Holomorphic Differential Forms Research Articles

The Hilbert series of the superspace coinvariant ring

Abstract Let $\Omega _n$ be the ring of polynomial-valued holomorphic differential forms on complex n-space, referred to in physics as the superspace ring of rank n. The symmetric group ${\mathfrak {S}}_n$ acts diagonally on $\Omega _n$ by permuting commuting and anticommuting generators simultaneously. We let $SI_n \subseteq \Omega _n$ be the ideal generated by ${\mathfrak {S}}_n$ -invariants with vanishing constant term and study the quotient $SR_n = \Omega _n / SI_n$ of superspace by this ideal. We calculate the doubly-graded Hilbert series of $SR_n$ and prove an ‘operator theorem’, which characterizes the harmonic space $SH_n \subseteq \Omega _n$ attached to $SR_n$ in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results that were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Colmenarejo, Li, Machacek, Sulzgruber, Swanson, Wallach and Zabrocki.

The Eisenstein cycles and Manin–Drinfeld properties

Abstract Let Γ be a subgroup of finite index of SL 2 ⁡ ( 𝐙 ) {\operatorname{SL}_{2}(\mathbf{Z})} . We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian J Γ {J_{\Gamma}} of the corresponding modular curve X Γ {X_{\Gamma}} . Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case Γ ⊂ Γ ⁢ ( 2 ) {\Gamma\subset\Gamma(2)} , where we can consider convenient generalized Jacobians instead of J Γ {J_{\Gamma}} . We relate the Eisenstein classes to the scattering constants attached to Eisenstein series. Finally, we illustrate our approach by considering Fermat curves.

Holomorphic differential forms on moduli spaces of stable curves

We prove that the space of holomorphic p-forms on the moduli space overline{mathcal {M}}_{g,n} of stable curves of genus g with n marked points vanishes for p=14, 16, 18 unconditionally and also for p=20 under a natural assumption in the case g=3. This result is consistent with the Langlands program and it is obtained by applying the Arbarello–Cornalba inductive approach to the cohomology of moduli spaces.

Vanishing for Hodge ideals on toric varieties

AbstractIn this article we construct a Koszul‐type resolution of the exterior power of the sheaf of holomorphic differential forms on smooth toric varieties and use this to prove a Nadel‐type vanishing theorem for Hodge ideals associated to effective ‐divisors on smooth projective toric varieties. This extends earlier results of Mustaţă and Popa.

Vector-Valued Modular Forms and the Gauss Map

We use the gradients of theta functions at odd two-torsion points – thought of as vector-valued modular forms – to construct holomorphic differential forms on the moduli space of principally polarized abelian varieties, and to characterize the locus of decomposable abelian varieties in terms of the Gauss images of two-torsion points.

Haar expectations of ratios of random characteristic polynomials

We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups K = O(N), SO(N), and USp(N). To that end, we start from the Clifford-Weyl algebera in its canonical realization on the complex of holomorphic differential forms for a C-vector space V. From it we construct the Fock representation of an orthosymplectic Lie superalgebra osp associated to V. Particular attention is paid to defining Howe's oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of sp in osp. In the process, by pushing the semigroup representation to its boundary and arguing by continuity, we provide a construction of the Shale-Weil-Segal representation of the metaplectic group. To deal with a product of n ratios of characteristic polynomials, we let V = C^n \otimes C^N where C^N is equipped with its standard K-representation, and focus on the subspace of K-equivariant forms. By Howe duality, this is a highest-weight irreducible representation of the centralizer g of Lie(K) in osp. We identify the K-Haar expectation of n ratios with the character of this g-representation, which we show to be uniquely determined by analyticity, Weyl group invariance, certain weight constraints and a system of differential equations coming from the Laplace-Casimir invariants of g. We find an explicit solution to the problem posed by all these conditions. In this way we prove that the said Haar expectations are expressed by a Weyl-type character formula for all integers N \ge 1. This completes earlier work by Conrey, Farmer, and Zirnbauer for the case of U(N).

Symmetric differential forms on complete intersection varieties and applications

In this paper we study the cohomology of the symmetric powers of the cotangent bundle of complete intersection varieties in projective space. We provide an explicit description of some of those cohomology groups in terms of the equations defining the complete intersection. As a first application, we give a new example illustrating the fact that the dimension of the space of holomorphic symmetric differential forms is not deformation invariant. Then, as our main application, we construct varieties with ample cotangent bundle, providing new results towards a conjecture of Debarre.

The Poincaré index and the χy-characteristic of Hirzebruch

In the paper, we discuss several methods for computing the homology of contravariant and covariant versions of the classical De Rham complex on analytic spaces. Our approach is based on the theory of holomorphic and regular meromorphic differential forms, and is applicable in different settings depending on concrete types of varieties. Among other things, we describe how to compute by elementary calculations the homological index of vector fields and differential forms given on Cohen–Macaulay curves, graded normal surfaces, complete intersections and some others. In these situations, making use of ideas of X. Gómez-Mont, we derive explicit expressions for the local topological index of Poincaré and its generalizations. Furthermore, applying similar methods to the study of certain other complexes, we investigate some challenges, relating to the computation of classical topological–analytical invariants, such as the Euler characteristic of the Milnor fibre of an isolated singularity, the multiplicity of the discriminant of the versal deformation, the dimension of torsion and cotorsion modules, and so on.

Analogues of Mathai–Quillen forms in sheaf cohomology and applications to topological field theory

We construct sheaf-cohomological analogues of Mathai-Quillen forms, that is, holomorphic bundle-valued differential forms whose cohomology classes are independent of certain deformations, and which are believed to possess Thom-like properties. Ordinary Mathai-Quillen forms are special cases of these constructions, as we discuss. These sheaf-theoretic variations arise physically in A/2 and B/2 pseudo-topological field theories, and we comment on their origin and role.

A weak version of the Lipman–Zariski conjecture

The complex-analytic version of the Lipman–Zariski conjecture says that a complex space is smooth if its tangent sheaf is locally free. We prove the following weak version of the conjecture: A normal complex space is smooth if its tangent sheaf is locally free and locally admits a basis consisting of pairwise commuting vector fields. The main tool used in the proof of our result is a new extension theorem for reflexive differential forms on a normal complex space. It says that a closed holomorphic differential form of degree one defined on the smooth locus of a normal complex space can be extended to a holomorphic differential form on any resolution of singularities of the complex space.

A three dimensional ball quotient

In this paper we determine a very particular example of a Picard modular variety of general type. On its non-singular models there exist many holomorphic differential forms. In a forthcoming paper we will show that one can construct Calabi–Yau manifolds by considering quotients of this variety and resolving singularities.

Non-abelian Reciprocity Laws on a Riemann Surface

On a Riemann surface, there are relations among the periods of holomorphic differential forms called Riemann’s relations. We prove them, using iterated integrals. In this paper, we generalize these relations to relations among generating series of iterated integrals. The generating series gives infinitely many relations—one for each coefficient of the generating series. The lower-order terms give the well-known classical relations. The new result is reciprocity for the higher degree terms, which give non-trivial relations among iterated integrals on a Riemann surface. As an application we refine the definition of Manin’s non-commutative modular symbol in order to include Eisenstein series. Finally, this paper contains some constructions needed for multi-dimensional reciprocity laws like a refinement of one of the Kato–Parshin reciprocity laws (Horozov, I. “A refinement of the Parshin symbol for surfaces”. Preprint, arXiv.org, AG, 1002.2698].

On the stability of the tangent bundle of a hypersurface in a Fano variety

Let $M$ be a complex projective Fano manifold whose Picard group is isomorphic to $\mathbb{Z}$ and the tangent bundle $TM$ is semistable. Let $Z \subset M$ be a smooth hypersurface of degree strictly greater than degree($TM$)$(\mathrm{dim}_{\mathbb{C}} Z-1)/(2\mathrm{dim}_{\mathbb{C}} Z-1)$ and satisfying the condition that the inclusion of $Z$ in $M$ gives an isomorphism of Picard groups. We prove that the tangent bundle of $Z$ is stable. A similar result is proved also for smooth complete intersections in $M$. The main ingredient in the proof of it is a vanishing result for the top cohomology of the twisted holomorphic differential forms on $Z$.

A sextic holomorphic form of affine surfaces with constant affine mean curvature

We extend an original idea of Calabi for affine maximal surfaces and define a sextic holomorphic differential form for affine surfaces with constant affine mean curvature. We get some rigidity results for affine complete surfaces by using this sextic holomorphic form.

Optimal Global Conformal Surface Parameterization for Visualization

All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenomena, such as heat diffusion, electric-magnetic fields on the surface. Good parameterization is crucial for simulation and visualization. This paper gives an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. It relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories. Algorithms are developed to modify topology, locate zero points, and determine cohomology types of differential forms. The implementation is based on finite dimensional optimization method. The optimal pa- rameterization is intrinsic to the geometry, preserving angular structure, and can play an important role in various applications including texture mapping, remeshing, morphing and simulation. The method is demonstrated by visu- alizing the Riemann surface structure of real surfaces represented as triangle meshes.

Meromorphic Functions on a Riemann Surface to a Locally Semi-convex Space

In this article, vector-valued holomorphic and meromorphic functions on a Riemann surface to a complete Hausdorff locally semi-convex space are discussed. By introducing the concepts of vector-valued holomorphic and meromorphic differential forms, Cauchy's theorem and the Residue theorem of a vector-valued differential form on a Riemann surface are proved. Using the theory on the operator and the theory of a cohomology of a sheaf, we give a proof of the Mittag-Leffler theorem for vector-valued meromorphic functions on a non-compact Riemann surface to a complete Hausdorff locally semi-convex space.

Second order contact of minimal surfaces

The minimal surface equation Q in the second order contact bundle of R 3, modulo translations, is provided with a complex structure and a canonical vector-valued holomorphic differential form Ω on Q\\0. The minimal surfaces M in R 3 correspond to the complex analytic curves C in Q, where the derivative of the Gauss map sends M to C, and M is equal to the real part of the integral of Ω over C. The complete minimal surfaces of finite topological type and with flat points at infinity correspond to the algebraic curves in Q.

Generalized WDVV equations for F4 pure N=2 super-Yang–Mills theory

An associative algebra of holomorphic differential forms is constructed associated with pure N=2 super-Yang–Mills theory for the Lie algebra F 4. Existence and associativity of this algebra, combined with the general arguments in the work of Marshakov, Mironov and Morozov, proves that the prepotential of this theory satisfies the generalized WDVV (Witten–Dijkgraaf–Verlinde–Verlinde) system.

Covariant symplectic structure of the complex Monge–Ampère equation

The complex Monge–Ampère equation is invariant under arbitrary holomorphic changes of the independent variables with unit Jacobian. We present its variational formulation where the action remains invariant under this infinite group. The new Lagrangian enables us to obtain the first symplectic 2-form for the complex Monge–Ampère equation in the framework of the covariant Witten–Zuckerman approach to symplectic structure. We base our considerations on a reformulation of the Witten–Zuckerman theory in terms of holomorphic differential forms. The first closed and conserved Witten–Zuckerman symplectic 2-form for the complex Monge–Ampère equation is obtained in arbitrary dimension and for all cases elliptic, hyperbolic and homogeneous.The connection of the complex Monge–Ampère equation with Ricci-flat Kähler geometry suggests the use of the Hilbert action principle as an alternative variational formulation. However, we point out that Hilbert's Lagrangian is a divergence for Kähler metrics and serves as a topological invariant rather than yielding the Euclideanized Einstein field equations. Nevertheless, since the Witten–Zuckerman theory employs only the boundary terms in the first variation of the action, Hilbert's Lagrangian can be used to obtain the second Witten–Zuckerman symplectic 2-form. This symplectic 2-form vanishes on shell, thus defining a Lagrangian submanifold. In its derivation the connection of the second symplectic 2-form with the complex Monge–Ampère equation is indirect but we show that it satisfies all the properties required of a symplectic 2-form for the complex elliptic, or hyperbolic Monge–Ampère equation when the dimension of the complex manifold is 3 or higher.The complex Monge–Ampère equation admits covariant bisymplectic structure for complex dimension 3, or higher. However, in the physically interesting case of n=2 we have only one symplectic 2-form.The extension of these results to the case of complex Monge–Ampère–Liouville equation is also presented.

Abelian differentials on singular varieties and variations on a theorem of Lie-Griffiths

We compare various notions of meromorphic and holomorphic differential forms on (singular) analytic varieties. In particular we find that every meromorphic form gives rise to a canonical principal value current with support in the variety. Demanding that this current be ∂¯-closed we obtain a useful analytic description of the dualizing sheaf. We then go on to generalize the Lie-Griffiths theorem: On one hand it is shown that a rational trace is enough to ensure algebraicity of the data. On the other hand we prove that zero trace implies that the form is holomorphic in the sense of currents. In fact, we prove a more general, local version of the theorem.