Hodge Type Research Articles

Modular Calabi-Yau fourfolds and connections to M-theory fluxes

In this work, we study the local zeta functions of Calabi-Yau fourfolds. This is done by developing arithmetic deformation techniques to compute the factor of the zeta function that is attributed to the horizontal four-form cohomology. This, in turn, is sensitive to the complex structure of the fourfold. Focusing mainly on examples of fourfolds with a single complex structure parameter, it is demonstrated that the proposed arithmetic techniques are both applicable and consistent. We present a Calabi-Yau fourfold for which a factor of the horizontal four-form cohomology further splits into two pieces of Hodge type (4, 0) + (2, 2) + (0, 4) and (3, 1) + (1, 3). The latter factor corresponds to a weight-3 modular form, which allows expressing the value of the periods in terms of critical values of the L-function of this modular form, in accordance with Deligne’s conjecture. The arithmetic considerations are related to M-theory Calabi-Yau fourfold compactifications with background four-form fluxes. We classify such background fluxes according to their Hodge type. For those fluxes associated to modular forms, we express their couplings in the low-energy effective action in terms of L-function values.

Mod p points on shimura varieties of parahoric level

Abstract We study the $\overline {\mathbb {F}}_{p}$ -points of the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level. We show that if the group is quasi-split, then every isogeny class contains the reduction of a CM point, proving a conjecture of Kisin–Madapusi–Shin. We, furthermore, show that the mod p isogeny classes are of the form predicted by the Langlands–Rapoport conjecture (cf. Conjecture 9.2 of [Rap05]) if either the Shimura variety is proper or if the group at p is unramified. The main ingredient in our work is a global argument that allows us to reduce the conjecture to the case of very special parahoric level. This case is dealt with in the Appendix by Zhou. As a corollary to our arguments, we determine the connected components of Ekedahl–Oort strata.

A local construction of zip period maps of Shimura varieties

Let S be the special fibre of a Shimura variety of Hodge type of good reduction at a fixed place above p . We give a local approach to the construction of the zip period map for S , which is used to define the Ekedahl–Oort strata of S . The method employed is p -adic and group theoretic in nature.

Connected components of affine Deligne–Lusztig varieties for unramified groups

For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.

G-valued crystalline deformation rings in the Fontaine–Laffaille range

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.

G-displays of Hodge type and formal p-divisible groups

G-displays of Hodge type and formal p-divisible groups

Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds

A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere S2 of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold M and a generic complex structure L∈S2, the complex manifold (M,L) has algebraic dimension 0. A stronger result is proven when the hypercomplex nilmanifold is abelian. Consider the Lie algebra of left-invariant vector fields of Hodge type (1,0) on the corresponding nilpotent Lie group with respect to some complex structure I∈S2. A hypercomplex nilmanifold is called abelian when this Lie algebra is abelian. We prove that all complex subvarieties of (M,L) for generic L∈S2 on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.

A geometric Jacquet-Langlands transfer for automorphic forms of higher weights

In this paper, we give a geometric construction of the Jacquet-Langlands transfer for automorphic forms of higher weights. Our method is by studying the geometry of the mod p p fibres of Hodge type Shimura varieties which satisfy a mild assumption and the cohomological correspondences between them.

Sections of Lagrangian fibrations on holomorphically symplectic manifolds and degenerate twistorial deformations

Let (M,I,Ω) be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration π:M↦X, and η a closed form of Hodge type (1,1)+(2,0) on X. We prove that Ω′:=Ω+π⁎η is again a holomorphically symplectic form, for another complex structure I′, which is uniquely determined by Ω′. The corresponding deformation of complex structures is called “degenerate twistorial deformation”. The map π is holomorphic with respect to this new complex structure, and X and the fibers of π retain the same complex structure as before. Let s be a smooth section of π. We prove that there exists a degenerate twistorial deformation (M,I′,Ω′) such that s is a holomorphic section.

On Orthogonal Local Models of Hodge Type

Abstract We study local models that describe the singularities of Shimura varieties of non-PEL type for orthogonal groups at primes where the level subgroup is given by the stabilizer of a single lattice. In particular, we use the Pappas–Zhu construction and we give explicit equations that describe an open subset around the “worst” point of orthogonal local models given by a single lattice. These equations display the affine chart of the local model as a hypersurface in a determinantal scheme. Using this, we prove that the special fiber of the local model is reduced and Cohen–Macaulay.

Honda–Tate theory for Shimura varieties

A Shimura variety of Hodge type is a moduli space for abelian varieties equipped with a certain collection of Hodge cycles. We show that the Newton strata on such varieties are nonempty provided that the corresponding group G is quasisplit at p, confirming a conjecture of Fargues and Rapoport in this case. Under the same condition, we conjecture that every mod p isogeny class on such a variety contains the reduction of a special point. This is a refinement of Honda–Tate theory. We prove a large part of this conjecture for Shimura varieties of PEL type. Our results make no assumption on the availability of a good integral model for the Shimura variety. In particular, the group G may be ramified at p.

On integral models of Shimura varieties

We show how to characterize integral models of Shimura varieties over places of the reflex field where the level subgroup is parahoric by formulating a definition of a “canonical” integral model. We then prove that in Hodge type cases and under a tameness hypothesis, the integral models constructed by the author and Kisin in previous work are canonical and, in particular, independent of choices. A main tool is a theory of displays with parahoric structure that we develop in this paper.

Vanishing theorems for Shimura varieties at unipotent level

We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite \Gamma_1(p^\infty) -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.

On two mod p period maps: Ekedahl–Oort and fine Deligne–Lusztig stratifications

Consider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime pge 5. Consider its perfectoid cover S^{text {ad}}(p^infty ) and the Hodge–Tate period map introduced by Caraiani and Scholze. We compare the pull-back to S^{text {ad}}(p^infty ) of the Ekedahl–Oort stratification on the mod p special fiber of a toroidal compactification of S and the pull back to S^text {ad}(p^infty ) of the fine Deligne–Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge–Tate period map. An application to the non-emptiness of Ekedhal–Oort strata is provided.

Stratifications in good reductions of Shimura varieties of abelian type

In this paper we study the geometry of good reductions of Shimura varieties of abelian type. More precisely, we construct the Newton stratification, Ekedahl-Oort stratification, and central leaves on the special fiber of a Shimura variety of abelian type at a good prime. We establish several basic properties of these stratifications, including the non-emptiness, closure relation and dimension formula, generalizing those previously known in the PEL and Hodge type cases. We also study the relations between these stratifications, both in general and in some special cases, such as those of fully Hodge-Newton decomposable type. We investigate the examples of quaternionic and orthogonal Shimura varieties in details.

Hecke eigensystems of automorphic forms (mod p) of Hodge type and algebraic modular forms

We show that the systems of prime-to-p Hecke eigenvalues arising from automorphic forms $$\pmod p$$ for a good prime p associated to an algebraic group $$G/{\mathbb {Q}}$$ of Hodge type are the same as those arising from algebraic modular forms $$\pmod p$$ associated to an inner form of G.

An analytic application of Geometric Invariant Theory

Given a compact Kähler manifold, Geometric Invariant Theory is applied to construct analytic GIT-quotients that are local models for a classifying space of (poly)stable holomorphic vector bundles containing the coarse moduli space of stable bundles as an open subspace. For local models invariant generalized Weil-Petersson forms exist on the parameter spaces, which are restrictions of symplectic forms on smooth ambient spaces. If the underlying Kähler manifold is of Hodge type, then the Weil-Petersson form on the moduli space of stable vector bundles is known to be the Chern form of a certain determinant line bundle equipped with a Quillen metric. It gives rise to a holomorphic line bundle on the classifying GIT space together with a continuous hermitian metric.

A note on a question of Markman

Let F be a vector bundle on a complex projective algebraic variety X. If F deforms along a first order deformation of X, its Mukai vector remains of Hodge type along this deformation. We prove an analogous statement for all polyvector fields, not only for those in H1(X,TX) corresponding to deformations of the complex structure. This answers a question of Markman. We also explore a Lie theoretic analogue of the statement above.

Some Remarks on Ekedahl–Oort Stratifications

We study independence of symplectic embeddings of the theory of Ekedahl–Oort stratifications on Shimura varieties of Hodge type, by comparing two different embeddings with a third one. The main results are as follows. 1. The Ekedahl–Oort stratification is independent of the choices of symplectic embeddings. 2. Under certain reasonable assumptions, there is certain functoriality for Ekedahl–Oort stratifications with respect to morphisms of Shimura varieties.

On Singular Distributions With Statistical Structure

In this paper, we extend our previous study regarding a Riemannian manifold endowed with a singular (or regular) distribution, generalizing Bochner’s technique and a statistical structure. Following the construction of an almost Lie algebroid, we define the central concept of the paper: The Weitzenböck type curvature operator on tensors, prove the Bochner–Weitzenböck type formula and obtain some vanishing results about the null space of the Hodge type Laplacian on a distribution.