The Thom-Sebastiani theorem for the Euler characteristic of cyclic L-infinity algebras

Abstract

Let $L$ be a cyclic $L_{\infty }$-algebra of dimension $3$ with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic $L_{\infty }$-algebra structure on the cohomology $H^{\ast }(L)$. The inner product plus the higher products of the cyclic $L_{\infty }$-algebra defines a superpotential function $f$ on $H^1(L)$. We associate with an analytic Milnor fiber for the formal function $f$ and define the Euler characteristic of $L$ is to be the Euler characteristic of the étale cohomology of the analytic Milnor fiber. In this paper we prove a Thom-Sebastiani type formula for the Euler characteristic of cyclic $L_{\infty }$-algebras. As applications we prove the Joyce-Song formulas about the Behrend function identities for semi-Schur objects in the derived category of coherent sheaves over Calabi-Yau threefolds. A motivic Thom-Sebastiani type formula and a conjectural motivic Joyce-Song formulas for the motivic Milnor fiber of cyclic $L_{\infty }$-algebras are also discussed.

33 pages, a reference of Berkovich updated, Proposition 5.5 on the relation between the Behrend function and the Euler characteristic of the analytic Milnor fiber clarified, comments are very welcome

References in corpus (7)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Desingularization of quasi-excellent schemes in characteristic zero
• A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications
• A trace formula for rigid varieties, and motivic Weil generating series for formal schemes
• Generalized Donaldson-Thomas theory over fields K $\ne$ C
• On motivic Joyce-Song formula for the Behrend function identities
• The motivic Thom-Sebastiani theorem for regular and formal functions

Cited by in corpus (5)

• Hall algebras in the derived category and higher rank DT invariants
• On motivic Joyce-Song formula for the Behrend function identities
• The moduli space of stable coherent sheaves via non-archimedean geometry
• Note on the motivic DT/PT correspondence and the motivic Flop formula
• Donaldson-Thomas invariants of Calabi-Yau orbifolds under flops