Bott Chern CohomologyEdit

Bott-Chern cohomology is a bi-graded invariant of complex manifolds that sits at the crossroads of several classical cohomology theories. Born out of the work of Bott and Chern, it captures geometric information that neither de Rham nor Dolbeault cohomology alone fully express, especially on non-Kähler manifolds. It uses the two canonical differential operators on a complex manifold, ∂ and ∂̄, to form a quotient that reflects the part of a form that is simultaneously closed with respect to both operators but not expressible as a ∂∂̄-exact form. Concretely, for a complex manifold X and integers p, q, the Bott-Chern cohomology group is defined as H^{p,q}_{BC}(X) = {α ∈ Ω^{p,q}(X) | ∂α = 0, ∂̄α = 0} / {∂∂̄β | β ∈ Ω^{p-1,q-1}(X)}. This construction yields a finite-dimensional vector space when X is compact, and its bi-grading reflects the complex structure more finely than the de Rham theory alone.

Introductory context and scope Bott-Chern cohomology fits into a family of invariants designed to detect complex-geometric structure beyond what de Rham cohomology (which is purely topological) and Dolbeault cohomology (which depends on the complex structure but discards some mixed information) can reveal. It is biholomorphically invariant, meaning that biholomorphic maps of complex manifolds induce isomorphisms on the Bott-Chern groups. This makes H^{p,q}_{BC} a natural tool in the study of complex structures and their deformations. In the larger ecosystem of complex geometry, Bott-Chern cohomology interacts with other invariants like Aeppli cohomology and the Frölicher spectral sequence, providing a complementary lens on the geometry of X.

Definition and basic properties - What it measures: H^{p,q}{BC}(X) records classes of (p,q)-forms that are closed under both ∂ and ∂̄ but not necessarily writable as ∂∂̄-exact. The presence of nontrivial BC classes signals subtleties in the complex structure that go beyond what Dolbeault cohomology alone detects. - Finite-dimensionality: On a compact complex manifold, each H^{p,q}{BC}(X) is finite-dimensional. This mirrors the well-behaved nature of the other standard cohomology theories while preserving sensitivity to the complex structure. - Functoriality: If f: X → Y is a biholomorphism, it induces isomorphisms on Bott-Chern cohomology. This underlines its status as an intrinsic invariant of the complex structure rather than a feature of a particular presentation.

Relation to other cohomologies - Dolbeault and de Rham connections: Bott-Chern cohomology sits in a network of natural maps to Dolbeault cohomology and de Rham cohomology. There are canonical maps H^{p,q}{BC}(X) → H^{p,q}{∂}(X) and to de Rham cohomology H^{k}_{dR}(X) after suitable summation of degrees. These maps become particularly informative when the ∂∂̄-lemma holds. - The ∂∂̄-lemma and Kähler manifolds: On Kähler manifolds (where the ∂∂̄-lemma holds), Bott-Chern cohomology aligns with other cohomologies in well-understood ways. In particular, the degeneracy of the Frölicher spectral sequence at E1 and the resulting decompositions imply strong equalities among BC, Dolbeault, and de Rham invariants. This reflects the classical harmony of Kähler geometry. - Non-Kähler geometry and the spectrum of invariants: On non-Kähler manifolds, Bott-Chern cohomology can reveal information that Dolbeault cohomology misses. For instance, the failure of the ∂∂̄-lemma is a hallmark of non-Kähler geometry, and BC cohomology often detects aspects of this failure in a way that complements Aeppli cohomology and the Frölicher spectral sequence.

Examples and computations - Kähler case: If X is Kähler, the ∂∂̄-lemma holds, and Bott-Chern cohomology matches expectations from the richer Hodge theory. In this setting, one often recovers familiar dimensions and decompositions, and the various cohomologies align in a coherent picture. - Non-Kähler examples: On non-Kähler surfaces and higher-dimensional manifolds, BC cohomology can be genuinely different from Dolbeault or de Rham data. Examples drawn from Hopf surfaces and other non-Kähler manifolds illustrate how BC groups can be nontrivial in degrees where Dolbeault data alone would suggest less structure, highlighting the utility of BC cohomology in distinguishing complex structures.

Applications and significance - Complex structure classification: Bott-Chern cohomology provides a refined invariant that helps in distinguishing non-equivalent complex structures on the same underlying smooth manifold, particularly when the ∂∂̄-lemma fails. - Hermitian geometry and metrics: The study of special Hermitian metrics (such as balanced or SKT metrics) interacts with Bott-Chern cohomology. Certain metric conditions impose constraints on BC cohomology, and BC classes can reflect geometric properties of the metric in question. - Theoretical physics: In string theory and related areas, complex geometry and flux compactifications make refined cohomological tools valuable. Bott-Chern cohomology contributes to the mathematical backbone that underpins certain compactification scenarios and the analysis of moduli spaces.

Controversies and debates - Utility versus over-specification: A live discussion in complex geometry concerns how much computational or conceptual payoff BC cohomology provides compared with the more classical invariants. Proponents argue that BC cohomology captures delicate features of non-Kähler manifolds that de Rham and Dolbeault data miss, which is crucial for a faithful account of complex structure in the non-Kähler realm. Critics sometimes point out that the finer data can be difficult to interpret globally and may vary in families, making it harder to use as a universal classifier. - Deformation behavior: The behavior of Bott-Chern groups under deformations of the complex structure is subtle. While some invariants are robust under small changes, BC cohomology can exhibit jumps in dimension in families of complex manifolds. This makes it a sensitive probe of geometric stability, but it also complicates attempts to use BC data as a rigid predictor of geometric features across moduli. - ∂∂̄-lemma and beyond: The contrast between the clean, topological flavor of de Rham cohomology and the complex-geometric flavor of BC cohomology highlights a broader methodological divide: should one emphasize invariants that behave well under broad classes of transformations (topological invariants), or invariants that faithfully reflect delicate analytic structure (complex-analytic invariants)? From a conservative mathematical perspective, the latter is essential for a true understanding of non-Kähler geometry, even if it comes with added complexity.

Woke criticisms and the intrinsic value of the mathematics - The most productive line here is to keep the discussion about mathematical content, not political framing. Some critics outside the discipline allege that certain lines of inquiry are overvalued or driven by fashion rather than substance. Supporters of Bott-Chern cohomology would argue that the theory is a natural outgrowth of classical complex analysis and differential geometry, grounded in well-defined operators and invariants. It offers a principled way to distinguish complex structures that are not captured by coarser invariants, and its development follows the tradition of rigorous, coordinate-free reasoning that has stood the test of time in mathematics.

See also - Bott-Chern cohomology - Dolbeault cohomology - de Rham cohomology - Aeppli cohomology - Frölicher spectral sequence - Kähler manifold - ∂∂̄-lemma - Non-Kähler manifold - Hermitian metric - Complex geometry - String theory