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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]

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]

In anabelian geometry, various strong/desired forms of Grothendieck Conjecture-type results for hyperbolic curves over relatively small arithmetic fields — for instance, finite fields, number fields, or p -adic local fields — have been obtained by many researchers, especially by A. Tamagawa and S. Mochizuki. Let us recall that, in their proofs, the Weil Conjecture or p -adic Hodge theory plays an essential role. Therefore, to obtain such Grothendieck Conjecture-type results, it appears that the condition that the cyclotomic characters of the absolute Galois groups of the base fields are highly nontrivial is indispensable. On the other hand, in an author's recent joint work with Y. Hoshi and S. Mochizuki, we introduced the notion of TKND-AVKF-field [concerning the divisible subgroups of the groups of rational points of semi-abelian varieties] and obtained the semi-absolute version of the Grothendieck Conjecture for higher dimensional (≥2) configuration spaces associated to hyperbolic curves of genus 0 over TKND-AVKF-fields contained in the algebraic closure of the field of rational numbers. For instance, every [possibly, infinite] cyclotomic extension field of a number field is such a TKND-AVKF-field. In particular, this Grothendieck Conjecture-type result suggests that the condition that the cyclotomic character of the absolute Galois group of the base field under consideration is sufficiently nontrivial is, in fact, not indispensable for strong/desired form of anabelian phenomena. In the present paper, to pose another evidence for this observation, we prove the relative birational version of the Grothendieck Conjecture for smooth curves over TKND-AVKF-fields with a certain mild condition that every cyclotomic extension field of a number field satisfies. From the viewpoint of the condition on base fields, this result may be regarded as a partial generalization of F. Pop and S. Mochizuki's results on the birational version of the Grothendieck Conjecture for smooth curves. [ABSTRACT FROM AUTHOR]

Using Buium's theory of arithmetic differential characters, we construct a filtered F -isocrystal H (A) K associated to an abelian scheme A over a p -adically complete discrete valuation ring with perfect residue field. As a filtered vector space, H (A) K admits a natural map to the usual de Rham cohomology of A , but the Frobenius operator comes from arithmetic differential theory and is not the same as the usual crystalline one. When A is an elliptic curve, we show that H (A) K has a natural integral model H (A) , which implies an integral refinement of a result of Buium's on arithmetic differential characters. The weak admissibility of H (A) K depends on the invertibility of an arithmetic-differential modular parameter. Thus the Fontaine functor associates to suitably generic A a local Galois representation of an apparently new kind. [ABSTRACT FROM AUTHOR]

Given a complex variety X , a linear algebraic group G and a representation ρ of the fundamental group π 1 (X , x) into G , we develop a framework for constructing a functorial mixed Hodge structure on the formal local ring of the representation variety of π 1 (X , x) into G at ρ using mixed Hodge diagrams and methods of L ∞ algebras. We apply it in two geometric situations: either when X is compact Kähler and ρ is the monodromy of a variation of Hodge structure, or when X is smooth quasi-projective and ρ has finite image. [ABSTRACT FROM AUTHOR]

Abstract We study the Milnor monodromies of non-isolated hypersurface singularities and show that the reduced cohomology groups of the Milnor fibers are concentrated in the middle degree for some eigenvalues of the monodromies. As an application of this result, we give an explicit formula for some parts of their Jordan normal forms. [ABSTRACT FROM AUTHOR]

Abstract In [13] , initially motivated by questions about the Hodge line bundle of a Hodge-type Shimura variety, we singled out a generalization of the notion of minuscule character which we termed quasi-constant. Here we prove that the character of the Hodge line bundle is always quasi-constant. Furthermore, we classify the quasi-constant characters of an arbitrary connected, reductive group over an arbitrary field. As an application, we observe that, if μ is a quasi-constant cocharacter of an F p -group G , then our construction of group-theoretical Hasse invariants in loc. cit. applies to the stack G - Zip μ , without any restrictions on p , even if the pair (G , μ) is not of Hodge type and even if μ is not minuscule. We conclude with a more speculative discussion of some further motivation for considering quasi-constant cocharacters in the setting of our program outlined in loc. cit. [ABSTRACT FROM AUTHOR]

Abstract We transform the positive-genus real Gromov–Witten invariants of many real-orientable symplectic threefolds into signed counts of curves. These integer invariants provide lower bounds for counts of real curves of a given genus that pass through conjugate pairs of constraints. We conclude with some implications and related conjectures for Hodge integrals. [ABSTRACT FROM AUTHOR]

We study how the geometry of a projective variety X is reflected in the positivity properties of the diagonal Δ X considered as a cycle on X × X . We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of X . For example, when the cohomology class of Δ X is big, we prove that the Hodge groups H k , 0 ( X ) vanish for k > 0 . We also classify varieties of low dimension where the diagonal is nef and big. [ABSTRACT FROM AUTHOR]

We prove that the Beilinson regulator, which is a map from K -theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of E ∞ -ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over R . The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual K -theory spectrum is the source. To prove this result we compute the space of maps from the motivic K -theory spectrum to the motivic spectrum that represents absolute Hodge cohomology using the motivic Snaith theorem. We identify those maps which admit an E ∞ -refinement and prove a uniqueness result for these refinements. [ABSTRACT FROM AUTHOR]

Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford–Tate domains, arise as open G R -orbits in flag varieties G / P . We investigate Hodge-theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, we relate the Griffiths–Yukawa coupling to the variety of lines on G / P (under a minimal homogeneous embedding), construct a large class of polarized G R -orbits in G / P , and compute the associated Hodge-theoretic boundary components. An emphasis is placed throughout on adjoint flag varieties and the corresponding families of Hodge structures of levels two and four. [ABSTRACT FROM AUTHOR]

The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau–Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau–Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main result shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers defined using a mixed Hodge structure on the hypercohomology of the perverse sheaf. [ABSTRACT FROM AUTHOR]

An F -nilpotent local ring is a local ring ( R , m ) of prime characteristic defined by the nilpotence of the Frobenius action on its local cohomology modules H m i ( R ) . A singularity in characteristic zero is said to be of F -nilpotent type if its modulo p reduction is F -nilpotent for almost all p . In this paper, we give a Hodge-theoretic interpretation of three-dimensional normal isolated singularities of F -nilpotent type. In the graded case, this yields a characterization of these singularities in terms of divisor class groups and Brauer groups. [ABSTRACT FROM AUTHOR]

Associated to any divisor in the Chow ring of a simplicial tropical fan, we construct a family of polytopal complexes, called normal complexes, which we propose as an analogue of the well-studied notion of normal polytopes from the setting of complete fans. We describe certain closed convex polyhedral cones of divisors for which the "volume" of each divisor in the cone—that is, the degree of its top power—is equal to the volume of the associated normal complexes. For the Bergman fan of any matroid with building set, we prove that there exists an open family of such cones of divisors with nonempty interiors. We view the theory of normal complexes developed in this paper as a polytopal model underlying the combinatorial Hodge theory pioneered by Adiprasito, Huh, and Katz. [ABSTRACT FROM AUTHOR]

We introduce a version of the P=W conjecture relating the Borel–Moore homology of the stack of representations of the fundamental group of a genus g Riemann surface with the Borel–Moore homology of the stack of degree zero semistable Higgs bundles on a smooth projective complex curve of genus g. In order to state the conjecture we propose a construction of a canonical isomorphism between these Borel–Moore homology groups. We relate the stacky P=W conjecture with the original P=W conjecture concerning the cohomology of smooth moduli spaces, and the PI=WI conjecture concerning the intersection cohomology groups of singular moduli spaces. In genus zero and one, we prove the conjectures that we introduce in this paper. [ABSTRACT FROM AUTHOR]

Motivated by definitions in mixed Hodge theory, we define the weight filtration and the monodromy weight filtration on the combinatorial intersection cohomology of a fan. These filtrations give a natural definition of the multivariable invariants of subdivisions of polytopes, lattice polytopes and fans, namely the mixed h -polynomial, the refined limit mixed h ⁎ -polynomial, and the mixed cd -index, defined by Katz–Stapledon and Dornian–Katz–Tsang. Previously, only the refined limit mixed h ⁎ -polynomial had a geometric interpretation, which came from filtrations on the cohomology of a schön hypersurface. Consequently, we generalize a positivity result on the mixed h -polynomial by Katz and Stapledon using the relative hard Lefschetz theorem of Karu. [ABSTRACT FROM AUTHOR]

Kato and Usui developed a theory of partial compactifications for quotients of period domains D by arithmetic groups Γ, in an attempt to generalize the toroidal compactifications of Ash-Mumford-Rapoport-Tai to non-classical cases. Their partial compactifications, which aim to fully compactify the images of period maps, rely on the choice of a fan which is strongly compatible with Γ. In particular, they conjectured the existence of a complete fan, which would serve to simultaneously compactify all period maps of a given type. In this article, we briefly review the theory, and construct a fan which compactifies the image of a period map arising from a particular two-parameter family of Calabi-Yau threefolds studied by Hosono and Takagi, with Hodge numbers (1 , 2 , 2 , 1). On the other hand, we disprove the existence of complete fans in some general cases, including the (1 , 2 , 2 , 1) case. [ABSTRACT FROM AUTHOR]

We construct a hermitian metric on the classifying spaces of graded-polarized mixed Hodge structures and prove analogs of the strong distance estimate [6] between an admissible period map and the approximating nilpotent orbit. We also consider the asymptotic behavior of the biextension metric introduced by Hain [12] , analogs of the norm estimates of [19] and the asymptotics of the naive limit Hodge filtration considered in [21] . [ABSTRACT FROM AUTHOR]

We study holonomic D -modules on SL n ( C ) × C n , called mirabolic modules , analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus. We introduce an analogue of Verdier's specialization functor for representations of Cherednik algebras which agrees, on category O , with the restriction functor of Bezrukavnikov and Etingof. In type A , we also consider a Verdier specialization functor on mirabolic D -modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic D -modules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known Hodge-theoretic purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof. [ABSTRACT FROM AUTHOR]

We prove birational boundedness results on complete intersections with trivial canonical class of base point free divisors in (some version of) Fano varieties. Our results imply in particular that Batyrev–Borisov toric construction produces only a bounded set of Hodge numbers in any given dimension, even as the codimension is allowed to grow. [ABSTRACT FROM AUTHOR]

We introduce an effective method to determine the ∂ ¯ -harmonic forms on the Kodaira-Thurston manifold endowed with an almost complex structure and an almost Hermitian metric. Using the Weil-Brezin transform, we reduce the elliptic PDE system to countably many linear ODE systems. By solving a fundamental problem on linear ODE systems, the problem of finding ∂ ¯ -harmonic forms is equivalent to a generalised Gauss circle problem. We demonstrate two remarkable applications. First, the dimension of the almost complex ∂ ¯ -Hodge numbers on the Kodaira-Thurston manifold could be arbitrarily large. Second, Hodge numbers vary with different choices of almost Hermitian metrics. This answers a question of Kodaira and Spencer in Hirzebruch's 1954 problem list. [ABSTRACT FROM AUTHOR]

Abstract: We prove a new formula for the Hirzebruch–Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern–Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusiński and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern–Milnor classes of complete intersections with isolated singularities. [Copyright &y& Elsevier]

Abstract: A category of motivic “sheaves” is constructed over a variety in characteristic 0 using Nori’s method. Although the relationship with many alternative constructions remains to be clarified, it does have many of the properties one expects. For example, it is abelian and has Betti, Hodge and -adic realizations, and it has a Tannakian subcategory of motivic local systems. [Copyright &y& Elsevier]

Abstract: We prove that suitably truncated topological log-syntomic-étale homotopy -theory of proper semistable schemes in mixed characteristic surjects onto the Kummer log-étale -adic -theory of the generic fiber. As a corollary we get that the log-syntomic-étale cohomology is a direct factor of the log-syntomic-étale cohomology of homotopy -theory sheaves. The proofs use -adic Hodge theory computations of -adic nearby cycles. [Copyright &y& Elsevier]

Abstract: We prove a representation-theoretic version of Borisov–Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex cohomology. We conjecture that the corresponding orbifold Hodge diamonds are also mirror. When is the Fermat quintic in , and is a -equivariant, toric resolution of its mirror , we deduce that for any subgroup of the alternating group , the -Hilbert schemes and are smooth Calabi–Yau threefolds with (explicitly computed) mirror Hodge diamonds. [Copyright &y& Elsevier]

Abstract: We prove a formula expressing the motivic integral (Loeser and Sebag, 2003) of a K3 surface over with semi-stable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces. [Copyright &y& Elsevier]

Abstract: The isomonodromic tau function of the Fuchsian differential equations associated to Frobenius structures on Hurwitz spaces can be viewed as a section of a line bundle on the space of admissible covers. We study the asymptotic behavior of the tau function near the boundary of this space and compute its divisor. This yields an explicit formula for the pullback of the Hodge class to the space of admissible covers in terms of the classes of compactification divisors. [Copyright &y& Elsevier]

Abstract: We prove that any compact complex homogeneous space with vanishing first Chern class, after an appropriate deformation of the complex structure, admits a homogeneous Calabi–Yau with torsion structure, provided that it also has an invariant volume form. A description of such spaces among the homogeneous C-spaces is given as well as many examples and a classification in the 3-dimensional case. We calculate the cohomology ring of some of the examples and show that in dimension 14 there are infinitely many simply-connected spaces admitting such a structure with the same Hodge numbers and torsional Chern classes. We provide also an example solving the Strominger''s equations in heterotic string theory. [Copyright &y& Elsevier]

Abstract: The Milnor–Hirzebruch class of a locally complete intersection X in an algebraic manifold M measures the difference between the (Poincaré dual of the) Hirzebruch class of the virtual tangent bundle of X and, respectively, the Brasselet–Schürmann–Yokura (homology) Hirzebruch class of X. In this note, we calculate the Milnor–Hirzebruch class of a globally defined algebraic hypersurface X in terms of the corresponding Hirzebruch invariants of vanishing cycles and singular strata in a Whitney stratification of X. Our approach is based on Schürmann''s specialization property for the motivic Hirzebruch class transformation of Brasselet–Schürmann–Yokura. The present results also yield calculations of Todd, Chern and L-type characteristic classes of hypersurfaces. [Copyright &y& Elsevier]

Abstract: Let be a link. We study the Heegaard Floer homology of the branched double-cover of , branched along L. When L is an alternating link, of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose term is a suitable variant of Khovanov''s homology for the link L, converging to the Heegaard Floer homology of . [Copyright &y& Elsevier]

This paper extends Dolbeault cohomology and its surrounding theory to arbitrary almost complex manifolds. We define a spectral sequence converging to ordinary cohomology, whose first page is the Dolbeault cohomology, and develop a harmonic theory which injects into Dolbeault cohomology. Lie-theoretic analogues of the theory are developed which yield important calculational tools for Lie groups and nilmanifolds. Finally, we develop applications to maximally non-integrable manifolds, including nearly Kähler 6-manifolds, and show Dolbeault cohomology can be used to prohibit the existence of nearly Kähler metrics. [ABSTRACT FROM AUTHOR]

It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily–Borel compactifications of appropriate ball quotients. In this paper, we show that their respective toroidal compactifications are isomorphic to moduli spaces of stable pairs as defined in the context of the MMP. Moreover, we give a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case of eight labeled points in the projective line. [ABSTRACT FROM AUTHOR]

We study the Hodge numbers f p , q of Landau–Ginzburg models as defined by Katzarkov, Kontsevich, and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau–Ginzburg mirrors of Fano threefolds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold X there is a natural LG mirror (Y , w) so that h p , q (X) = f 3 − q , p (Y , w). [ABSTRACT FROM AUTHOR]

This paper develops a theory of analytic geometry over the field with one element. The approach used is the analytic counter-part of the Toën-Vaquié theory of schemes over F 1 , i.e. the base category relative to which we work out our theory is the category of sets endowed with norms (or families of norms). Base change functors to analytic spaces over Banach rings are studied and the basic spaces of analytic geometry (e.g. polydisks) are recovered as a base change of analytic spaces over F 1. We conclude by discussing some applications of our theory to the theory of the Fargues-Fontaine curve and to the ring of Witt vectors. [ABSTRACT FROM AUTHOR]