相关论文: Period Integrals and the Riemann-Hilbert Correspon…
A tautological system, introduced in \cite{LSY}\cite{LY}, arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with…
We study period integrals of CY hypersurfaces in a partial flag variety. We construct a holonomic system of differential equations which govern the period integrals. By means of representation theory, a set of generators of the system can…
Period mappings were introduced in the sixties [G] to study variation of complex structures of families of algebraic varieties. The theory of tautological systems was introduced recently [LSY,LY] to understand period integrals of algebraic…
Tautological systems developed in [6],[7] are Picard-Fuchs type systems to study period integrals of complete intersections in Fano varieties. We generalize tautological systems to local complete intersections, which are zero loci of global…
Let X be a complex manifold. The classical Riemann-Hilbert correspondence associates to a regular holonomic system M the C-constructible complex of its holomorphic solutions. Denote by t the affine coordinate in the complex projective line.…
We verify a formula on the solution rank of the tautological system arising from ample complete intersections in a projective homogeneous space of a semisimple group conjectured by Huang--Lian--Yau--Yu arXiv:1801.01194. As an application,…
Many hypergeometric differential systems that arise from a geometric setting can be endowed with the structure of mixed Hodge modules. We generalize this fundamental result to the tautological systems associated to homogeneous spaces by…
We develop a global Poincar\'e residue formula to study period integrals of families of complex manifolds. For any compact complex manifold $X$ equipped with a linear system $V^*$ of generically smooth CY hypersurfaces, the formula…
We develop a formula for tautological integrals over geometric subsets of the Hilbert scheme of points on complex manifolds. As an illustration of the theory, we derive a new iterated residue formula for the number of nodal curves in…
Let $X$ be a smooth complete intersection over $\mathbb{C}$ of dimension $n-k$ in the projective space $\mathbf{P}^n_{\mathbb{C}}$, for given positive integers $n$ and $k$. For a given integral homology cycle $[\gamma] \in…
We develop the theory of relative regular holonomic D-modules with a smooth complex manifold S of arbitrary dimension as parameter space, together with their main functorial properties. In particular, we establish in this general setting…
A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra, a Vessiot-Guldberg Lie algebra,…
In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of…
In a previous paper, the author introduced a Z-structure in quantum cohomology defined by the K-theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds. Applying the quantum Lefschetz principle…
On the product of a complex manifold $X$ by a complex curve $S$ considered as a parameter space, we show a Riemann-Hilbert correspondence between regular holonomic relative $\mathcal D$-modules (resp. complexes) on the one hand and relative…
In 2023, two striking, nearly simultaneous, mathematical discoveries have excited their respective communities, one by Greenfeld and Tao, the other (the Hat tile) by Smith, Myers, Kaplan and Goodman-Strauss, which can both be summed up as…
We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the…
The group of automorphisms of the geometry of an integrable system is considered. The geometrical structure used to obtain it is provided by a normal form representation of integrable systems that do not depend on any additional geometrical…
Marvin Knopp developed the theory of automorphic integrals, which generalize automorphic forms; each automorphic integral has an additional period function in its automorphic relation. The period functions satisfy relations that arise from…
There have been a number of recent works on the theory of period polynomials and their zeros. In particular, zeros of period polynomials have been shown to satisfy a "Riemann Hypothesis" in both classical settings and for cohomological…