Yoshikawa in [Invent. Math. 156 (2004), 53–117] introduces a holomorphic torsion invariant of K3$K3$ surfaces with involution. In this paper we completely determine its structure as an automorphic function on the moduli space of such K3$K3$ surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds’ conjecture.

We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family X \longrightarrow B$X \longrightarrow B$ with singular fibre over b_0\in B$b_0\in B$ yields a family \mathscr {M}(X/B,\beta ) \longrightarrow B$\mathscr {M}(X/B,\beta ) \longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over b_0$b_0$ in terms of rigid tropical maps to the tropicalization of X/B$X/B$. This generalizes one aspect of known results in the case that the fibre X_{b_0}$X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.

Étant donné un groupe réductif G$G$ sur une extension de degré fini de \mathbb {Q}_p$\mathbb {Q}_p$ on classifie les G$G$-fibrés sur la courbe introduite dans Fargues and Fontaine [Courbes et fibrés vectoriels en théorie de Hodgep$p$-adique, Astérisque 406 (2018)]. Le résultat est interprété en termes de l'ensemble B(G)$B(G)$ de Kottwitz. On calcule également la cohomologie étale de la courbe à coefficients de torsion en lien avec la théorie du corps de classe local.

We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a polydifferential endofunctor {\mathcal {D}}${\mathcal {D}}$ in the category of augmented props with the property that for any representation of a prop {\mathcal {P}}${\mathcal {P}}$ in a vector space V$V$ the associated prop {\mathcal {D}}{\mathcal {P}}${\mathcal {D}}{\mathcal {P}}$ admits an induced representation on the graded commutative algebra \odot ^\bullet V$\odot ^\bullet V$ given in terms of polydifferential operators. Applying this functor to the minimal resolution \widehat {\mathcal {L}\textit{ieb}}_\infty$\widehat {\mathcal {L}\textit{ieb}}_\infty$ of the genus completed prop \widehat {\mathcal {L}\textit{ieb}}$\widehat {\mathcal {L}\textit{ieb}}$ of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in one-to-one correspondence with morphisms of dg props

F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty $$F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty$$

\mathcal {A}\textit{ssb}_\infty$\mathcal {A}\textit{ssb}_\infty$

\mathfrak{A}$\mathfrak{A}$

{\mathcal {L}} ie_\infty${\mathcal {L}} ie_\infty$

{\mathcal {L}} ie_\infty${\mathcal {L}} ie_\infty$

\mathsf {Def}({\mathcal {A}} ss{\mathcal {B}}_\infty \rightarrow {\mathcal {E}} nd_{\odot ^\bullet V})$\mathsf {Def}({\mathcal {A}} ss{\mathcal {B}}_\infty \rightarrow {\mathcal {E}} nd_{\odot ^\bullet V})$

\mathsf {Def}({\mathcal {L}} ie{\mathcal {B}}\rightarrow {\mathcal {E}} nd_V)$\mathsf {Def}({\mathcal {L}} ie{\mathcal {B}}\rightarrow {\mathcal {E}} nd_V)$

\odot V$\odot V$

V$V$

\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat {\mathcal {L}\textit{ieb}}_\infty )$\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat {\mathcal {L}\textit{ieb}}_\infty )$

\{F_\mathfrak{A}\}$\{F_\mathfrak{A}\}$

GRT=GRT_1\rtimes {\mathbb {K}}^*$GRT=GRT_1\rtimes {\mathbb {K}}^*$

\{\mathfrak{A}\}$\{\mathfrak{A}\}$

We introduce the notion of a Y$Y$-pattern with coefficients and its geometric counterpart: an \mathcal {X}$\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed \mathcal {X}$\mathcal {X}$-cluster variety \widehat {\mathcal {X} }$\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed \mathcal {X}$\mathcal {X}$-varieties encoded by \operatorname {Star}(\tau )$\operatorname {Star}(\tau )$ for each cone \tau$\tau$ of the \mathbf {g}$\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to \mathcal {A}_{\mathrm {prin}}$\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to \mathcal {A} _t$\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of \operatorname {Gr}_2(\mathbb {C} ^{5})$\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.