International Journal of Mathematics

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions _nF_m. We prove that, if n = m + 1, these functions are analytic for |z| < 1 and we give a necessary condition for the convergence on the unit circle |z| = 1.

In this paper, assuming Ω ∈ H¹(S^n-1), we prove that the singular integral T_Ω and the maximal singular integral are all bounded on Triebel–Lizorkin spaces, homogeneous or inhomogeneous.

Let S^[n] be the Hilbert scheme of length n subschemes of a K3 surface S. H²(S^[n],ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL[H*(S^[n],ℤ)] generated by monodromy operators, and let Mon² be its image in OH²(S^[n],ℤ). We prove that Mon² is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon² does not surject onto OH²(S^[n],ℤ)/(±1), when n - 1 is not a prime power.

As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKähler varieties X deformation equivalent to S^[n]. The weight 2 Hodge structure on H²(X,ℤ) does not determine the bimeromorphic class of X, whenever n - 1 is not a prime power (the first case being n = 7). There are at least 2^{ρ(n - 1) - 1} distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n - 1) is the Euler number of n - 1.

The second main result states, that if a monodromy operator acts as the identity on H²(S^[n],ℤ), then it acts as the identity on H^k(S^[n],ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon², if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6).

Let U be a complex space of complex dimension n ≥ 2, P a point of U, π : Ũ → U a modification such that Ũ is nonsingular and D = π^-1 (P) is a divisor with normal crossings. A Bochner–Martinelli form on U\{P} is a -closed differential form ω on Ũ\D, of pure type (n,n - 1), logarithmic along D. Such form detects a cohomology class of H^{2n - 1} (U\{P},ℂ) on the singular space U\{P}. Thanks to a general residue formula we prove that the forms ω give rise to an integral formula of Bochner–Martinelli type for holomorphic functions.

If U satisfies the following assumption that {there exists a compact complex space X bimeromorphic to a Kähler manifold, and a closed subspace T ⊂ X, such that X\T = U (an affine, or a quasi-projective variety satisfies the above property), we relate Bochner–Martinelli forms to the mixed Hodge structure carried by H^2n-1 (U\{P},ℂ). Most of our results hold for complex spaces which are not Stein.

Let G/ℚ be an inner form of Sp₄/ℚ which does not split over ℝ. Consequently, it is not quasi-split. In this paper we determine completely the automorphic cohomology of G. That is, we describe the Eisenstein and the cuspidal cohomology of congruence subgroups Γ of G with respect to arbitrary coefficient systems. In particular we establish precise nonvanishing results for cuspidal cohomology. In addition, we calculate the residual spectrum of G.