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2012•Journal of Differential GeometryOpen access

Stable pairs on local $K3$ surfaces

Yukinobu Toda

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Abstract

We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local $K3$ surfaces to counting invariants of semistable sheaves on them. Our formula generalizes Kawai- Yoshioka’s formula for stable pairs with irreducible curve classes to arbitrary curve classes. We also propose a conjectural multiple cover formula of sheaf counting invariants which, combined with our main result, leads to an Euler characteristic version of Katz- Klemm-Vafa conjecture for stable pairs.

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What this paper is about

We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local $K3$ surfaces to counting invariants of semistable sheaves on them. Our formula generalizes Kawai- Yoshioka’s formula for stable pairs with irreducible curve classes to arbitrary curve classes. We also propose a conjectural multiple cover formula of sheaf counting invariants which, combined with our main result, leads to an Euler characteristic version of Katz- Klemm-Vafa conjecture for stable pairs.

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Available abstract

We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local $K3$ surfaces to counting invariants of semistable sheaves on them. Our formula generalizes Kawai- Yoshioka’s formula for stable pairs with irreducible curve classes to arbitrary curve classes. We also propose a conjectural multiple cover formula of sheaf counting invariants which, combined with our main result, leads to an Euler characteristic version of Katz- Klemm-Vafa conjecture for stable pairs.

Key concepts: Mathematics, Euler characteristic, Pure mathematics, Conjecture, Sheaf, Moduli space, Moduli, Euler's formula

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