Bong H. Lian
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The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Bong H. Lian has a total of 30 co-authors. Here are the top co-authors by number of shared publications:
- Shing-Tung Yau
Beijing Institute of Mathematical Sciences and Applications, Beijing, China
31 shared publications - Keiji Oguiso
Mathematical Sciences, the University of Tokyo, Tokyo, Japan
8 shared publications - Shinobu Hosono
University of Tokyo
8 shared publications - An Huang
Brandeis University
5 shared publications - Kefeng Liu
Center of Mathematical Sciences, Zhejiang University, Hangzhou, China
5 shared publications - Tsung-Ju Lee
Center of Mathematical Sciences and Applications, Cambridge, Massachusetts, U.S.A.
5 shared publications
Latest Publications
- Open Access
Non-commutative resolutions as mirrors of singular Calabi--Yau varieties
It has been conjectured that the hemisphere partition function arXiv:1308.2217, arXiv:1308.2438 in a gauged linear sigma model (GLSM) computes the central charge arXiv:math/0212237 of an object in the bounded derived category of coherent sheaves for Calabi--Yau (CY) manifolds. There is also evidence in arXiv:alg-geom/ 9511001, arXiv:hep-th/0007071. On the other hand, non-commutative resolutions of singular CY varieties have been studied in the context of abelian GLSMs arXiv:0709.3855. In this paper, we study an analogous construction of abelian GLSMs for non-commutative resolutions and propose they can be used to study a class of recently discovered mirror pairs of singular CY varieties. Our main result shows that the hemisphere partition functions (a.k.a.~$A$-periods) in the new GLSM are in fact period integrals (a.k.a.~$B$-periods) of the singular CY varieties. We conjecture that the two are completely equivalent: $B$-periods are the same as $A$-periods. We give some examples to support this conjecture and formulate some expected homological mirror symmetry (HMS) relation between the GLSM theory and the CY. As shown in arXiv:2003.07148, the $B$-periods in this case are precisely given by a certain fractional version of the $B$-series of arXiv:alg-geom/9511001. Since a hemisphere partition function is defined as a contour integral in a cone in the complexified secondary fan (or FI-theta parameter space) arXiv:1308.2438, it can be reduced to a sum of residues (by theorems of Passare-Tsikh-Zhdanov and Tsikh-Zhdanov). Our conjecture shows that this residue sum may now be amenable to computations in terms of the $B$-series.
Open AccessNon-commutative resolutions as mirrors of singular Calabi--Yau varieties
It has been conjectured that the hemisphere partition function arXiv:1308.2217, arXiv:1308.2438 in a gauged linear sigma model (GLSM) computes the central charge arXiv:math/0212237 of an object in the bounded derived category of coherent sheaves for Calabi--Yau (CY) manifolds. There is also evidence in arXiv:alg-geom/ 9511001, arXiv:hep-th/0007071. On the other hand, non-commutative resolutions of singular CY varieties have been studied in the context of abelian GLSMs arXiv:0709.3855. In this paper, we study an analogous construction of abelian GLSMs for non-commutative resolutions and propose they can be used to study a class of recently discovered mirror pairs of singular CY varieties. Our main result shows that the hemisphere partition functions (a.k.a.~$A$-periods) in the new GLSM are in fact period integrals (a.k.a.~$B$-periods) of the singular CY varieties. We conjecture that the two are completely equivalent: $B$-periods are the same as $A$-periods. We give some examples to support this conjecture and formulate some expected homological mirror symmetry (HMS) relation between the GLSM theory and the CY. As shown in arXiv:2003.07148, the $B$-periods in this case are precisely given by a certain fractional version of the $B$-series of arXiv:alg-geom/9511001. Since a hemisphere partition function is defined as a contour integral in a cone in the complexified secondary fan (or FI-theta parameter space) arXiv:1308.2438, it can be reduced to a sum of residues (by theorems of Passare-Tsikh-Zhdanov and Tsikh-Zhdanov). Our conjecture shows that this residue sum may now be amenable to computations in terms of the $B$-series.
- Open Access
On a conjecture of Huang--Lian--Yau--Yu
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, we prove the existence of the rank one point for such a system, where mirror symmetry is expected.
Open AccessOn a conjecture of Huang--Lian--Yau--Yu
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, we prove the existence of the rank one point for such a system, where mirror symmetry is expected.
- 1 September 2022
- Published by Springer Nature in Mathematische Annalen
- Vol. 387 (1-2) , 145-173
Hasse–Witt matrices, unit roots and period integrals
- 1 September 2022
- Published by Springer Nature in Mathematische Annalen
- Vol. 387 (1-2) , 145-173
Hasse–Witt matrices, unit roots and period integrals
- Open Access
On Calabi--Yau fractional complete intersections
In this article, we study mirror symmetry for pairs of singular Calabi--Yau manifolds which are double covers of toric manifolds. Their period integrals can be seen as certain `fractional' analogues of those of ordinary complete intersections. This new structure can then be used to solve their Riemann--Hilbert problems. The latter can then be used to answer definitively questions about mirror symmetry for this class of Calabi--Yau manifolds.
Open AccessOn Calabi--Yau fractional complete intersections
In this article, we study mirror symmetry for pairs of singular Calabi--Yau manifolds which are double covers of toric manifolds. Their period integrals can be seen as certain `fractional' analogues of those of ordinary complete intersections. This new structure can then be used to solve their Riemann--Hilbert problems. The latter can then be used to answer definitively questions about mirror symmetry for this class of Calabi--Yau manifolds.
- 1 January 2022
- Published by International Press of Boston in Pure and Applied Mathematics Quarterly
- Vol. 18 (1) , 317-342
On Calabi–Yau fractional complete intersections
- 1 January 2022
- Published by International Press of Boston in Pure and Applied Mathematics Quarterly
- Vol. 18 (1) , 317-342
On Calabi–Yau fractional complete intersections
- 1 January 2021
- Published by International Press of Boston in Mathematical Research Letters
- Vol. 28 (2) , 415-434
Period integrals of vector bundle sections and tautological systems
- 1 January 2021
- Published by International Press of Boston in Mathematical Research Letters
- Vol. 28 (2) , 415-434
Period integrals of vector bundle sections and tautological systems
- Open Access
Mirror symmetry for double cover Calabi--Yau varieties
The presented paper is a continuation of the series of papers arXiv:1810.00606 and arXiv:1903.09373. In this paper, utilizing Batyrev and Borisov's duality construction on nef-partitions, we generalize the recipe in arXiv:1810.00606 and arXiv:1903.09373 to construct a pair of singular double cover Calabi--Yau varieties $(Y,Y^{\vee})$ over toric manifolds and compute their topological Euler characteristics and Hodge numbers. In the $3$-dimensional cases, we show that $(Y,Y^{\vee})$ forms a topological mirror pair, i.e., $h^{p,q}(Y)=h^{3-p,q}(Y^{\vee})$ for all $p,q$.Open AccessMirror symmetry for double cover Calabi--Yau varieties
The presented paper is a continuation of the series of papers arXiv:1810.00606 and arXiv:1903.09373. In this paper, utilizing Batyrev and Borisov's duality construction on nef-partitions, we generalize the recipe in arXiv:1810.00606 and arXiv:1903.09373 to construct a pair of singular double cover Calabi--Yau varieties $(Y,Y^{\vee})$ over toric manifolds and compute their topological Euler characteristics and Hodge numbers. In the $3$-dimensional cases, we show that $(Y,Y^{\vee})$ forms a topological mirror pair, i.e., $h^{p,q}(Y)=h^{3-p,q}(Y^{\vee})$ for all $p,q$. - 1 January 2020
- Published by International Press of Boston in Communications in Number Theory and Physics
- Vol. 14 (4) , 739-783
K3 surfaces from configurations of six lines in $\mathbb{P}^2$ and mirror symmetry IInternational Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 1 January 2020
- Published by International Press of Boston in Communications in Number Theory and Physics
- Vol. 14 (4) , 739-783
K3 surfaces from configurations of six lines in $\mathbb{P}^2$ and mirror symmetry IInternational Press of Boston - publishers of scholarly mathematical and scientific journals and books
- Open Access
Period Integrals of CY and General Type Complete Intersections
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 expresses period integrals in terms of a canonical global meromorphic top form on $X$. Two important ingredients of our construction are the notion of a CY principal bundle, and a classification of such rank one bundles. We also generalize our construction to CY and general type complete intersections. When $X$ is an algebraic manifold having a sufficiently large automorphism group $G$ and $V^*$ is a linear representation of $G$, we construct a holonomic D-module that governs the period integrals. The construction is based in part on the theory of tautological systems we have developed in the paper \cite{LSY1}, joint with R. Song. The approach allows us to explicitly describe a Picard-Fuchs type system for complete intersection varieties of general types, as well as CY, in any Fano variety, and in a homogeneous space in particular. In addition, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety.Open AccessPeriod Integrals of CY and General Type Complete Intersections
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 expresses period integrals in terms of a canonical global meromorphic top form on $X$. Two important ingredients of our construction are the notion of a CY principal bundle, and a classification of such rank one bundles. We also generalize our construction to CY and general type complete intersections. When $X$ is an algebraic manifold having a sufficiently large automorphism group $G$ and $V^*$ is a linear representation of $G$, we construct a holonomic D-module that governs the period integrals. The construction is based in part on the theory of tautological systems we have developed in the paper \cite{LSY1}, joint with R. Song. The approach allows us to explicitly describe a Picard-Fuchs type system for complete intersection varieties of general types, as well as CY, in any Fano variety, and in a homogeneous space in particular. In addition, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety. - Open AccessK3 surfaces from configurations of six lines in $\mathbb{P}^2$ and mirror symmetry IFrom the viewpoint of mirror symmetry, we revisit the hypergeometric system $E(3,6)$ for a family of K3 surfaces. We construct a good resolution of the Baily-Borel-Satake compactification of its parameter space, which admits special boundary points (LCSLs) given by normal crossing divisors. We find local isomorphisms between the $E(3,6)$ systems and the associated GKZ systems defined locally on the parameter space and cover the entire parameter space. Parallel structures are conjectured in general for hypergeometric system $E(n,m)$ on Grassmannians. Local solutions and mirror symmetry will be described in a companion paper \cite{HLTYpartII}, where we introduce a K3 analogue of the elliptic lambda function in terms of genus two theta functions.Open AccessK3 surfaces from configurations of six lines in $\mathbb{P}^2$ and mirror symmetry IFrom the viewpoint of mirror symmetry, we revisit the hypergeometric system $E(3,6)$ for a family of K3 surfaces. We construct a good resolution of the Baily-Borel-Satake compactification of its parameter space, which admits special boundary points (LCSLs) given by normal crossing divisors. We find local isomorphisms between the $E(3,6)$ systems and the associated GKZ systems defined locally on the parameter space and cover the entire parameter space. Parallel structures are conjectured in general for hypergeometric system $E(n,m)$ on Grassmannians. Local solutions and mirror symmetry will be described in a companion paper \cite{HLTYpartII}, where we introduce a K3 analogue of the elliptic lambda function in terms of genus two theta functions.