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Let k be a field of characteristic zero with a fixed embedding into the field of complex numbers. Given a k-variety X, we use the triangulated category of étale motives with rational coefficients on X to construct an abelian category M.X/of perverse mixed motives. We show that over Spec.k/the category obtained is canonically equivalent to the usual category of Nori motives and that the derived categories Db.M.X//are equipped with the four operations of Grothendieck (for morphisms of quasi-projective k-varieties) as well as nearby and vanishing cycles functors and a formalism of weights. In particular, as an application, we show that many classical constructions done with perverse sheaves, such as intersection cohomology groups or Leray spectral sequences, are motivic and therefore compatible with Hodge theory. This recovers and strengthens work by Zucker, Saito, Arapura and de Cataldo-Migliorini and provides an arithmetic proof of the pureness of intersection cohomology with coefficients in a geometric variation of Hodge structures. [ABSTRACT FROM AUTHOR]

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds. [ABSTRACT FROM AUTHOR]

In 2010, Brasselet, Schürmann, and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky–MacPherson |$L$| -class |$L_{*}(X)$| and the Hirzebruch homology class |$T_{1,*}(X)$| for a compact complex algebraic variety |$X$| that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions that we find of independent interest. [ABSTRACT FROM AUTHOR]

Let |${\mathcal{M}}_{\operatorname{Dol}}(X,G)$| denote the hyperkähler moduli space of |$G$| -Higgs bundles over a smooth projective curve |$X$|⁠. In the context of four dimensional supersymmetric Yang–Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of |${\mathcal{M}}_{\operatorname{Dol}}(X,G)$|⁠. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne–Hitchin twistor space |$\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$|⁠. Following Gaiotto's suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack , a derived analytic stack with corresponding moduli space |$\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$|⁠ , defined as a gluing between two analytic Hodge stacks along the Riemann–Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence. [ABSTRACT FROM AUTHOR]

Let H = ((H , F •) , L) be a polarized variation of Hodge structure on a smooth quasi-projective variety U. By M. Saito's theory of mixed Hodge modules, the variation of Hodge structure H can be viewed as a polarized Hodge module M ∈ H M (U). Let X be a compactification of U , and j : U ↪ X is the natural map. In this paper, we use local cohomology with mixed Hodge module theory to study j + M ∈ D b M H M (X). In particular, we study the graded pieces of the de Rham complex G r p F D R (j + M) ∈ D c o h b (X) , and the Hodge structure of H i (U , L) for i in sufficiently low degrees. [ABSTRACT FROM AUTHOR]

For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb {Q}_p$ we give a geometrisation of the $p$ -adic Simpson correspondence of rank one in terms of analytic moduli spaces: the $p$ -adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$ -line bundles. As an application, we study a major open question in $p$ -adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the $p$ -adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters $\pi ^{{\mathrm {\acute {e}t}}}_1(X)\to K^\times$ in terms of moduli spaces: for projective $X$ over $K=\mathbb {C}_p$ , it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group $\operatorname {Pic}(X)$. [ABSTRACT FROM AUTHOR]

We equip integral graded-polarized mixed period spaces with a natural Ralg-definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures is definable in Ran,exp with respect to this structure. As a consequence we re-prove that the zero loci of admissible normal functions are algebraic. [ABSTRACT FROM AUTHOR]

Let |$(M,I,J,K,\Omega)$| be a compact HKT manifold, and let us denote with |$\partial$| the conjugate Dolbeault operator with respect to I , |$\partial_J:=J^{-1}\overline\partial J$|⁠ , |$\partial^\Lambda:=[\partial,\Lambda]$|⁠ , where Λ is the adjoint of |$L:=\Omega\wedge-$|⁠. Under suitable assumptions, we study Hodge theory for the complexes |$(A^{\bullet,0},\partial,\partial_J)$| and |$(A^{\bullet,0},\partial,\partial^\Lambda)$| showing a similar behavior to Kähler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT |$\mathrm{SL}(n,\mathbb{H})$| -manifold, the differential graded algebra |$(A^{\bullet,0},\partial)$| is formal and this will lead to an obstruction for the existence of an HKT |$\mathrm{SL}(n,\mathbb{H})$| structure |$(I,J,K,\Omega)$| on a compact complex manifold (M ,  I). Finally, balanced HKT structures on solvmanifolds are studied. [ABSTRACT FROM AUTHOR]

In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. They arise when we focus on the purely topological properties of good Geometric Invariant Theory (GIT) quotients regardless of their algebraic properties. The flexibility granted by their topological nature enables an easier identification in geometric constructions than classical GIT quotients. We obtain several results about the interplay between pseudo-quotients and good quotients. Additionally, we show that in characteristic zero pseudo-quotients are unique up to virtual class in the Grothendieck ring of algebraic varieties. As an application, we compute the virtual class of S L 2 (k) -character varieties for free groups and surface groups as well as their parabolic counterparts with punctures of Jordan type. [ABSTRACT FROM AUTHOR]

The K-moduli theory provides different compactifications of various moduli spaces, including moduli of curves. As a general genus six curve can be canonically embedded into the smooth quintic del Pezzo surface, we study in this paper the K-moduli spaces \overline {M}^K(c) of the quintic log Fano pairs. We classify the strata of genus six curves C appearing in the K-moduli by explicitly describing the wall-crossing structure. The K-moduli spaces interpolate between two birational moduli spaces constructed by Geometric Invariant Theory (GIT) and moduli of K3 surfaces via Hodge theory. [ABSTRACT FROM AUTHOR]

Let U be a smooth connected complex algebraic variety, and let f: U → C∗ be an algebraic map. To the pair (U, f) one can associate an infinite cyclic cover Uf, and (homology) Alexander modules are defined as the homology groups of this cover. In two recent works, the first of which is joint with C. Geske, L. Maxim and B. Wang, we developed two different ways to put a mixed Hodge structure on Alexander modules. Since they are not finite dimensional in general, each approach replaces the Alexander module by a different finite dimensional module: one of them takes the torsion submodule, the other takes finite dimensional quotients, and the constructions are not directly comparable. In this note, we show that both constructions are compatible, in the sense that the map from the torsion to the quotients is a mixed Hodge structure morphism. [ABSTRACT FROM AUTHOR]

We solve Prill's problem, originally posed by David Prill in the late 1970s and popularized in Arbarello, Cornalba, Griffiths and Harris's "Geometry of Algebraic Curves." That is, for any curve Y of genus 2, we produce a finite 'etale degree 36 connected cover f: X → Y where, for every point y ∈ Y, the preimage f-1(y) moves in a pencil. [ABSTRACT FROM AUTHOR]

We establish a connection between the algebraic geometry of the type B$B$ permutohedral toric variety and the combinatorics of delta‐matroids. Using this connection, we compute the volume and lattice point counts of type B$B$ generalized permutohedra. Applying tropical Hodge theory to a new framework of "tautological classes of delta‐matroids," modeled after certain vector bundles associated to realizable delta‐matroids, we establish the log‐concavity of a Tutte‐like invariant for a broad family of delta‐matroids that includes all realizable delta‐matroids. Our results include new log‐concavity statements for all (ordinary) matroids as special cases. [ABSTRACT FROM AUTHOR]

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds.

Smooth minimal surfaces of general type with K2=1$K^2=1$, pg=2$p_g=2$, and q=0$q=0$ constitute a fundamental example in the geography of algebraic surfaces, and the 28‐dimensional moduli space M$\mathbf {M}$ of their canonical models admits a modular compactification M¯$\overline{\mathbf {M}}$ via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parameterizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of M$\mathbf {M}$ and the Hodge theory of the degenerate surfaces that the eight divisors parameterize. [ABSTRACT FROM AUTHOR]

In this paper, we study the cohomology of semisimple local systems in the spirit of classical Hodge theory. On the one hand, we construct a generalized Weil operator from the complex conjugate of the cohomology of a semisimple local system to the cohomology of its dual local system, which is functorial with respect to smooth restrictions. This operator allows us to study the Poincaré pairing, usually not positive definite, in terms of a positive definite Hermitian pairing. On the other hand, we prove a global invariant cycle theorem for semisimple local systems. As an application, we give a new proof of Sabbah's Decomposition Theorem for the direct images of semisimple local systems under proper algebraic maps, by adapting the method of de Cataldo-Migliorini, without using the category of polarizable twistor D -modules. This answers a question of Sabbah. [ABSTRACT FROM AUTHOR]

Given a polarizable ℤ-variation of Hodge structures V over a complex smooth quasi-projective base S , a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge tensors appear) is a countable union of irreducible algebraic subvarieties of S , called the special subvarieties for V . Our main result in this paper is that, if the level of V is at least 3, this Hodge locus is in fact a finite union of such special subvarieties (hence is algebraic), at least if we restrict ourselves to the Hodge locus factorwise of positive period dimension (Theorem 1.5). For instance the Hodge locus of positive period dimension of the universal family of degree d smooth hypersurfaces in P C n + 1 , n ≥ 3 , d ≥ 5 and (n , d) ≠ (4 , 5) , is algebraic. On the other hand we prove that in level 1 or 2, the Hodge locus is analytically dense in S an as soon as it contains one typical special subvariety. These results follow from a complete elucidation of the distribution in S of the special subvarieties in terms of typical/atypical intersections, with the exception of the atypical special subvarieties of zero period dimension. [ABSTRACT FROM AUTHOR]

Matroids are combinatorial abstractions of independence, a ubiquitous notion that pervades many branches of mathematics. June Huh and his collaborators recently made spectacular breakthroughs by developing a Hodge theory of matroids that resolved several long-standing conjectures in matroid theory. We survey the main results in this development and ideas behind them. [ABSTRACT FROM AUTHOR]

Using Hodge theory and Banach fixed point theorem, Liu and Zhu developed a global method to deal with various problems in deformation theory. In this note, the authors generalize Liu-Zhu's method to treat two deformation problems for non-Kähler manifolds. They apply the -Hodge theory to construct a deformation formula for (p, q)-forms of compact complex manifold under deformations, which can be used to study the Hodge number of complex manifold under deformations. In the second part of this note, by using the -Hodge theory, they provide a simple proof of the unobstructed deformation theorem for the non-Kähler Calabi-Yau -manifolds. [ABSTRACT FROM AUTHOR]

A translation surface is a multifaceted object that can be studied with the tools of dynamics, analysis, or algebraic geometry. Moduli spaces of translation surfaces exhibit equally rich features. This survey provides an introduction to the subject and describes some developments that make use of Hodge theory to establish algebraization and finiteness statements in moduli spaces of translation surfaces. [ABSTRACT FROM AUTHOR]

We show that for a variety which admits a quasi-finite period map, finiteness (resp. non-Zariski-density) of S-integral points implies finiteness (resp. non-Zariski-density) of points over all Z -finitely generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, and Bakker-Brunebarbe-Tsimerman. We give straightforward applications to arithmetic locally symmetric varieties, the moduli space of smooth hypersurfaces in projective space, and the moduli of smooth divisors in an abelian variety. [ABSTRACT FROM AUTHOR]

We give a formula comparing the E-series of the moduli stacks of rank 2 degree 0 semistable Higgs bundles in genus g ≥ 2 to intersection E-polynomials of its coarse moduli space. A parallel formula holds in various 2-Calabi–Yau settings, for example for sheaves on K3 surfaces, or preprojective algebras of g -loop quivers. As a consequence we provide evidence for a conjecture of Davison on the BPS cohomology of Higgs bundles, which has implications for non-abelian Hodge theory for stacks. We apply the formula to cohomological χ -independence tests for BPS cohomology of Higgs bundles and K3 surfaces. [ABSTRACT FROM AUTHOR]

We construct a Leray model for a discrete polymatroid with arbitrary building set and we prove a generalized Goresky–MacPherson formula. The first row of the model is the Chow ring of the polymatroid; we prove Poincaré duality, Hard Lefschetz, and Hodge–Riemann theorems for the Chow ring. Furthermore, we provide a relative Lefschetz decomposition with respect to the deletion of an element. [ABSTRACT FROM AUTHOR]

We study integrable deformations of two-dimensional non-linear σ-models and present a new class of classical solutions to critical bi-Yang-Baxter models for general groups. For the simplest example, namely the SL(2, ℝ) bi-Yang-Baxter model, we show that our solutions can be mapped to the known complex uniton solutions of the SU(2) bi-Yang-Baxter model. In general, our solutions are constructed from so-called Sl(2)-orbits that play a central role in the study of asymptotic Hodge theory. This provides further evidence for a close relation between integrable non-linear σ-models and the mathematical principles underlying Hodge theory. We have also included a basic introduction to the relevant aspects of asymptotic Hodge theory and have provided some simple examples. [ABSTRACT FROM AUTHOR]

We construct a category of Breuil-Kisin GK-modules to classify integral semi-stable Galois representations. Our theory uses Breuil-Kisin modules and Breuil-Kisin-Fargues modules with Galois actions, and can be regarded as the algebraic avatar of the integral p-adic cohomology theories of Bhatt-Morrow-Scholze and Bhatt-Scholze. As a key ingredient, we classify Galois representations that are of finite E.u/-height. [ABSTRACT FROM AUTHOR]

We propose a new conjecture on the relation between the exact Dirac zero-modes of free and massless lattice fermions and the topology of the manifold on which the fermion action is defined. Our conjecture claims that the maximal number of exact Dirac zero-modes of fermions on finite-volume and finite-spacing lattices defined by a discretizing torus, hyperball, their direct-product space, and hypersphere is equal to the summation of the Betti numbers of their manifolds if several specific conditions on lattice formulations are satisfied. We start with reconsidering exact Dirac zero-modes of naive fermions on the lattices whose topologies are a torus, hyperball, and their direct-product space (T D × B d ). We find that the maximal number of exact zero-modes of free Dirac fermions is in exact agreement with the sum of Betti numbers |$\sum ^{D}_{r=0} \beta _{r}$| for these manifolds. Indeed, the 4 D lattice fermion on a torus has up to 16 zero-modes while the sum of Betti numbers of T 4 is 16. This coincidence holds also for the D -dimensional hyperball and their direct-product space T D × B d . We study several examples of lattice fermions defined on a certain discretized hypersphere (S D ), and find that it has up to two exact zero-modes, which is the same number as the sum of Betti numbers of S D . From these facts, we conjecture the equivalence of the maximal number of exact Dirac zero-modes and the summation of Betti numbers under specific conditions. We discuss a program for proof of the conjecture in terms of Hodge theory and spectral graph theory. [ABSTRACT FROM AUTHOR]

Let $G$ be a split reductive group over the ring of integers in a $p$ -adic field with residue field $\mathbf {F}$. Fix a representation $\overline {\rho }$ of the absolute Galois group of an unramified extension of $\mathbf {Q}_p$ , valued in $G(\mathbf {F})$. We study the crystalline deformation ring for $\overline {\rho }$ with a fixed $p$ -adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for $G$ -valued representations. In particular, we give a root theoretic condition on the $p$ -adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups. [ABSTRACT FROM AUTHOR]

We settle the last open case of Kuznetsov's conjecture on the derived categories of Fano threefolds. Contrary to the original conjecture, we prove the Kuznetsov components of quartic double solids and Gushel–Mukai threefolds are never equivalent, as recently shown independently by Zhang. On the other hand, we prove the modified conjecture asserting their deformation equivalence. Our proof of nonequivalence combines a categorical Enriques-K3 correspondence with the Hodge theory of categories. Along the way, we obtain a categorical description of the periods of Gushel–Mukai varieties, which we use to resolve a conjecture of Kuznetsov and the second author on the birational categorical Torelli problem, as well as to give a simple proof of a theorem of Debarre and Kuznetsov on the fibers of the period map. Our proof of deformation equivalence relies on results of independent interest about obstructions to enhancing group actions on categories. [ABSTRACT FROM AUTHOR]

We introduce the conormal fan of a matroid \operatorname {M}, which is a Lagrangian analog of the Bergman fan of \operatorname {M}. We use the conormal fan to give a Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of \operatorname {M}. This allows us to express the h-vector of the broken circuit complex of \operatorname {M} in terms of the intersection theory of the conormal fan of \operatorname {M}. We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations. The Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of \operatorname {M}, when combined with the Hodge–Riemann relations for the conormal fan of \operatorname {M}, implies Brylawski's and Dawson's conjectures that the h-vectors of the broken circuit complex and the independence complex of \operatorname {M} are log-concave sequences. [ABSTRACT FROM AUTHOR]

We give a new proof of Faltings's $p$ -adic Eichler–Shimura decomposition of the modular curves via Bernstein–Gelfand–Gelfand (BGG) methods and the Hodge–Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was used in the original proof. Then we construct overconvergent Eichler–Shimura maps for the modular curves providing 'the second half' of the overconvergent Eichler–Shimura map of Andreatta, Iovita and Stevens. We use higher Coleman theory on the modular curve developed by Boxer and Pilloni to show that the small-slope part of the Eichler–Shimura maps interpolates the classical $p$ -adic Eichler–Shimura decompositions. Finally, we prove that overconvergent Eichler–Shimura maps are compatible with Poincaré and Serre pairings. [ABSTRACT FROM AUTHOR]

We develop a Fredholm theory for the Hodge Laplacian in weighted spaces on ALG∗ manifolds in dimension four. We then give several applications of this theory. First, we show the existence of harmonic functions with prescribed asymptotics at infinity. A corollary of this is a non-existence result for ALG∗ manifolds with non-negative Ricci curvature having group Γ = { e } at infinity. Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG∗ manifold. A corollary of this is vanishing of the first Betti number for any ALG∗ manifold with non-negative Ricci curvature. Another application of our analysis is to determine the optimal order of ALG∗ gravitational instantons. [ABSTRACT FROM AUTHOR]

The nonabelian Hodge correspondence (also known as the Corlette-Simpson correspondence), between the polystable Higgs bundles with vanishing Chern classes on a compact Kähler manifold X and the completely reducible flat connections on X, is extended to the Fujiki class \mathcal C manifolds. [ABSTRACT FROM AUTHOR]

A Teichmüller curve V \subset \mathcal {M}_g is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems. [ABSTRACT FROM AUTHOR]

We show that the description of Deligne--Beilinson cohomology is improved by using log Hodge theory. We consider the log relative version of it, and also present a fundamental conjecture in log Hodge theory. [ABSTRACT FROM AUTHOR]

We demonstrate that the celebrated Stückelberg formalism is modified in the case of a massive four (3 + 1)-dimensional (4D) Abelian 2-form theory due to the presence of a self-duality discrete symmetry in the theory. The latter symmetry entails upon the modified 4D massive Abelian 2-form gauge theory to become a massive model of Hodge theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism where there is the existence of a set of (anti-)co-BRST transformations corresponding to the usual nilpotent (anti-)BRST transformations. The latter exist in any arbitrary dimension of spacetime for the usual Stückelberg-modified massive Abelian 2-form gauge theory. The modification in the Stückelberg technique is backed by the precise mathematical arguments from the differential geometry where the exterior derivative and Hodge duality operator play the decisive roles. The modified version of the Stückelberg technique remains invariant under the discrete duality transformations which also establish a precise and deep connection between the off-shell nilpotent (anti-)BRST and (anti-)co-BRST transformations. We have clarified a simple trick of using the equations of motion to remove the higher derivative terms in the appropriate Lagrangian densities so that our 4D theory can become consistent. [ABSTRACT FROM AUTHOR]

This is the first of a series of papers in which we study the plurigenera, the Kodaira dimension, and more generally the Iitaka dimension on compact almost complex manifolds. Based on the Hodge theory on almost complex manifolds, we introduce the plurigenera, Kodaira dimension and Iitaka dimension on compact almost complex manifolds. We show that plurigenera and the Kodaira dimension are birational invariants in almost complex category, at least in dimension 4, where a birational morphism is defined to be a degree one pseudoholomorphic map. However, they are no longer deformation invariants, even in dimension 4 or under tameness assumption. On the way to establish the birational invariance, we prove the Hartogs extension theorem in the almost complex setting by the foliation-by-disks technique. Some interesting phenomena of these invariants are shown through examples. In particular, we construct non-integrable compact almost complex manifolds with large Kodaira dimensions. Hodge numbers and plurigenera are computed for the standard almost complex structure on the six sphere S6, which are different from the data of a hypothetical complex structure. [ABSTRACT FROM AUTHOR]

Our workshop focused on recent results in our main field (complex geometry) and its connection with other branches of mathematics. The main theme of an important proportion of the talks was Hodge theory, combined with differential-geometric methods in the study of singular spaces. One special lecture was a very comprehensive introduction in Scholze-Clausen's theory of condensed mathematics. [ABSTRACT FROM AUTHOR]

The purpose of this Arbeitsgemeinschaft is to introduce the notion of twistor D-modules and their main properties. The guiding principle leading this discussion is Simpson's "meta-theorem", which gives a heuristic for generalizing (mixed) Hodge-theoretic results into (mixed) twistortheoretic results. The strength of the twistor approach is that it enables to enlarge the scope of Hodge theory not only to arbitrary semi-simple perverse sheaves, equivalently semi-simple regular holonomic D-modules via the Riemann-Hilbert correspondence, but also to possibly semi-simple irregular holonomic D-modules. An overarching goal for this session is Mochizuki's proof of the decomposition theorem for semi-simple holonomic D-modules on a smooth complex projective variety, first conjectured by Kashiwara in 1996. [ABSTRACT FROM AUTHOR]

We show that the vector bundle on the moduli stack Mell$M_\mathrm{ell}$ of elliptic curves associated to the 2‐cell complex Cν$C\nu$ is isomorphic to the de Rham cohomology sheaf HdR1(E/Mell)$\mathrm{H}^1_\mathrm{dR}(\mathcal {E}/M_\mathrm{ell})$ of the universal elliptic curve E→Mell$\mathcal {E}\rightarrow M_\mathrm{ell}$. We use this to calculate the homotopy groups of the E1$\mathbf {E}_{1}$‐quotient tmf//ν$\mathrm{tmf} /\!\!/ \nu$ of tmf$\mathrm{tmf}$ by ν$\nu$, called the spectrum of 'topological quasimodular forms', by relating its Adams–Novikov spectral sequence to the cohomology of the moduli stack of cubic curves with a chosen splitting of the Hodge–de Rham filtration. [ABSTRACT FROM AUTHOR]

We show that crystalline points are Zariski dense in the deformation space of a representation of the absolute Galois group of a p-adic field. We also show that these points are dense in the subspace parameterizing deformations with the determinant equal to a fixed crystalline character. Our proof is purely local and works for all p-adic fields and all residual Galois representations. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable ∞-category Cohb(ℝM) of complexes of sheaves with bounded coherent cohomology on a derived moduli stack ℝM. In the surface case, ℝM is a suitable derived enhancement of the moduli stack M of coherent sheaves on the surface. This construction categorifies the K-theoretical and cohomological Hall algebras of coherent sheaves on a surface of Zhao and Kapranov-Vasserot. In the curve case, we define three categorified Hall algebras associated with suitable derived enhancements of the moduli stack of Higgs sheaves on a curve X, the moduli stack of vector bundles with flat connections on X, and the moduli stack of finite-dimensional local systems on X, respectively. In the Higgs sheaves case we obtain a categorification of the K-theoretical and cohomological Hall algebras of Higgs sheaves on a curve of Minets and Sala-Schiffmann, while in the other two cases our construction yields, by passing to K0, new K-theoretical Hall algebras, and by passing to H*BM, new cohomological Hall algebras. Finally, we show that the Riemann-Hilbert and the non-abelian Hodge correspondences can be lifted to the level of our categorified Hall algebras of a curve. [ABSTRACT FROM AUTHOR]

We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite Γ1(p∞)-level (defined with respect to a Borel subgroup) vanishes above the middle degree, under the assumption that the group of the Shimura datum splits at p. This generalizes and strengthens the vanishing result proved in [A. Caraiani et al., Compos. Math. 156 (2020)]. As an application of this vanishing theorem, we prove a result on the codimensions of ordinary completed homology for the same groups, analogous to conjectures of Calegari-Emerton for completed (Borel-Moore) homology. [ABSTRACT FROM AUTHOR]

Sasakian manifolds are odd-dimensional counterpart to Kähler manifolds. They can be defined as contact manifolds equipped with an invariant Kähler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition on Vaisman and Sasakian manifolds. We construct a Lie superalgebra associated to a Sasakian manifold in the same way as the Kähler supersymmetry algebra is associated to a Kähler manifold. We use this construction to produce a self-contained, coordinate-free proof of the results by Tachibana, Kashiwada and Sato on the decomposition of harmonic forms and cohomology of Sasakian and Vaisman manifolds. In the last section, we compute the supersymmetry algebra of Sasakian manifolds explicitly. [ABSTRACT FROM AUTHOR]

With G=GL(n,C)$G=GL(n,\mathbb {C})$, let XΓG$\mathcal {X}_{\Gamma }G$ be the G‐character variety of a given finitely presented group Γ, and let XΓirrG⊂XΓG$\mathcal {X}_{\Gamma }^{irr}G\subset \mathcal {X}_{\Gamma }G$ be the locus of irreducible representation conjugacy classes. We provide a concrete relation, in terms of plethystic functions, between the generating series for E‐polynomials of XΓG$\mathcal {X}_{\Gamma }G$ and the one for XΓirrG$\mathcal {X}_{\Gamma }^{irr}G$, generalizing a formula of Mozgovoy–Reineke. The proof uses a natural stratification of XΓG$\mathcal {X}_{\Gamma }G$ coming from affine GIT, the combinatorics of partitions, and the formula of MacDonald–Cheah for symmetric products; we also adapt it to the so‐called Cartan brane in the moduli space of Higgs bundles. Combining our methods with arithmetic ones yields explicit expressions for the E‐polynomials, and Euler characteristics, of the irreducible stratum of GL(n,C)$GL(n,\mathbb {C})$‐character varieties of some groups Γ, including surface groups, free groups, and torus knot groups, for low values of n. [ABSTRACT FROM AUTHOR]

We prove that the rational Chow motive of a six-dimensional hyper-Kähler variety obtained as symplectic resolution of O'Grady type of a singular moduli space of semistable sheaves on an Abelian surface A belongs to the tensor category of motives generated by the motive of A. We in fact give a formula for the rational Chow motive of such a variety in terms of that of the surface. As a consequence, the conjectures of Hodge and Tate hold for many hyper-Kähler varieties of OG6-type. [ABSTRACT FROM AUTHOR]

In this paper, we study various hyperbolicity properties for a quasi-compact Kähler manifold U which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that U is algebraically hyperbolic and that the generalized big Picard theorem holds for U. In the second part, we prove that there is a finite étale cover Ũ of U from a quasi-projective manifold Ũ such that any projective compactification X of Ũ is Picard hyperbolic modulo the boundary X - Ũ, and any irreducible subvariety of X not contained in X - Ũ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices. [ABSTRACT FROM AUTHOR]

We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a $p$ -adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms. [ABSTRACT FROM AUTHOR]

We introduce a twisted cohomology cocycle over the Teichmüller flow and prove a "spectral gap" for its Lyapunov spectrum with respect to the Masur-Veech measures. We then derive Hölder estimates on spectral measures and bounds on the speed of weak mixing for almost all translation flows in every stratum of Abelian differentials on Riemann surfaces, as well as bounds on the deviation of ergodic averages for product translation flows on the product of a translation surface with a circle. [ABSTRACT FROM AUTHOR]

We generalize the construction of Rouquier complexes to the setting of one‐sided singular Soergel bimodules. Singular Rouquier complexes are defined by taking minimal complexes of restricted Rouquier complexes. We show that they retain many of the properties of ordinary Rouquier complexes: they are Δ$\Delta$‐split, they satisfy a vanishing formula, and when Soergel's conjecture holds they are perverse. As an application, we establish Hodge theory for singular Soergel bimodules. [ABSTRACT FROM AUTHOR]

Let |$E$| be a holomorphic vector bundle over a compact Kähler manifold |$(X,\omega)$| with negative sectional curvature |$sec\leq -K<0$| and |$D_{E}$| be the Chern connection on |$E$|⁠. In this article, we show that if |$C:=|[\Lambda ,i\Theta (E)]|\leq c_{n}K$|⁠ , then |$(X,E)$| satisfy a family of Chern number inequalities. The main idea in our proof is to study the |$L^{2}$| |$\bar {\partial }_{\tilde {E}}$| -harmonic forms on lifting bundle |$\tilde {E}$| over the universal covering space |$\tilde {X}$|⁠. We also observe that there is a close relationship between the eigenvalue of the Laplace–Beltrami operator |$\Delta _{\bar {\partial }_{\tilde {E}}}$| and the Euler characteristic of |$X$|⁠. Precisely, if there is a line bundle |$L$| on |$X$| such that |$\chi ^{p}(X,L^{\otimes m})$| is not constant for some integers |$p\in [0,n]$|⁠ , then the Euler characteristic of |$X$| satisfies |$(-1)^{n}\chi (X)\geq (n+1)+\lfloor \frac {c_{n}K}{2nC} \rfloor $|⁠. [ABSTRACT FROM AUTHOR]