorphic sectional curvature κ. In this talk, based on our previous work, we prove that if M is Stein and κ ≤ 0, then M is biholomorphic to B_n &setminus; E, where B_n is the unit ball in ℂⁿ and E is a relatively closed pluripolar subset of B_n. For κ > 0, we discuss some previous work and an ongoing project.
X. Ma University of Paris, France	To be announced.	To be announced.
N. Mir Texas A&M University at Qatar, Qatar	Artin approximation and CR geometry	We explore the validity of Artin’s approximation theorem under additional constraints arising from the CR equations of a real-analytic CR manifold. We will discuss both positive and negative results along these lines, including recent joint work with B. Lamel. Ultimately, these results highlight the crucial role played by the CR geometry of the underlying manifold.
N. Mok University of Hong Kong, Hong Kong	To be announced.	To be announced.
D. W. Nystrom University of Gothenburg, Sweden	Extremal simply connected subdomains of Riemann surfaces and metric graphs/tropical curves	Given a compact Riemann surface S with genus g > 0 and with a marked point p, there is a canonical simply connected subdomain of S containing p, that e.g. can be characterized in terms of its Green’s function. This follows from a classical result of Strebel, which itself built on earlier work of Teichmüller and Jenkins. This extremal subdomain is interesting because it shows how S can be constructed from the unit disc by gluing parts of the unit circle together in a simple way, and it can thus be used to study the associated moduli space. Some time ago Fredrik Viklund and I proposed a way to find the gluing parameters for a given Riemann surface, up to some small error, by using a probabilistic model that we call interface erosion (it is closely related to another probabilistic model called competitive erosion, introduced by Propp in the early 2000s). Trying to prove that this actually works led us (now also joined by Levi Hanschmid-Sibitz) to consider some related questions for metric graphs/tropical curves, and we then managed to prove in that setting the result analogous to that of Strebel’s. In my very non-technical talk, I will try to explain parts of this story, and time permitting maybe also discuss what might happen in higher dimensions.
A. Raich University of Arkansas, USA	A Generalized Bochner–Martinelli Plemelj Jump Formula and Integral Kernel Methods to Solve ∂̄_M	Solving the tangential Cauchy-Riemann equations on higher codimension CR manifolds is a relatively unexplored topic. In this talk, I have two goals. The first is to discuss a Bochner–Martinelli Plemelj jump formula for smooth, embedded, and generic CR manifolds. The jump formula reduces to the standard jump formula in the case the CR manifold is a hypersurface. The second goal is to use the new jump formula and integral kernel techniques to solve the tangential Cauchy-Riemann equations on smooth, embedded q-convex CR manifolds. This work is joint with Al Boggess of Arizona State University.
N. Savale Trinity College, Ireland	To be announced.	To be announced.
C. Schnell State University of New York at Stony Brook, USA	Meromorphic groups and their actions on compact Kähler spaces	I am going to review Fujiki’s theory of meromorphic groups, and then apply it to prove a new “freeness theorem” for the cohomology of compact Kähler spaces with a meromorphic action by a meromorphic group. This is joint work with Mark de Cataldo and Yoonjoo Kim.
Aeryeong Seo Kyungpook National University, Korea	On the Pseudoconvexity and Completeness of Holomorphic Fiber Bundles	This talk examines the intermediate pseudoconvexity and weakly 1-completeness of holomorphic fiber bundles. For holomorphic fiber bundles over compact Kähler manifolds with fibers given by bounded symmetric domains, we discuss the conditions under which these spaces are weakly 1-complete, particularly when the holonomy representation is reductive. Building on this, we analyze locally trivial holomorphic ℬⁿ-bundles over compact Riemann surfaces of genus greater than one. By assuming the existence of a harmonic section with a point of maximal rank, we show that these bundles are 1-convex, while their complements in the associated ℂPⁿ-bundles exhibit n-convexity—a result derived through an analysis of the leafwise positivity of the normal bundle of the foliation induced on the boundary. This is joint work with Masanori Adachi and Seungjae Lee.
Y. T. Siu Harvard University, USA	Some Problems in Several Complex Variables	We will discuss the following three problems in several complex variables. (1) Finite generation problem to determine the contexts for the finite generation of the canonical ring to hold. (2) Global non-deformability problem for compact Hermitian symmetric manifolds without assuming deformation to be Kähler. (3) Differential relations derived from failure of certain conditions. For example, differential relations satisfied by Kohn’s subelliptic multipliers and Donaldson’s holomorphicity of destabilizing subsheaves for nonexistence of Hermitian-Einstein metric for vector bundles.
L. Stolovitch Université Côte d’Azur, France	Recent results on equivalence of neighborhoods of embedded compact complex manifolds and higher codimension foliations	In this talk, we will survey recent work done in collaboration with Xianghong Gong, Takayuki Koike, and Xiaojun Wu. We consider an embedded n-dimensional compact complex manifold C in (n + d)-dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods when the normal bundle is flat. 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. We also give conditions ensuring the existence of a holomorphic foliation in a neighborhood of C in M having C as a compact leaf, extending Ueda’s theory to the higher dimension and codimension cases. Both problems appear as a kind of “linearization problem” involving appropriate notions of resonances and a small-divisor condition.
E. Straube Texas A&M University, USA	Diederich–Fornæss index and regularity of the complex Green operator	Let Ω be a smooth bounded pseudoconvex domain in ℂⁿ. We show that if Ω satisfies a certain comparability condition on the Levi eigenvalues of its boundary, then Diederich–Fornæss index one implies regularity of the complex Green operators and the associated canonical operators on bΩ. The particular comparability condition needed is relevant only in dimension n ≥ 3, it never holds on domains in ℂ². (This situation is rather in contrast to that for the analogous results for the ∂̄-Neumann operators: the relevant comparability condition there is relevant only in dimension n ≥ 3 because it always holds on domains in ℂ².) We show that nevertheless, DF-index one implies regularity of the canonical operators on the boundary for domains in ℂ².
D. V. Vu Cologne University, Germany	Uniform estimates for singular Kähler metrics in big cohomology classes	We generalize Guo-Phong-Song-Sturm’s uniform diameter estimates and local non-vanishing of volumes for Kähler metrics to the case of big cohomology classes. Main ingredients of the proof are a uniform diameter estimate for a family of smooth Kähler metrics only involving an integrability condition and stability properties of complex Monge–Ampère equations with prescribed singularities. This is a joint work with Duc-Bao Nguyen (NUS, Singapore).
J. Wang Academia Sinica, Taiwan	On Campana’s conjecture for covering of toric varieties	In joint work with Ji Guo, Khoa D. Nguyen, and Chia-Liang Sun, we extend results of Corvaja–Zannier, Turchet, and Capuano–Turchet to establish new cases of the Lang–Vojta Conjecture for varieties of log general type arising as ramified covers of algebraic tori over function fields. The main technical ingredient is a function field analogue of Vojta’s generalized abc conjecture for algebraic tori with explicitly computable exceptional sets. This is proved via a greatest common divisor theorem for multivariable polynomials evaluated at S-unit points. I will then explain how these methods can be further developed to prove cases of Campana’s orbifold conjecture for toric varieties with sufficiently large boundary multiplicities, both over function fields and in the complex analytic setting. These results are based on joint work with Carlo Gasbarri and Ji Guo in the function field case, and with Min Ru in the complex analytic setting.
M. Xiao University of California at San Diego, USA	Bergman Metrics of Constant Holomorphic Sectional Curvature	A classical theorem of Lu states that a bounded domain with a complete Bergman metric of constant holomorphic sectional curvature must be biholomorphic to the unit ball. In this talk, we discuss some recent progress in the more general setting of complex manifolds whose Bergman metric is not necessarily complete but has constant holomorphic sectional curvature.
J. Xie Beijing University, China	To be announced.	To be announced.
R. Zhang UCSD, USA	To be announced.	To be announced.
X. Zhou Chinese Academy of Sciences, China	To be announced.	To be announced.