On motivic vanishing cycles of critical loci

Abstract

Let U U be a smooth scheme over an algebraically closed field K \mathbb K of characteristic zero and let f : U → A 1 f:U\rightarrow \mathbb A^1 be a regular function, and write X = Crit ( f ) X=\textrm {Crit}(f) , as a closed K \mathbb K -subscheme of U U . The motivic vanishing cycle M F U , f mot,\phi MF_{U,f}^\textrm {mot,\phi } is an element of the μ ^ \hat \mu -equivariant motivic Grothendieck ring M X μ ^ \mathcal M^{\hat \mu }_X , defined by Denef and Loeser, and Looijenga, and used in Kontsevich and Soibelman’s theory of motivic Donaldson–Thomas invariants. We prove three main results: (a) M F U , f mot,\phi MF_{U,f}^\textrm {mot,\phi } depends only on the third-order thickenings U ( 3 ) , f ( 3 ) U^{(3)},f^{(3)} of U , f U,f . (b) If V V is another smooth K \mathbb K -scheme, g : V → A 1 g:V\rightarrow \mathbb A^1 is regular, Y = Crit ( g ) Y=\textrm {Crit}(g) , and Φ : U → V \Phi :U\rightarrow V is an embedding with f = g ∘ Φ f=g\circ \Phi and Φ | X : X → Y \Phi \vert _X:X\rightarrow Y an isomorphism, then Φ | X ∗ ( M F V , g mot, \phi ) = M F U , f mot, \phi ⊙ Υ ( P Φ ) \Phi \vert _X^*\bigl (MF^\textrm {mot, \phi }_{V,g}\bigr )=MF^\textrm {mot, \phi }_{U,f}\odot \Upsilon (P_\Phi ) in a certain quotient ring M ¯ X μ ^ \,\,\overline {\!\!\mathcal M\!}\,^{\hat \mu }_X of M X μ ^ \mathcal M^{\hat \mu }_X , where P Φ → X P_\Phi \rightarrow X is a principal Z 2 \mathbb Z_2 -bundle associated to Φ \Phi and Υ : { \Upsilon :\{ principal Z 2 \mathbb Z_2 -bundles on X } → M ¯ X μ ^ X\}\rightarrow \,\,\overline {\!\!\mathcal M\!}\,^{\hat \mu }_X a natural morphism. (c) If ( X , s ) (X,s) is an oriented algebraic d-critical locus in the sense of Joyce, there is a natural motive M F X , s ∈ M ¯ X μ ^ MF_{X,s}\in \,\,\overline {\!\!\mathcal M\!}\,^{\hat \mu }_X , such that if ( X , s ) (X,s) is locally modelled on Crit ( f : U → A 1 ) \textrm {Crit}(f:U\rightarrow \mathbb A^1) , then M F X , s MF_{X,s} is locally modelled on M F U , f mot,\phi MF_{U,f}^\textrm {mot,\phi } . Using results of Pantev, Toën, Vaquié, and Vezzosi, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi–Yau 3-fold equipped with “orientation data”, as required in Kontsevich and Soibelman’s motivic Donaldson–Thomas theory, and on intersections L ∩ M L\cap M of oriented Lagrangians L , M L,M in an algebraic symplectic manifold ( S , ω ) (S,\omega ) .