Open research questions in Geometry and complex manifolds

What the literature leaves open

• The authors note that there is a lack of knowledge about the existence of astheno-Kähler metrics on these manifolds.

  Aeppli-Bott-Chern Massey products on non-Kähler solvmanifolds · 2026 · DOI

• The gap is the lack of understanding of Bergman spaces on singular Riemann surfaces. The gap is the need for a generalization of Wiegerinck's theorem to the case of certain singular metrics.

  Bergman spaces on algebraic curves · 2026 · DOI

• It remains to be shown that: (α)1 (under the condition (i)) there exists a proper flag B(q) ∈ F(B; V ∗) relative to the identity map idY (namely, SZ ∗ (B(q); B) has pure codimension qβq in Y).

  On the geometric Segre and Chern classes of vector bundles and generalized Schubert cycles · 2026 · DOI

• The stability of the pluriclosed flow and generalized Ricci flow was not well understood. The paper aims to fill this gap by establishing the local stability of pluriclosed steady solitons.

  Dynamical stability of pluriclosed and generalized Ricci solitons · 2026 · DOI

• There is a gap in the existing literature regarding the stability of Kähler structures under deformations. There is a need to extend the results of prior work to more general settings.

  Degenerations and Stability of Kähler Structures on Calabi--Yau Manifolds · 2026

• The authors note that prior work has not characterized the integrability of B_n-generalized almost complex and pseudo-Hermitian structures in terms of generalized connections.

  Integrability of generalized structures on odd exact Courant algebroids using generalized connections · 2026

• The paper also identifies the need for a sheaf-theoretical proof of the Chantraine-Murphy theorem as a challenge.

  Derived sheaves in locally conformally symplectic geometry · 2026

• The paper assumes that the manifolds are compact and Kähler. The results are limited to manifolds of algebraic dimension zero. The paper does not provide a complete classification of all compact Kähler manifolds.

  Bogomolov Decomposition and Compact K{ä}hler Manifolds of Algebraic Dimension Zero · 2026

• Future research can focus on extending the results to manifolds of higher algebraic dimension. Future research can focus on developing new techniques to analyze the properties of compact Kähler manifolds. Future research can focus on applying the results to other areas of mathematics and physics.

  Bogomolov Decomposition and Compact K{ä}hler Manifolds of Algebraic Dimension Zero · 2026

• The study of the Hodge-theoretic properties of complex singularities is an active area of research. The paper identifies a gap in the understanding of the properties of the Deligne-Du Bois complexes.

  Higher singularities for hypersurfaces · 2026

• The paper suggests future research directions, including the further study of connections between Kobayashi geometry and the Dirichlet problem for the complex Monge-Ampère equation. The authors propose exploring the properties of domains with nice boundary geometry and their relationship to the complex Monge-Ampère equation.

  On some connections between Kobayashi geometry and pluripotential theory · 2026 · DOI

• The authors highlight the need for further research in this area and provide a direction for future studies.

  On some connections between Kobayashi geometry and pluripotential theory · 2026 · DOI

• Future research could explore the applications of the paper's results to other areas of complex geometry. The authors suggest studying the degeneracy sets of random sections in more detail.

  Tian’s theorem for Grassmannian embeddings and degeneracy sets of random sections · 2026 · DOI

• There is a need to generalize Tian's theorem for Chern forms of any degree. The paper identifies a gap in the understanding of the Kodaira map and its relation to meromorphic transforms.

  Tian’s theorem for Grassmannian embeddings and degeneracy sets of random sections · 2026 · DOI

• The lack of understanding of the behavior of the zeros of random sections in the context of geometric quantization. The lack of a semiclassical approach to the inverse problem. The lack of a computation of the expectation of the distribution of zeros of random sections.

  Berezin–Toeplitz operators, Kodaira maps, and random sections · 2026 · DOI

• The lack of a complete answer to the conjecture that if two germs of holomorphic functions are topologically equivalent, then the Milnor fibres of their initial forms are homotopy equivalent. The need for a better understanding of the relationship between the Milnor fibres of initial forms of holomorphic functions and their topological equivalence.

  On the Milnor fibres of initial forms of topologically equivalent holomorphic functions · 2026 · DOI

• The gap in prior work is the lack of a computation of a₂ for EIX. The gap in prior work is the lack of a comparison of the EIX value with the round sphere S₁₁₂ value.

  The second Seeley–DeWitt coefficient of the scalar Laplacian on the E₈ Wolf space EIX · 2026 · DOI

• The paper also identifies a gap in the study of solutions that are radially symmetric in one variable.

  Non-quadratic solutions to the Monge–Ampère equation · 2026 · DOI

• Prior work has shown existence results for projective varieties, but with additional hypotheses. The paper fills the gap by showing that no additional hypotheses are necessary for rational homogeneous varieties.

  Deformed Hermitian Yang–Mills equation on rational homogeneous varieties · 2026 · DOI

• The behavior of warped QAC Calabi-Yau metrics on affine quadrics is not well understood. A conjecture by Yang Li is unverified.

  A gluing theorem for collapsing warped-QAC Calabi–Yau manifolds · 2026 · DOI

• The symmetry groups of Monge-Ampère equation and affine maximal type equation were not classified before. The Lie’s theory was not developed to fully nonlinear PDEs before.

  Complete classification of the symmetry groups of Monge-Ampère equation and affine maximal type equation · 2026 · DOI

• Further study of the geometry and topology of complex manifolds. Application of the results to string theory and mirror symmetry. Extension of the results to more general settings, including non-Kähler manifolds.

  Degenerations and Stability of Kähler Structures on Calabi--Yau Manifolds · 2026

• In this final section we mention some open questions open questions for further…

  Solutions to Monge-Ampère Equations with Low Codimensional singularities · 2026

• The study of the behavior of the zeros of random sections in more general contexts. The development of a numerical implementation of the results. The application of the results to the study of quantum chaos and random matrix theory.

  Berezin–Toeplitz operators, Kodaira maps, and random sections · 2026 · DOI

• The construction of non-Abelian Euclidean Yang-Mills theories is a major open problem - Significant gaps remain in our understanding of lattice Yang-Mills…

  On the Leading Order Term of the Lattice Yang-Mills Free Energy · 2026 · DOI

Most-cited papers in Geometry and complex manifolds

• Unbundling the local loop · European Economic Review · 2003 · 81 citations
• Three versions of the bundle theory · Philosophical Studies · 1985 · 59 citations
• A Maslov-type index theory for symplectic paths · Topological Methods in Nonlinear Analysis · 1997 · 47 citations
• Dimorphic Structure and the Parasocial Element · Journal of Near Eastern Studies · 1977 · 33 citations
• An extreme case of cherubism · British Journal of Oral and Maxillofacial Surgery · 2002 · 31 citations
• On the concept of orientability for Fredholm maps between real Banach manifolds · Topological Methods in Nonlinear Analysis · 2000 · 29 citations
• Retraction: Tangent structures and analytical mechanics; pp. 98–103 · Proceedings of the Estonian Academy of Sciences · 2011 · 24 citations
• Properties of the nearly kähler S3×S3 · Publications de l Institut Mathematique · 2018 · 20 citations
• The coincidence Reidemeister classes of maps on nilmanifolds · Topological Methods in Nonlinear Analysis · 1998 · 18 citations
• A cohomology complex for manifolds with boundary · Topological Methods in Nonlinear Analysis · 1995 · 16 citations

Most recent work

• Stability of Gaussian Poincaré inequalities and Heisenberg uncertainty principle with monomial weights · Mathematische Zeitschrift · 2026
• Harmonic metrics and semi-simpleness · Advances in Mathematics · 2026
• Hyperuniformity of random measures on Euclidean and hyperbolic spaces · Mathematische Annalen · 2026
• Z-critical equations for holomorphic vector bundles on Kähler surfaces · Mathematische Annalen · 2026
• Kähler metrics of the negative holomorphic (bi)sectional curvature on a compact relative Kähler fibration · Bulletin of the London Mathematical Society · 2026
• Quasi‐positive mixed curvature, vanishing theorems, and rational connectedness · Bulletin of the London Mathematical Society · 2026
• Homogeneous structures of 3-dimensional Lie groups · Monatshefte für Mathematik · 2026
• Rigidity of Kobayashi isometries of a class of 2-dimensional Lempert manifolds · Archiv der Mathematik · 2026
• Log del Pezzo C∗-Surfaces, Kähler–Einstein Metrics, Kähler–Ricci Solitons and Sasaki–Einstein Metrics · Michigan Mathematical Journal · 2026
• Almost toric fibrations on symplectic blowups · Proceedings of the American Mathematical Society · 2026

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