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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]

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]

The mirror P=W conjecture, formulated by Harder-Katzarkov-Przyjalkowski [27] , predicts a correspondence between weight and perverse filtrations in the context of mirror symmetry. In this paper, we revisit this conjecture through the lens of mirror symmetry for a Fano pair (X , D) , where X is a smooth Fano variety and D is a simple normal crossing divisor. We introduce its mirror object as a multi-potential analogue of a Landau-Ginzburg (LG) model, which we call the hybrid LG model. This model is expected to capture the mirrors of all irreducible components of D. We study the topological aspects, particularly the perverse filtration, and the Hodge theory of hybrid LG models, building upon the work of Katzarkov-Kontsevich-Pantev [32]. As an application, we discover an interesting upper bound on the multiplicativity of the perverse filtration for a hybrid LG model. Additionally, we propose a relative version of the homological mirror symmetry conjecture and explain how the mirror P=W conjecture naturally emerges from it. [ABSTRACT FROM AUTHOR]

On the basis of (i) the discrete and continuous symmetries (and corresponding conserved charges), (ii) the ensuing algebraic structures of the symmetry operators and conserved charges, and (iii) a few basic concepts behind the subject of differential geometry, we show that the celebrated Friedberg-Lee-Pang-Ren (FLPR) quantum mechanical model (describing the motion of a single non-relativistic particle of unit mass under the influence of a general spatial 2D rotationally invariant potential) provides a tractable physical example for the Hodge theory within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where the symmetry operators and conserved charges lead to the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. We concisely mention the Hodge decomposition theorem in the quantum Hilbert space of states and choose the harmonic states as the real physical states of our theory. We discuss the physicality criteria w.r.t. the conserved and nilpotent versions of the (anti-)BRST and (anti-)co-BRST charges and the physical consequences that ensue from them. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a coherent subsheaf of Saito's S -sheaf, which combines the S -sheaf and the multiplier ideal sheaf. We construct its L 2 -Dolbeault resolution, which generalizes MacPherson's conjecture on the L 2 resolution of the Grauert-Riemenschneider sheaf. We also prove various transcendentally-based vanishing theorems for the S -sheaf, including Saito's vanishing theorem, Kawamata-Viehweg vanishing theorem, and new ones such as the Nadel vanishing theorem. Finally, we discuss the applications of our results on the relative version of Fujita's conjecture, such as Kawamata's conjecture. [ABSTRACT FROM AUTHOR]

The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential d ω = d + ω ∧ , where ω is a closed 1-form. We study Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy invariance and then extend it to more general type of forms on a Riemannian foliation. We study the Laplacian and Hodge decompositions for the corresponding differential operators on reduced leafwise Morse-Novikov complexes. In the case of Riemannian foliations, we prove that the reduced leafwise Morse-Novikov cohomology groups satisfy the Hodge theorem and Poincaré duality. The resulting isomorphisms yield a Hodge diamond structure for leafwise Morse-Novikov cohomology. [ABSTRACT FROM AUTHOR]