The BGMN conjecture via stable pairs

Jenia Tevelev; Sebastián Torres

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arXiv · 2108.11951

The BGMN conjecture via stable pairs

Jenia Tevelev·Sebastián Torres

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Abstract

Let $C$ be a smooth projective curve of genus $g\ge2$ and let $N$ be the moduli space of stable rank $2$ vector bundles on $C$ of odd degree. We construct a semi-orthogonal decomposition of the bounded derived category of $N$ conjectured by Narasimhan and by Belmans, Galkin and Mukhopadhyay. It has two blocks for each $i$-th symmetric power of $C$ for $i=0,\ldots,g-2$ and one block for the $(g-1)$-st symmetric power. We conjecture that the subcategory generated by our blocks has a trivial semi-orthogonal complement, proving the full BGMN conjecture. Our proof is based on an analysis of wall-crossing between moduli spaces of stable pairs, combining classical vector bundles techniques with the method of windows.

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Jenia Tevelev, Sebastián Torres. 2021-08-26. The BGMN conjecture via stable pairs. https://arxiv.org/abs/2108.11951

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