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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]

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]

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]

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]

Let G be a complex reductive algebraic group with Lie algebra \mathfrak {g} and let G_{\mathbb {R}} be a real form of G with maximal compact subgroup K_{\mathbb {R}}. Associated to G_{\mathbb {R}} is a K \times \mathbb {C}^{\times }-invariant subvariety \mathcal {N}_{\theta } of the (usual) nilpotent cone \mathcal {N} \subset \mathfrak {g}^*. In this article, we will derive a formula for the ring of regular functions \mathbb {C}[\mathcal {N}_{\theta }] as a representation of K \times \mathbb {C}^{\times }. Some motivation comes from Hodge theory. In [ Hodge theory and unitary representations of reductive Lie groups, Frontiers of Mathematical Sciences , Int. Press, Somerville, MA, 2011, pp. 397–420], Schmid and Vilonen use ideas from Saito's theory of mixed Hodge modules to define canonical good filtrations on many Harish-Chandra modules (including all standard and irreducible Harish-Chandra modules). Using these filtrations, they formulate a conjectural description of the unitary dual. If G_{\mathbb {R}} is split, and X is the spherical principal series representation of infinitesimal character 0, then conjecturally gr(X) \simeq \mathbb {C}[\mathcal {N}_{\theta }] as representations of K \times \mathbb {C}^{\times }. So a formula for \mathbb {C}[\mathcal {N}_{\theta }] is an essential ingredient for computing Hodge filtrations. [ABSTRACT FROM AUTHOR]

We prove a filtered version of the Homotopy Transfer Theorem which gives an A-infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the geometry and topology of complex manifolds, using the Hodge filtration, as well as to complex algebraic varieties, using mixed Hodge theory. [ABSTRACT FROM AUTHOR]

We describe the E-infinity algebra structure on the complex of singular cochains of a topological space, in the context of sheaf theory. As a first application, for any algebraic variety we define a weight filtration compatible with its E-infinity structure. This naturally extends the theory of mixed Hodge structures in rational homotopy to p-adic homotopy theory. The spectral sequence associated to the weight filtration gives a new family of algebraic invariants of the varieties for any coefficient ring, carrying Steenrod operations. As a second application, we promote Deligne's intersection complex computing intersection cohomology, to a sheaf carrying E-infinity structures. This allows for a natural interpretation of the Steenrod operations defined on the intersection cohomology of any topological pseudomanifold. [ABSTRACT FROM AUTHOR]

We give explicit harmonic representatives of Dolbeault cohomology of Oeljeklaus-Toma manifolds and show that they are geometrically Dolbeault formal. We also give explicit harmonic representatives of Bott-Chern cohomology of Oeljeklaus-Toma manifolds of type (s,1) and study the Angella-Tomassini inequality. [ABSTRACT FROM AUTHOR]

We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give a simple necessary and sufficient condition for this, whose proof relies on the Decomposition Theorem and Saito's theory of mixed Hodge modules. We use it to generalize the theorem of Greb-Kebekus-Kovács-Peternell to complex spaces with rational singularities, and to prove the existence of a functorial pull-back for reflexive differentials on such spaces. We also use our methods to settle the "local vanishing conjecture" proposed by Mustaţă, Olano, and Popa. [ABSTRACT FROM AUTHOR]

Following a suggestion of Peter Scholze, we construct an action of Gm on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the p-adic modular curve whose ring of functions is Serre's space of p-adic modular functions. This action is a local, p-adic analog of a global, archimedean action of the circle group S1 on the lattice-unstable locus of the modular curve over C. To construct the Gm-action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates q; along the way we also prove a natural generalization of Dwork's equation τ = log q for extensions of Qp/Zp by μp∞ valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of Gm integrates the differential operator \theta coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and p-adic L-functions. [ABSTRACT FROM AUTHOR]

We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface Y of degree 2 over Q together with a 3-torsion Brauer class α that is unramified at all primes except for 3, but ramifies at all 3-adic points of Y. Motivated by Hodge theory, the pair (Y, α) is constructed from a cubic fourfold X of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for α. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is insolubility of the fourfold X (and hence the fibers) over Q3 and local solubility at all other primes. [ABSTRACT FROM AUTHOR]

We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of period integrals to an initial value problem for ordinary differential equations of Picard-Fuchs type. In this way, the periods can be computed to extreme precision in order to study their arithmetic properties. The initial conditions are obtained by an exact determination of the cohomology pairing on Fermat hypersurfaces with respect to a natural basis. [ABSTRACT FROM AUTHOR]

Mixed boundary conditions are introduced to finite element exterior calculus. We construct smoothed projections from Sobolev de Rham complexes onto finite element de Rham complexes which commute with the exterior derivative, preserve homogeneous boundary conditions along a fixed boundary part, and satisfy uniform bounds for shape-regular families of triangulations and bounded polynomial degree. The existence of such projections implies stability and quasi-optimal convergence of mixed finite element methods for the Hodge Laplace equation with mixed boundary conditions. In addition, we prove the density of smooth differential forms in Sobolev spaces of differential forms over weakly Lipschitz domains with partial boundary conditions. [ABSTRACT FROM AUTHOR]

The standard mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex are based on proper discrete subcomplexes. As a consequence, the exterior derivatives, which are local operators, are computed exactly. However, the approximations of the associated coderivatives are nonlocal. In fact, this nonlocal property is an inherent consequence of the mixed formulation of these methods, and can be argued to be an undesired effect of these schemes. As a consequence, it has been argued, at least in special settings, that more local methods may have improved properties. In the present paper, we construct such methods by relying on a careful balance between the choice of finite element spaces, degrees of freedom, and numerical integration rules. Furthermore, we establish key convergence estimates based on a standard approach of variational crimes. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of ghost classes in the cohomology of the Shimura variety associated to the group of symplectic similitudes GSp4. The existence of ghost classes for the trivial coefficient system is known. We show that ghost classes only exist for the trivial coefficient system and they lie in the cohomology group in degree 2. Moreover we prove that the weight of the mixed Hodge structure associated to the space of ghost classes is the middle weight. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of ghost classes in the cohomology of the Shimura variety associated to the group of symplectic similitudes GSp4. The existence of ghost classes for the trivial coefficient system is known. We show that ghost classes only exist for the trivial coefficient system and they lie in the cohomology group in degree 2. Moreover we prove that the weight of the mixed Hodge structure associated to the space of ghost classes is the middle weight. [ABSTRACT FROM AUTHOR]

If G is a finite graph, a proper coloring of G is a way to color the vertices of the graph using n colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts that if G is planar, then there is at least one proper coloring of G with four colors.) By a classical result of Birkhoff, the number of proper colorings of G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read's conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh's result was subsequently refined and generalized by Huh and Katz (also in 2012), again using substantial doses of algebraic geometry. Both papers in fact establish log-concavity of the coefficients, which is stronger than unimodality. The breakthroughs of the Huh and Huh-Katz papers left open the more general Rota-Welsh conjecture, where graphs are generalized to (not necessarily representable) matroids, and the chromatic polynomial of a graph is replaced by the characteristic polynomial of a matroid. The Huh and Huh- Katz techniques are not applicable in this level of generality, since there is no underlying algebraic geometry to which to relate the problem. But in 2015 Adiprasito, Huh, and Katz announced a proof of the Rota-Welsh conjecture based on a novel approach motivated by but not making use of any results from algebraic geometry. The authors first prove that the Rota-Welsh conjecture would follow from combinatorial analogues of the hard Lefschetz theorem and Hodge-Riemann relations in algebraic geometry. They then implement an elaborate inductive procedure to prove the combinatorial hard Lefschetz theorem and Hodge-Riemann relations using purely combinatorial arguments. We will survey these developments. [ABSTRACT FROM AUTHOR]

We prove a surjectivity theorem for the Deligne canonical extension of a polarizable variation of Hodge structure with quasi-unipotent monodromy at infinity along the lines of Esnault-Viehweg. We deduce from it several injectivity theorems and vanishing theorems for pure Hodge modules. We also give an inductive proof of Kawamata-Viehweg vanishing for the lowest graded piece of the Hodge filtration of a pure Hodge module using mixed Hodge modules of nearby cycles. [ABSTRACT FROM AUTHOR]

The Serre-Tate correspondence contains a lot of Tate's work in a casual form. We present some excerpts that show how some of Tate's best known contributions came into being. [ABSTRACT FROM AUTHOR]

We discuss the space of sections and certain bisections on a quadric surfaces bundle X over a smooth curve. The Abel-Jacobi from these spaces to the intermediate Jacobian will be shown to be dominant with rationally connected fibers. As an application, we prove that the integral Hodge conjecture holds for degree 4 integral Hodge classes (IHC) of fourfolds fibered by quadric bundles over a smooth curve. This gives an alternative proof of a result of Colliot-Thélène and Voisin. [ABSTRACT FROM AUTHOR]

An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. In this paper, we study the cohomology of the complement of an abelian arrangement. For unimodular abelian arrangements, we provide a combinatorial presentation for a differential graded algebra whose cohomology is isomorphic to the rational cohomology of the complement. Moreover, this DGA has a bi-grading that allows us to compute the mixed Hodge numbers. [ABSTRACT FROM AUTHOR]

We study the Newton polygon jumping locus of a Mumford family in char p. Our main result says that, under a mild assumption on p, the jumping locus consists of only supersingular points and its cardinality is equal to (pr - 1)(g - 1), where r is the degree of the defining field of the base curve of a Mumford family in char p and g is the genus of the curve. The underlying technique is the p-adic Hodge theory. [ABSTRACT FROM AUTHOR]

We use simple methods from harmonic maps to investigate singularities of period mappings at infinity. More precisely, we derive a harmonic map version of Schmid's nilpotent orbit theorem. [ABSTRACT FROM AUTHOR]

We study the homotopy theory of a certain type of diagram category whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is applied to the category of mixed Hodge diagrams of differential graded algebras. Using Sullivan's minimal models, we prove a multiplicative version of Beilinson's Theorem on mixed Hodge complexes. As a consequence, we obtain functoriality for the mixed Hodge structures on the rational homotopy type of complex algebraic varieties. In this context, the mixed Hodge structures on homotopy groups obtained by Morgan's theory follow from the derived functor of the indecomposables of mixed Hodge diagrams. [ABSTRACT FROM AUTHOR]

In this article we study some necessary and sufficient conditions for the existence of solutions in W01,∞ (Ω;Λk) of the differential inclusion dΩ ∈ E a.e. in Ω where E ⊂ Λk+1 is a prescribed set. [ABSTRACT FROM AUTHOR]

We consider finite energy and L2 differential forms associated with strongly local regular Dirichlet forms on compact connected topologically onedimensional spaces. We introduce notions of local exactness and local harmonicity and prove the Hodge decomposition, which in our context says that the orthogonal complement to the space of all exact 1-forms coincides with the closed span of all locally harmonic 1-forms. Then we introduce a related Hodge Laplacian and define a notion of harmonicity for finite energy 1-forms. As a corollary, under a certain capacity-separation assumption, we prove that the space of harmonic 1-forms is nontrivial if and only if the classical Cech cohomology is nontrivial. In the examples of classical self-similar fractals these spaces typically are either trivial or infinitely dimensional. Finally, we study Navier-Stokes type models and prove that under our assumptions they have only steady state divergence free solutions. In particular, we solve the existence and uniqueness problem for the Navier-Stokes and Euler equations for a large class of fractals that are topologically one-dimensional but can have arbitrary Hausdorff and spectral dimensions. [ABSTRACT FROM AUTHOR]

We discuss the stabilization of symmetric products Symn (X) of a smooth projective variety X in the Grothendieck ring of varieties. For smooth projective surfaces X with non-zero h0(X, ωX ), these products do not stabilize; we conditionally show that they do not stabilize in another related sense, in response to a question of R. Vakil and M. Wood. There are analogies between such stabilization, the Dold-Thom theorem, and the analytic class number formula. Finally, we discuss conjectural Hodge-theoretic obstructions to the stabilization of symmetric products. We provide evidence for these obstructions by showing that the Newton polygon of the motivic zeta function associated to a curve equals the Hodge polygon of the curve. [ABSTRACT FROM AUTHOR]

We study the kernel and cokernel of the Frobenius map on the p-typical Witt vectors of a commutative ring, not necessarily of characteristic p. We give many equivalent conditions to surjectivity of the Frobenus map on both finite and infinite length Witt vectors. In particular, surjectivity on finite Witt vectors turns out to be stable under certain integral extensions; this provides a clean formulation of a strong generalization of Faltings's almost purity theorem from p-adic Hodge theory, incorporating recent improvements by Kedlaya and Liu, and by Scholze. [ABSTRACT FROM AUTHOR]

We introduce the notion of a symmetric Whitney tower cobordism between bordered 3-manifolds, aiming at the study of homology cobordism and link concordance. It is motivated by the symmetric Whitney tower approach to slicing knots and links initiated by T. Cochran, K. Orr, and P. Teichner. We give amenable Cheeger-Gromov ?-invariant obstructions to bordered 3- manifolds being Whitney tower cobordant. Our obstruction is related to and generalizes several prior known results, and also gives new interesting cases. As an application, our method applied to link exteriors reveals new structures on (Whitney tower and grope) concordance between links with nonzero linking number, including the Hopf link. [ABSTRACT FROM AUTHOR]

The subject of the present work is the de Rham part of noncommutative Hodge structures on the periodic cyclic homology of differential graded algebras and categories. We discuss explicit formulas for the corresponding connection on the periodic cyclic homology viewed as a bundle over the punctured formal disk. Our main result says that for the category of matrix factorizations of a polynomial the formulas reproduce, up to a certain shift, a well-known connection on the associated twisted de Rham cohomology which plays a central role in the geometric approach to the Hodge theory of isolated singularities. [ABSTRACT FROM AUTHOR]

We propose a new numerical approach for two-dimensional Maxwell's equations that is based on the Hodge decomposition for divergencefree vector fields. In this approach an approximate solution for Maxwell's equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P1 finite element method. [ABSTRACT FROM AUTHOR]

We propose a new numerical approach for two-dimensional Maxwell's equations that is based on the Hodge decomposition for divergencefree vector fields. In this approach an approximate solution for Maxwell's equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P1 finite element method. [ABSTRACT FROM AUTHOR]

The article describes the spectrum of the k-form Laplacian on conformally cusp Riemannian manifolds. It analyzes the essential spectrum using positive commutator methods and also establish a limiting absorption principle. Further it concludes that spectrum of the Laplacian on forms on hyperbolic form finite volume.

We give upper and lower bounds of the first eigenvalue of the Hodge Laplacian acting on smooth $ p$-th longest principal axis of the ellipsoid of maximal volume included in the domain (the John ellipsoid). Using John's theorem, we then give a spectral geometric interpretation of the bounds and relate the eigenvalues with the largest volume of a $ p$ [ABSTRACT FROM AUTHOR]

We give the complete description of the spectrum of $ (Spin(10,2),SL(2,\mathbb{R}))$, and we find that it only consists of a discrete spectrum. This is interesting as both groups are noncompact. Also, most of the representations occurring in the spectrum are unitary representations with nonzero cohomology. The representations we obtained have different Hodge types from the representations we obtain from classical theta liftings. [ABSTRACT FROM AUTHOR]

We study the analyticity of the semigroup generated by the Stokes operator equipped with Neumann-type boundary conditions on $L^p$ spaces in Lipschitz domains. Our strategy is to regularize this operator by considering the Hodge Laplacian, which has the additional property that it commutes with the Leray projection. [ABSTRACT FROM AUTHOR]

We study the deformations of the minimally elliptic surface singularity $N_{16}$. A standard argument reduces the study of the deformations of $N_{16}$ to the study of the moduli space of pairs $(C,L)$ consisting of a plane quintic curve and a line. We construct this moduli space in two ways: via the periods of $K3$ surfaces and by using geometric invariant theory (GIT). The GIT construction depends on the choice of the linearization. In particular, for one choice of linearization we recover the space constructed via $K3$ surfaces and for another we obtain the full deformation space of $N_{16}$. The two spaces are related by a series of explicit flips. In conclusion, by using the flexibility given by GIT and the standard tools of Hodge theory, we obtain a good understanding of the deformations of $N_{16}$. [ABSTRACT FROM AUTHOR]

We prove a stability of weakly almost conformal mappings in $W^{1, p}(\Omega;\mathbf {R}^n)$ for $p$ not too far below the dimension $ n$ by studying the $ W^{1, p}$-quasiconvex hull of the set $\mathcal C_n $ of conformal matrices. The study is based on coercivity estimates from the nonlinear Hodge decompositions and reverse Hölder inequalities from the Ekeland variational principle. [ABSTRACT FROM AUTHOR]