arXiv subjects

Nicolò Sibilla

Publications and source records attributed to Nicolò Sibilla.

At least 19 recordsLinked to original sources

arXiv2026

Higher Koszul duality and $n$-affineness

In this paper we study $\mathbb{E}_n$-Koszul duality in the topological setting, and the closely related question of \emph{$n$-affineness} for Betti stacks. The $\mathbb{E}_n$-Koszul dual of the algebra of chains on the $n$-fold loop space of a space $X$ is the algebra of cochains on $X$. It was expected that $\mathbb{E}_n$-Koszul duality should induce a kind of Morita equivalence between categories of iterated modules, but even the precise formulation of such a statement was not known. We give a rigorous formulation, and a proof, of such an $\mathbb{E}_n$-Koszul duality in the topological setting as an equivalence of $(\infty,n)$-categories. Conceptually, our main innovation is highlighting the coaffine stack defined by the \emph{cospectrum} of $\mathrm{C}^{\bullet}(X;\Bbbk)$ as a key geometric object supporting Koszul duality. Our result is new already in the classical case $n=1$, although it can be seen to recover well known formulations of $\mathbb{E}_1$-Koszul duality as a Morita equivalence of module categories (up to appropriate completions of the $t$-structures). We also investigate (higher) affineness properties of Betti stacks. We give a complete characterization of $n$-affine Betti stacks, in terms of the $0$-affineness of their iterated loop space. As a consequence, we prove that $n$-truncated Betti stacks are $n$-affine; and that $π_{n+1}(X)$ is an obstruction to $n$-affineness.

math.AG↗

arXiv2026

Equivariant Elliptic Cohomology and Mapping Stacks

We introduce a new cohomology theory for stacks called elliptic Hochschild homology, prove some fundamental properties and compute it in some classes of examples. We then introduce its periodic cyclic version and show that, over the complex numbers and for a quotient stack, this recovers Grojnowski's equivariant elliptic cohomology of the analytification.

math.AG↗

arXiv2026

An Orlov theorem for matrix factorizations with multiple factors

We prove a generalization of Orlov's theorem for matrix factorizations with $n$ steps. Let $X$ be a regular scheme, $W\colon X\to \mathbb{A}^1$ a flat morphism and $D:=W^{-1}(0)$ its central fiber. We construct an appropriate triangulated category of matrix factorizations with $n$-steps and show that it is equivalent to the singularity category of the root stack $\sqrt[n]{(X, D)}$. We also show that this category admits a semiorthogonal decomposition into $n-1$ copies of the usual (absolute derived) category of matrix factorizations with $2$ steps.

math.AG↗

arXiv2026

Hochschild-Kostant-Rosenberg isomorphism for derived Deligne-Mumford stacks

We prove a Hochschild--Konstant--Rosenberg (HKR) theorem for arbitrary derived Deligne--Mumford (DM) stacks, extending the results of Arinkin-Căldăraru-Hablicsek in the smooth, global quotient case, although with different methods. To formulate our result, we introduce the notion of orbifold inertia stack of a derived DM stack; this supplies a finely tuned derived enhancement of the classical inertia stack, which does not always coincide with the classical truncation of the free loop space. We show that, in characteristic 0, given a derived DM stack, the shifted tangent bundle of its orbifold inertia stack is equivalent to its free loop space. This yields a canonical HKR isomorphism of algebras between the Hochschild homology of a derived DM stack and the cohomology of differential forms on its orbifold inertia stack. Moreover, this isomorphism intertwines the natural circle action and the de Rham differential. Similarly, HKR theorems for derived DM stacks are established for Hochschild cohomology, cyclic homology, negative cyclic homology, and periodic cyclic homology. As applications, we provide a rich supply of computations of Hochschild homology and Hochschild cohomology for interesting derived DM stacks, such as weighted projective lines, root stacks, quotients by algebraic groups, and mapping stacks, among others.

math.AG↗

arXiv2025

Fourier--Mukai equivalences for formal groups and elliptic Hochschild homology

This paper establishes a unifying framework for various forms of twisted Hochschild homology by comparing two definitions of elliptic Hochschild homology: one introduced by Moulinos--Robalo--Toën and the other by Sibilla--Tomasini. Central to our approach is a new Fourier--Mukai duality for formal groups. We prove that when $\widehat{E}$ is the formal group associated to an elliptic curve $E$, the resulting $\widehat{E}$-Hochschild homology coincides with the mapping stack construction of Sibilla--Tomasini. This identification also recovers ordinary and Hodge Hochschild homology as degenerate limits corresponding to nodal and cuspidal cubics, respectively. Building on this, we introduce global versions of elliptic Hochschild homology over the moduli stacks of elliptic and cubic curves, which interpolate between these theories and suggest a universal form of TMF-Hochschild homology.

math.AG↗

arXiv2025

Speculations on higher Fukaya categories

We investigate a possible theory of higher Fukaya categories associated to $n$-shifted symplectic stacks, where $n \geq 0$. We consider two paradigmatic cases, the shifted cotangent stack of a smooth manifold and the coadjoint stack of a compact Lie group, drawing connections to the work of Teleman and 3D mirror symmetry. Our evidence includes some new results in $1$-shifted symplectic geometry.

math.SG↗

arXiv2025

Higher local systems and the categorified monodromy equivalence

We study local systems of $(\infty,n)$-categories on spaces. We prove that categorical local systems are captured by (higher) monodromy data: in particular, if $X$ is $(n+1)$-connected, then local systems of $(\infty,n)$-categories over $X$ can be described as $\mathbb{E}_{n+1}$-modules over the iterated loop space $Ω_{n+1}X$. This generalizes the classical monodromy equivalence presenting ordinary local systems as modules over the based loop spaces. Along the way we revisit from the perspective of $\infty$-categories Teleman's influential theory of topological group actions on categories, and we extend it to topological actions on $(\infty,n)$-categories. Finally, we show that the group of invertible objects in the category of local systems of $(\infty,n)$-categories over an $n$-connected space $X$ is isomorphic to the group of characters of $π_n(X)$. This should be thought of as a topological analogue of the higher Brauer group of the space $X$. We conclude the paper with applications of the theory of categorical local systems to the fiberwise Fukaya category of symplectic fibrations.

math.AT↗

arXiv2024

Homological mirror symmetry for complete intersections in algebraic tori

We prove one direction of homological mirror symmetry for complete intersections in algebraic tori, in all dimensions. The mirror geometry is not a space but a LG model, i.e. a pair given by a space and a regular function. We show that the Fukaya category of the complete intersection is equivalent to the category of matrix factorizations of the LG pair. Our approach yields new results also in the hypersurface setting, which was treated earlier by Gammage and Shende. Our argument depends on breaking down the complete intersection into smaller more manageable pieces, i.e. finite covers of products of higher dimensional pairs-of-pants, thus implementing a program first suggested by Seidel.

math.SG↗

arXiv2022

Singularity categories of normal crossings surfaces, descent, and mirror symmetry

Given a smooth 3-fold $Y$, a line bundle $L \to Y$, and a section $s$ of $L$ such that the vanishing locus of $s$ is a normal crossings surface $X$ with graph-like singular locus, we present a way to reconstruct the singularity category of $X$ as a homotopy limit of several copies of the category of matrix factorizations of $xyz : \mathbb{A}^{3} \to \mathbb{A}^{1}$ (the mirror to the Fukaya category of the pair of pants). This extends our previous result for the case where $L$ is trivialized. The key technique is the classification of non-two-periodic autoequivalences of the category of matrix factorizations. We also present a conjectural mirror for these singularity categories in terms of the Rabinowitz wrapped Fukaya categories of Ganatra-Gao-Venkatesh for certain symplectic four-manifolds, and relate this construction to work of Lekili-Ueda and Jeffs.

math.AG↗

arXiv2021

Fukaya categories of higher-genus surfaces and pants decompositions

In this paper we prove a local-to-global principle for the Fukaya category of a closed Riemann surface $Σ$ of genus $g \geq 2$. We show that $\mathrm{Fuk}(Σ)$ can be glued from the Fukaya categories of the pairs-of-pants making up a pants decomposition of $Σ$. This extends our earlier results for the case of punctured Riemann surfaces. Our result has several interesting consequences: we obtain simple proofs of old and new HMS statements for Riemann surfaces, and establish a geometrization theorem for the objects of $\mathrm{Fuk}(Σ)$.

math.SG↗

arXiv2020

The categorified Grothendieck-Riemann-Roch theorem

In this paper we prove a categorification of the Grothendieck-Riemann-Roch theorem. Our result implies in particular a Grothendieck-Riemann-Roch theorem for Toën and Vezzosi's secondary Chern character. As a main application, we establish a comparison between the Toën-Vezzosi Chern character and the classical Chern character, and show that the categorified Chern character recovers the classical de Rham realization.

math.KT↗

arXiv2020

Parabolic semi-orthogonal decompositions and Kummer flat invariants of log schemes

We construct semi-orthogonal decompositions on triangulated categories of parabolic sheaves on certain kinds of logarithmic schemes. This provides a categorification of the decomposition theorems in Kummer flat K-theory due to Hagihara and Nizioł. Our techniques allow us to generalize Hagihara and Nizioł's results to a much larger class of invariants in addition to K-theory, and also to extend them to more general logarithmic stacks.

math.AG↗

arXiv2020

Gluing semi-orthogonal decompositions

We introduce preordered semi-orthogonal decompositions (psod-s) of dg-categories. We show that homotopy limits of dg-categories equipped with compatible psod-s carry a natural psod. This gives a way to glue semi-orthogonal decompositions along faithfully-flat covers, extending some results of [4]. As applications we will construct semi-orthogonal decompositions for root stacks of log pairs (X,D) where D is a (not necessarily simple) normal crossing divisors, generalizing results from [17] and [3]. Further we will compute the Kummer flat K-theory of general log pairs (X,D), generalizing earlier results of Hagihara and Nizioł in the simple normal crossing case [15], [23].

math.AG↗

arXiv2019

On the profinite homotopy type of log schemes

We complete the program, initiated in [6], to compare the many different possible definitions of the underlying homotopy type of a log scheme. We show that, up to profinite completion, they all yield the same result, and thus arrive at an unambiguous definition of the profinite homotopy type of a log scheme. Specifically, in [6], we define this to be the profinite étale homotopy type of the infinite root stack, and show that, over $\mathbb{C},$ this agrees up to profinite completion with the Kato-Nakayama space. Other possible candidates are the profinite shape of the Kummer étale site $X_{\mbox{két}},$ or of the representable étale site of $\sqrt[\infty]{X}.$ Our main result is that all of these notions agree, and moreover the profinite étale homotopy type of the infinite root stack is not sensitive to whether or not it is viewed as a pro-system in stacks, or as an actual stack (by taking the limit of the pro-system). We furthermore show that in the log regular setting, all these notions also agree with the étale homotopy type of the classical locus $X^{\mbox{triv}}$ (up to an appropriate completion). We deduce that, over an arbitrary locally Noetherian base, the étale homotopy type of $\mathbb{G}_m^N$ agrees with that of $Bμ_\infty^N$ up to completion.

math.AG↗

arXiv2019

Topological Fukaya category and mirror symmetry for punctured surfaces

In this paper we establish a version of homological mirror symmetry for punctured Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured surface $Σ$ via the topological Fukaya category. We prove that the topological Fukaya category of $Σ$ is equivalent to the category of matrix factorizations of the mirror LG model $(X,W)$. Along the way we establish new gluing results for the topological Fukaya category of punctured surfaces which might be of independent interest.

math.AT↗

arXiv2018

On a logarithmic version of the derived McKay correspondence

We globalize the derived version of the McKay correspondence of Bridgeland-King-Reid, proven by Kawamata in the case of abelian quotient singularities, to certain log algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible log blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under log blow-up, up to Morita equivalence.

math.AG↗

arXiv2017

Higher traces, noncommutative motives, and the categorified Chern character

We propose a categorification of the Chern character that refines earlier work of Toën and Vezzosi and of Ganter and Kapranov. If X is an algebraic stack, our categorified Chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over X, which we introduce, to S1-equivariant perfect complexes on the derived free loop stack LX. As an application of the theory, we show that Toën and Vezzosi's secondary Chern character factors through secondary K-theory. Our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal (infinity,n)-categories, which is of independent interest.

math.KT↗

arXiv2016

Kato-Nakayama spaces, infinite root stacks, and the profinite homotopy type of log schemes

For a log scheme locally of finite type over $\mathbb{C}$, a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Finally, for a log scheme not necessarily over $\mathbb{C}$, another natural candidate is the profinite étale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over $\mathbb{C}$, these three notions agree. In particular, we construct a comparison map from the Kato-Nakayama space to the underlying topological stack of the infinite root stack, and prove that it induces an equivalence on profinite completions. In light of these results, we define the profinite homotopy type of a general fine saturated log scheme as the profinite étale homotopy type of its infinite root stack.

math.AG↗