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arXiv · 2411.10348

Integral-integral affine geometry, geometric quantization, and Riemann-Roch

Mark Hamilton·Yael Karshon·Takahiko Yoshida
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Abstract

We give a simple proof that, for a pre-quantized compact symplectic manifold with a Lagrangian torus fibration, its Riemann-Roch number coincides with its number of Bohr-Sommerfeld fibres. This can be viewed as an instance of the "independence of polarization" phenomenon of geometric quantization. The base space for such a fibration acquires a so-called integral-integral affine structure. The proof uses the following simple fact, whose proof is trickier than we expected: on a compact integral-integral affine manifold, the total volume is equal to the number of integer points.

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Mark Hamilton, Yael Karshon, Takahiko Yoshida. 2024-11-15. Integral-integral affine geometry, geometric quantization, and Riemann-Roch. https://arxiv.org/abs/2411.10348

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