Hermitian Quasi-Modular Forms

• Hermitian quasi-modular forms are extensions of automorphic forms that recover modularity through controlled non-holomorphic corrections and Hodge-theoretic structures.
• They are constructed via diverse methods including Movasati’s Hodge filtration framework, theta-series completions, and differential operators on U(n,n) preserving bounded depth.
• Applications range from providing algebraic coordinates on Shimura varieties to generating series of special cycles with explicit boundary corrections.

Searching arXiv for the cited papers and closely related work on Hermitian quasi-modular forms. arxiv_search(query="^^^^^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^^^^^"Hermitian quasi-modular forms\"^^^^^^^^2 OR \2^^^^^^^^"quasi-modular forms attached to Hodge structures\"", max_results=^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^) arxiv_search(query="(&&&^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^&&&)", max_results=5) Searching arXiv by title to confirm the core source on Hodge-theoretic quasi-modular forms. Hermitian quasi-modular forms are extensions of automorphic forms on Hermitian symmetric domains in which modularity is recovered only after allowing controlled correction terms, or, in a Hodge-theoretic formulation, after pulling back regular functions from a universal parameter space along a local inverse of a period map. The subject appears in several closely related settings: Movasati’s construction of quasi-automorphic forms attached to Hodge structures on Hermitian symmetric period domains (&&&^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^&&&), the theory of polynomial weighted theta functions and cycle-valued generating series on unitary Shimura varieties (&&&^^^^2 OR \2^^^^&&&), and the almost holomorphic framework produced by explicit differential operators on Hermitian modular forms for PRESERVED_PLACEHOLDER_^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^ (Takeda, 23 Jun 2025). Across these settings, the common structure is a holomorphic object together with a canonical non-holomorphic completion, differential stability, and a modular transformation law governed by Hermitian upper half-spaces, Weil representations, or Hodge-theoretic monodromy data.

^^^^2 OR \2^^^^. Hodge-theoretic origin on Hermitian symmetric period domains

In Movasati’s framework, one begins with a fixed complex vector space PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^, a weight $m$, Hodge numbers $h^{i,m-i}$, a Hodge filtration

$$F_0:\{0\}=F^{m+1}\subset F^m\subset \cdots \subset F^1\subset F^0=V_0,$$

and a non-degenerate bilinear form $Y_0$ satisfying the polarization condition $Y_0(F^j,F^k)=0$ whenever $j+k>m$. Instead of working directly with the Griffiths period domain $D$, the construction introduces a space $U$ of polarized lattices in PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^ compatible with the fixed filtration and polarization. The algebraic group

PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2^^^^

acts on PRESERVED_PLACEHOLDER_^^^^2 OR \22^^^^, and the quotient is canonically identified with the classical moduli of polarized Hodge structures, namely PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^3 (&&&^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^&&&).

This reformulation is significant in the Hermitian symmetric case, because PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^4 is then one of the classical Hermitian domains arising from Shimura data. The examples explicitly indicated are the Siegel upper half-spaces PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^5, type IV domains attached to signatures PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^6, and complex balls for PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^7. In the classical picture, automorphic forms on PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^8 give algebraic coordinates on the quotient via the Baily–Borel theorem; in the modified picture, the same moduli problem is expressed through PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^9, where the filtration-compatible coordinates are adapted to period matrices and the Gauss–Manin connection (&&&^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^&&&).

A central structural ingredient is Griffiths transversality. In a filtration-compatible basis with Gauss–Manin connection matrix $m$^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^, the horizontal distribution on $m$^^^^2 OR \2^^^^ is described by vanishing conditions on certain entries of $m$2. This encodes the differential constraints satisfied by period maps and provides the mechanism through which quasi-automorphic functions acquire canonical derivations.

2. Quasi-automorphic forms from universal period maps

The Hodge-theoretic definition of quasi-automorphic forms is formulated through universal enhanced families. Assuming the existence of an affine variety $m$3 carrying a full $m$4-equivariant universal family and assuming local Torelli, a geometric period map

$m$5

admits a local inverse on a suitable open subset $m$6. The quasi-automorphic algebra is then defined by

$m$7

the pullback of regular functions on $m$8 to holomorphic functions on $m$9 (&&&^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^&&&).

This construction generalizes the Kaneko–Zagier viewpoint on elliptic quasi-modular forms. The key point is that the functions in $h^{i,m-i}$^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^ are not arbitrary holomorphic functions on the Hermitian domain: they are holomorphic realizations of algebraic coordinates on a universal parameter space, transported through period maps. In this sense, automorphic forms occur as the depth-zero part, while the larger algebra incorporates differential or jet-like corrections. The paper does not formalize a general jet-bundle theory or a depth filtration, but it explicitly states that the resulting ring is closed under natural derivations induced by the Gauss–Manin connection (&&&^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^&&&).

In the elliptic case, the universal parameters $h^{i,m-i}$^^^^2 OR \2^^^^ transform under

$h^{i,m-i}$2

by

$h^{i,m-i}$3

and the pullback algebra recovers the quasi-modular ring generated by the Eisenstein series corresponding to $h^{i,m-i}$4. In the Siegel case, the same philosophy defines a ring of Siegel quasi-modular forms through the composition $h^{i,m-i}$5, and the algebra is closed under the derivations $h^{i,m-i}$6 (&&&^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^&&&).

The resulting notion is broader than the classical scalar theory. It is attached not merely to a discrete group acting on a Hermitian domain, but to the Hodge-theoretic data of filtration, polarization, period map, and Gauss–Manin connection.

3. Hermitian modular forms, weighted theta series, and the formal definition

In the unitary setting of signature $h^{i,m-i}$7, Hermitian quasi-modular forms are developed on the Hermitian upper half-space

$h^{i,m-i}$8

with fractional linear action

$h^{i,m-i}$9

by $$F_0:\{0\}=F^{m+1}\subset F^m\subset \cdots \subset F^1\subset F^0=V_0,$$^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^. A scalar-valued Hermitian modular form of weight $$F_0:\{0\}=F^{m+1}\subset F^m\subset \cdots \subset F^1\subset F^0=V_0,$$^^^^2 OR \2^^^^ is a holomorphic $$F_0:\{0\}=F^{m+1}\subset F^m\subset \cdots \subset F^1\subset F^0=V_0,$$2 satisfying

$$F_0:\{0\}=F^{m+1}\subset F^m\subset \cdots \subset F^1\subset F^0=V_0,$$3

and the vector-valued theory is defined using the genus-$$F_0:\{0\}=F^{m+1}\subset F^m\subset \cdots \subset F^1\subset F^0=V_0,$$4 Weil representation $$F_0:\{0\}=F^{m+1}\subset F^m\subset \cdots \subset F^1\subset F^0=V_0,$$5 on $$F_0:\{0\}=F^{m+1}\subset F^m\subset \cdots \subset F^1\subset F^0=V_0,$$6 (&&&^^^^2 OR \2^^^^&&&).

The quasi-modular theory is built from the finite-dimensional spaces

$$F_0:\{0\}=F^{m+1}\subset F^m\subset \cdots \subset F^1\subset F^0=V_0,$$7

These weighted polynomial spaces carry lowering operators $$F_0:\{0\}=F^{m+1}\subset F^m\subset \cdots \subset F^1\subset F^0=V_0,$$8, a global lowering operator $$F_0:\{0\}=F^{m+1}\subset F^m\subset \cdots \subset F^1\subset F^0=V_0,$$9, a raising operator $Y_0$^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^, and an operator

$Y_0$^^^^2 OR \2^^^^

satisfying

$Y_0$2

Thus $Y_0$3 is an $Y_0$4-triple on $Y_0$5 (&&&^^^^2 OR \2^^^^&&&).

Shimura’s modularity criterion then produces completed theta series. For $Y_0$6,

$Y_0$7

transforms as a Hermitian modular form of weight $Y_0$8 with representation $Y_0$9. Its holomorphic part

$Y_0(F^j,F^k)=0$^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^

is, by definition, a Hermitian quasi-modular form; the space spanned by such holomorphic parts is denoted $Y_0(F^j,F^k)=0$^^^^2 OR \2^^^^ (&&&^^^^2 OR \2^^^^&&&).

The depth is encoded by the completion: for genus $Y_0(F^j,F^k)=0$2, the expansion contains terms up to $Y_0(F^j,F^k)=0$3. This gives a direct higher-rank analogue of the phenomenon that the classical $Y_0(F^j,F^k)=0$4 is not modular, but becomes modular after adding a non-holomorphic correction.

4. Boundary geometry, special cycles, and the Kudla program

A major source of Hermitian quasi-modular forms is the generating series of special cycles on compactified unitary Shimura varieties. Let $Y_0(F^j,F^k)=0$5 have signature $Y_0(F^j,F^k)=0$6, let $Y_0(F^j,F^k)=0$7 be an $Y_0(F^j,F^k)=0$8-lattice, and let $Y_0(F^j,F^k)=0$9 be the resulting complex ball quotient. For $j+k>m$^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^, one forms cycle-valued generating series

$j+k>m$^^^^2 OR \2^^^^

Kudla–Millson showed that the cohomology class of $j+k>m$2 is a holomorphic Hermitian modular form of weight $j+k>m$3, and the 2^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \22^^^^5 development extends this to compactifications and boundary corrections (&&&^^^^2 OR \2^^^^&&&).

The boundary geometry is explicit. For an isotropic line $j+k>m$4, the boundary divisor $j+k>m$5 is an abelian variety $j+k>m$6, where $j+k>m$7. Its normal bundle is anti-ample, and

$j+k>m$8

For $j+k>m$9, the boundary cycles $D$^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^ admit harmonic representatives built from the Gram polynomial

$D$^^^^2 OR \2^^^^

with

$D$2

This provides the geometric input for the non-holomorphic completion (&&&^^^^2 OR \2^^^^&&&).

The crucial completion theorem states that the boundary generating series

$D$3

admits a non-holomorphic correction

$D$4

such that the sum transforms as a Hermitian modular form of weight $D$5. Here

$D$6

The global consequence is that the generating series of Zariski closures of special cycles in the toroidal compactification is a Hermitian quasi-modular form: it becomes modular after adding precisely these boundary-induced non-holomorphic terms (&&&^^^^2 OR \2^^^^&&&).

Theorems ^^^^2 OR \2^^^^.2 and ^^^^2 OR \2^^^^.3 establish this up to the middle codimension $D$7. The mechanism relies on the Splitting Lemma, the Lefschetz decomposition on boundary cohomology, and the $D$8-calculus on the polynomial weights.

5. Almost holomorphic realizations and differential operators on $D$9

A complementary viewpoint treats Hermitian quasi-modularity through explicit differential operators on Hermitian modular forms for $U$^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^. The Hermitian upper half-space of degree $U$^^^^2 OR \2^^^^ is

$U$2

with action

$U$3

for $U$4. A vector-valued form of weight $U$5 transforms by the automorphy factor

$U$6

through

$U$7

In this setting, an almost holomorphic form of depth $U$8 is a function admitting an expansion

$U$9

where PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^, each PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2 OR \2^^^^ is holomorphic, and PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \22^^^^ is a polynomial in the entries of PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^3. The exposition associated with the paper identifies Hermitian quasi-modular forms with these almost holomorphic forms of bounded depth (Takeda, 23 Jun 2025).

The operators are indexed by two-variable spherical pluriharmonic polynomials. For PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^4 in the appropriate polynomial space, one defines

PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^5

These operators are constructed using explicit bases of pluriharmonic polynomials, including monomial and descending bases, and are compatible with the representation theory of PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^6. Their effect on Fourier expansions is direct: PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^7 for homogeneous PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^8 of degree PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^9 (Takeda, 23 Jun 2025).

The connection with quasi-modularity arises from the archimedean transformation formula

PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^

Because the entries of PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2 OR \2^^^^ are rational in PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \22^^^^ and PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2^^^^3, and hence yield polynomial dependence on PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2^^^^4 after normalization, the images of holomorphic modular forms under PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2^^^^5 are almost holomorphic of depth bounded by PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2^^^^6. This gives a Hermitian analogue of the Maass–Shimura and Ibukiyama paradigms: the differential operators preserve a quasi-modular space and control its depth (Takeda, 23 Jun 2025).

The same framework produces exact pullback formulas for Hermitian Eisenstein series and explicit archimedean constants, but its main conceptual relevance for quasi-modularity is the systematic production of bounded-depth PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2^^^^7-expansions from holomorphic input.

6. Examples, comparisons, and unresolved directions

The elliptic case remains the basic model. In Movasati’s construction, the universal parameter space PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2^^^^8 for the Weierstrass family yields generators corresponding to PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2^^^^9, and the non-holomorphic transformation law of PRESERVED_PLACEHOLDER_^^^^2 OR \22(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^ displays the defining quasi-modular anomaly. The same paper states that the Hodge-theoretic construction recovers the Kaneko–Zagier algebra in genus PRESERVED_PLACEHOLDER_^^^^2 OR \22 OR \2^^^^, while the Siegel case furnishes higher-dimensional Hermitian symmetric examples with closure under the derivations PRESERVED_PLACEHOLDER_^^^^2 OR \222^^^^ (&&&^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^&&&).

In the unitary theta-series framework, the genus-PRESERVED_PLACEHOLDER_^^^^2 OR \22^^^^3 correction term is completely explicit: PRESERVED_PLACEHOLDER_^^^^2 OR \22^^^^4 The corrected divisor class

PRESERVED_PLACEHOLDER_^^^^2 OR \22^^^^5

has a holomorphic generating series of weight PRESERVED_PLACEHOLDER_^^^^2 OR \22^^^^6. The analogy with the classical completion

PRESERVED_PLACEHOLDER_^^^^2 OR \22^^^^7

is stated directly: in the Hermitian setting, the scalar correction is replaced by a modular coefficient PRESERVED_PLACEHOLDER_^^^^2 OR \22^^^^8, the inverse imaginary part PRESERVED_PLACEHOLDER_^^^^2 OR \22^^^^9 is replaced by PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010)2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^, and the boundary polarization PRESERVED_PLACEHOLDER_^^^^2 OR \2(Movasati, 2010)2 OR \2^^^^ enters through wedge products (&&&^^^^2 OR \2^^^^&&&).

Several misconceptions are clarified by the existing literature. First, “Hermitian quasi-modular form” is not restricted to a single definition. In the Hodge-theoretic setting it means a pullback algebra PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^32; in the unitary theta-series setting it means the holomorphic part of a completed Hermitian modular form; and in the PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^33 differential-operator setting it is modeled by almost holomorphic expansions in PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^34. Second, the abstract of (&&&^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^&&&) uses “Hermition,” but the correct geometric term is “Hermitian symmetric domain” (&&&^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^&&&).

The current theory is developed unevenly across domains. The elliptic case is explicit, the unitary case provides a full completion theory for polynomial weighted theta functions and special cycles up to middle degree, and the PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^35 operator calculus gives explicit bases, generating functions, and pullback formulas. By contrast, explicit generators, higher-rank Ramanujan-type systems, Fourier–Jacobi expansions, and Hecke operators for Hermitian quasi-modular forms are not developed in the cited works, and broader verification of the universal algebraization conjectures in the Hodge-theoretic setting remains open (&&&^^^^2(Movasati, 2010) OR (Greer et al., 17 Jul 2025) OR (Takeda, 23 Jun 2025) OR \2^^^^&&&). A plausible implication is that future work will consolidate these perspectives into a unified theory in which boundary corrections, period maps, and PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^36-expansions are understood as different realizations of the same quasi-modular phenomenon.