Core Concepts

On a non-spin spin$^c$ manifold, the existence of a generalized positive scalar curvature metric is equivalent to the vanishing of a certain index, and there's a trichotomy theorem for generalized scalar curvature analogous to the Kazdan-Warner theorem for scalar curvature.

Abstract

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arxiv.org

Botvinnik, B., & Rosenberg, J. (2024). Generalized positive scalar curvature on spinc manifolds. arXiv preprint arXiv:2404.00703.

This paper investigates the concept of generalized positive scalar curvature on spin$^c$ manifolds, particularly its relationship with the index of the spin$^c$ Dirac operator and its topological implications. The authors aim to establish a trichotomy theorem for generalized scalar curvature, mirroring the Kazdan-Warner theorem for scalar curvature.

Key Insights Distilled From

by Boris Botvin... at **arxiv.org** 10-10-2024

Deeper Inquiries

The generalized positive scalar curvature, represented by the condition $R_{gen}(g,A) > 0$, establishes significant connections with other geometric invariants of spin$^c$ manifolds, offering insights into the manifold's underlying structure and topology. Here's a breakdown of these relationships:
Index of the spin$^c$ Dirac Operator: As emphasized in the paper, the positivity of $R_{gen}(g,A)$ directly implies the vanishing of the index ($\alpha_c(M,L)$) of the spin$^c$ Dirac operator $D_L$. This connection stems from the Lichnerowicz-Schrӧdinger formula, which links the square of $D_L$ to $R_{gen}(g,A)$. Consequently, the existence of a metric and connection pair $(g,A)$ satisfying $R_{gen}(g,A) > 0$ imposes topological constraints on the manifold.
Eigenvalues of the Modified Conformal Laplacian: The paper introduces the modified conformal Laplacian $L(g,A)$, whose first eigenvalue $\mu_1(L(g,A))$ provides a lower bound for the eigenvalues of $D_L^2$. The condition $R_{gen}(g,A) > 0$ ensures $\mu_1(L(g,A)) > 0$, implying the invertibility of $D_L$. This relationship highlights the interplay between the generalized scalar curvature and the spectral properties of operators on spin$^c$ manifolds.
Existence of Parallel Spinors: Theorem 1.5 in the paper reveals a fascinating connection between the generalized scalar curvature and the existence of parallel spinors. If a non-spin spin$^c$ manifold $(M,L)$ with non-vanishing $\alpha_c(M,L)$ admits a metric $g$ and connection $A$ such that $\mu_1(L(g,A)) = 0$, then $(M,L)$ with a conformally related metric allows a parallel spinor. This connection suggests that the generalized scalar curvature can provide information about the existence of special geometric structures on spin$^c$ manifolds.
Kӓhler Geometry: The paper further explores the case when $\mu_1(L(g,A)) = 0$. Theorem 1.5 states that such manifolds, under the same assumptions as above, are conformally equivalent to Riemannian products of Kӓhler manifolds and simply connected spin manifolds with parallel spinors. This result reveals a deep connection between generalized scalar curvature and Kӓhler geometry, particularly when the manifold does not decompose into a product, implying it's conformally Kӓhler.
In summary, the generalized positive scalar curvature on spin$^c$ manifolds intertwines with various geometric invariants, including the index of the spin$^c$ Dirac operator, eigenvalues of the modified conformal Laplacian, existence of parallel spinors, and Kӓhler structures. These connections provide a rich framework for investigating the topology and geometry of spin$^c$ manifolds.

Yes, it's conceivable that alternative definitions of generalized scalar curvature on spin$^c$ manifolds could lead to different trichotomy results. The specific trichotomy theorem presented in the paper (Theorem 1.4) arises from the chosen definition of $R_{gen}(g,A) = R_g - 2|\Omega|_{op}$. Modifying this definition could alter the classes of manifolds defined by the generalized scalar curvature and lead to different geometric and topological implications.
Here are some potential avenues for exploring alternative definitions:
Different Norms on Curvature: The current definition uses the operator norm ($|\Omega|_{op}$) of the curvature 2-form $\Omega$. Exploring alternative norms, such as the $L^p$-norms for $p \neq \infty$, could lead to different curvature conditions and potentially different trichotomy results. The geometric and analytic properties associated with these norms might highlight different aspects of the spin$^c$ structure.
Incorporating Other Characteristic Classes: The current definition primarily focuses on the first Chern class $c_1(L)$ of the line bundle $L$. One could explore incorporating higher Chern classes or other characteristic classes of the spin$^c$ structure into the definition of generalized scalar curvature. This could lead to finer classifications of spin$^c$ manifolds and potentially reveal new geometric insights.
Twisting by Other Bundles: The notion of generalized scalar curvature in the paper arises from twisting the spin Dirac operator by a line bundle $L$. One could consider twisting by higher-rank vector bundles or even more general principal bundles. This generalization could lead to a richer theory of generalized scalar curvature with connections to different geometric structures.
It's important to note that the existence of a meaningful trichotomy theorem relies on the interplay between the chosen definition of generalized scalar curvature and the analytic and geometric properties of the spin$^c$ manifold. Exploring alternative definitions would require careful consideration of these factors to establish meaningful and insightful results.

The non-triviality of the spin$^c$ index difference homomorphism, if proven true as conjectured in the paper, could have profound implications for the study of exotic spheres and their smooth structures. Here's why:
Exotic Spheres and Positive Scalar Curvature: Exotic spheres are smooth manifolds that are homeomorphic but not diffeomorphic to the standard sphere. The existence of exotic spheres is intimately tied to the existence of metrics of positive scalar curvature. The spin$^c$ index difference homomorphism provides a tool to study the space of metrics with positive generalized scalar curvature on spin$^c$ manifolds.
Detecting Smooth Structures: The conjecture suggests that the spin$^c$ index difference homomorphism can distinguish between different smooth structures on a given topological spin$^c$ manifold. If two exotic spheres admit different spin$^c$ structures with non-trivial index difference, it would imply that they cannot be equipped with the same smooth structure.
Obstructions to Smooth Structures: The non-triviality of the index difference could lead to new obstructions to the existence of certain smooth structures on exotic spheres. If a particular spin$^c$ structure on an exotic sphere leads to a non-trivial index difference, it might obstruct the existence of a smooth structure compatible with that spin$^c$ structure.
Connections to Surgery Theory: The proof of the conjecture would likely involve techniques from surgery theory, which provides a way to modify the smooth structure of a manifold in a controlled manner. The index difference homomorphism could provide new insights into how surgeries affect the generalized scalar curvature and the spin$^c$ structure of a manifold.
In essence, the non-triviality of the spin$^c$ index difference homomorphism would provide a powerful tool for investigating the relationship between the topology and smooth structures of exotic spheres. It could potentially lead to new methods for detecting and classifying exotic spheres, as well as for understanding the constraints on the existence of certain smooth structures.

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