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Open research questions in Geometry and complex manifolds

61 unresolved questions extracted from the limitations and future-work sections of 332 Geometry and complex manifolds papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • The paper identifies the challenge of constructing and studying non-equivalent Lagrangian tori in dimension four. The paper identifies the challenge of proving that these tori remain distinct under embeddings into geometrically bounded symplectic four-manifolds.

    Semi-Local Exotic Lagrangian Tori in Dimension Four · 2026 · DOI

  • The complexity of the topology of cohomogeneity one manifolds. The need for new techniques to analyze the homotopy fiber of the inclusion of a principal orbit.

    Cohomogeneity one manifolds with singly generated rational cohomology II · 2026 · DOI

  • Future research could explore the applications of the new criterion for isotriviality. It could also investigate the relationship between deformation rigidity and other geometric properties of complex manifolds.

    Deformation rigidity for projective manifolds and isotriviality of smooth families · 2026 · DOI

  • The paper identifies a gap in the understanding of eternal classes in symplectic cohomology. The paper identifies a need for a spectral pseudo-metric on the universal cover of the group of contactomorphisms.

    Remarks on eternal classes in symplectic cohomology · 2026 · DOI

  • The paper identifies a gap in the understanding of compact Hermitian surfaces with constant mixed curvature. The paper also identifies a gap in the classification of compact locally conformal Kähler manifolds with constant mixed curvature.

    On mixed curvature for Hermitian manifolds · 2026 · DOI

  • The equation (1.2) is a much more complicated fourth order fully nonlinear equation. The Lie’s theory needs to be developed to fully nonlinear PDEs.

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

  • The equation is a fully nonlinear elliptic equation. The paper needs to describe the Lagrangian phase in terms of Lie theory.

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

  • Geometric structures have branched into multiple frameworks - The need for a unified understanding

    Symplectic Geometry and Curvature as Manifestations of the FRLT Pivot · 2026 · DOI

  • A general vanishing theorem is not available in this context. The prior work does not provide a unified approach to holomorphic Morse inequalities.

    Nakano-Griffiths inequality, holomorphic Morse inequalities, and extension theorems for q-concave domains · 2026 · DOI

  • The lack of a general criterion for isotriviality of smooth families is a gap in current research. The paper seeks to address this gap.

    Deformation rigidity for projective manifolds and isotriviality of smooth families · 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

  • The lack of understanding of the geometric Segre and Chern classes in the context of vector bundles and value distribution theory. The need for a geometric definition of the Segre and Chern forms.

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

  • 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 paper identifies a gap in understanding the relationships between Kobayashi geometry and the Dirichlet problem for the complex Monge-Ampère equation. 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 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 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 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 identifies a gap in the literature, namely the study of non-quadratic solutions to the Monge–Ampère equation. 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

  • The study of degenerate Monge-Ampère equations on non-compact complex manifolds. The application of the results of the paper to other areas of mathematics and physics.

    Monge-Ampère equations with prescribed singularities on compact Hermitian manifolds · 2026 · DOI

  • The lack of a sufficient and necessary condition for the existence of weak solutions to degenerate Monge-Ampère equations on compact Hermitian manifolds with prescribed singularities. The need for a new tool for the study of degenerate Monge-Ampère equations.

    Monge-Ampère equations with prescribed singularities on compact Hermitian manifolds · 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

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61 open questions have been extracted from the limitations and future-work passages of 332 Geometry and complex manifolds papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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