Related papers: Irregularity of hypergeometric systems via slopes …
We study the irregularity of hypergeometric D-modules $\mathcal{M}_A (\beta )$ via the explicit construction of Gevrey series solutions along coordinate subspaces in $X =\mathbb{C}^n$. As a consequence, we prove that along coordinate…
We prove that a holonomic binomial $D$--module $M_A (I,\beta)$ is regular if and only if certain associated primes of $I$ determined by the parameter vector $\beta\in \CC^d$ are homogeneous. We further describe the slopes of $M_A(I,\beta)$…
A modified $A$-hypergeometric system is a system of differential equations for the function $f(t^w \cdot x)$ where $f(y)$ is a solution of an $A$-hypergeometric system in $n$ variables and $w$ is an $n$ dimensional integer vector, which is…
The holonomic rank of the A-hypergeometric system M_A(\beta) is the degree of the toric ideal I_A for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this…
We study $A$-hypergeometric systems $H_A(\beta)$ in the sense of Gelfand, Kapranov and Zelevinsky under two aspects: the structure of their holonomically dual system, and reducibility of their rank module. We prove first that rank-jumping…
We consider homological mirror symmetry in the context of hypertoric varieties, showing that appropriate categories of B-branes (that is, coherent sheaves) on an additive hypertoric variety match a category of A-branes on a Dolbeault…
In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ($\mathcal{D}$-modules), and reformulate the…
For each braid $\beta\in Br_n$ we construct a $2$-periodic complex $\mathbb{S}_\beta$ of quasi-coherent $\mathbb{C}^*\times \mathbb{C}^*$-equivariant sheaves on the non-commutative nested Hilbert scheme $Hilb_{1,n}^{free}$. We show that the…
We define a congruence module $\Psi_A(M)$ associated to a surjective $\mathcal O$-algebra morphism $\lambda\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra…
The type $A_n$-singularity $\mathbb{C}^2/\mathbb{Z}_{n+1}$ can be resolved by hyper-K\"ahler manifolds $X_{\zeta}$ with underlying smooth manifolds diffeomorphic to the resolution of singularities $X_{\text{res}}$, whose hyper-K\"ahler…
In this monograph, we lay some foundations of a theory of infinite dimensional Euclidean lattices - and more generally, of infinite dimensional Hermitian vector bundles over some "arithmetic curve" ${\rm Spec}\,\mathcal{O}_K$ attached to…
Motivated by classical Alexander invariants of affine hypersurface complements, we endow certain finite dimensional quotients of the homology of abelian covers of complex algebraic varieties with a canonical and functorial mixed Hodge…
The lattice of flats $\mathcal L_M$ of a matroid $M$ is combinatorially well-behaved and, when $M$ is realizable, admits a geometric model in the form of a "Schubert variety of hyperplane arrangement". In contrast, the lattice of flats of a…
Recently L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (whose link $M$ is a rational homology sphere) with the Seiberg-Witten invariant of $M$…
By giving a thin width of $0<\delta\ll 1$ to both a unit circle and a unit line, we set an annulus and a tube on the Euclidean plane $\mathbb{R}^2$. Consider the maximal means $M_\delta$ over dilations of the annulus, and $N_\delta$ over…
We study algebras of meromorphic modular forms whose poles lie on Heegner divisors for orthogonal and unitary groups associated to root lattices. We give a uniform construction of $147$ hyperplane arrangements on type IV symmetric domains…
We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for…
The dimension of the space of holomorphic solutions at nonsingular points (also called the holonomic rank) of a $A$--hypergeometric system $M_A (\beta)$ is known to be bounded above by $ 2^{2d}\operatorname{vol}(A)$, where $d$ is the rank…
This paper solves the global moduli problem for regular holonomic D-modules with normal crossing singularities on a nonsingular complex projective variety. This is done by introducing a level structure (which gives rise to…
We study a Lie algebra of formal vector fields $W_n$ with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi-Yau manifold with boundaries in the B-model. We show that equivalent…