Publications | Tsung-Ju Lee

2025

arXiv

Non-commutative resolutions and pre-quotients of Calabi-Yau double covers

Tsung-Ju Lee, Bong H. Lian, Mauricio Romo, and 1 more author

Jul 2025

Abs

Following an earlier proposal arXiv:2307.02038 to apply the GLSM formalism to understand the so-called non-commutative resolution, this paper takes one important step further to extend this formalism to a much larger class of non-commutative resolutions. The proposal was initially motivated by the discovery of a new class of mirror pairs singular Calabi-Yau varieties arXiv:2003.07148, given by certain branched double covers over toric varieties of MPCP type. The overarching problem was to understand these mirror pairs from the viewpoint of homological mirror symmetry arXiv:alg-geom/9411018. In the present paper, we propose two main results along this line. First, one new insight is that the ‘gauge-fixing’ condition on the branching locus of the double cover used in arXiv:2003.07148 can be relaxed in an interesting way. This turns out to produce GLSMs that describe a much larger class of non-commutative resolutions, leading to A-periods for a larger class of non-commutative resolutions, as well as the GKZ systems for their A-periods. Second, we show that the A-periods can also be realized as A-periods of a certain smooth CICY family in a toric variety of MPCP type, such that a suitable finite quotient of this family recovers the double cover CY we have started with. We call this CICY family the ‘pre-quotient’ of the double cover CY. This realization strongly suggests that pre-quotient may provide an important approach for understanding homological mirror symmetry for singular double cover CY varieties and non-commutative resolutions.
arXiv

Mirror symmetry for singular double cover Calabi–Yau varieties: quantum test

Tsung-Ju Lee, Bong H. Lian, and Shing-Tung Yau

Oct 2025

Abs

We continue our study on the pairs of singular Calabi–Yau varieties arising from double covers over semi-Fano toric manifolds. In this paper, we first investigate singular CY double covers of \mathbbP^3 branched along (1) a union of eight hyperplanes in general position, and (2) a union of four hyperplanes and a quartic in generation. Our previous construction produces hypothetical singular mirror partners. We prove that they are mirror pairs in the sense that the B-model of one (variation of Hodge structure) is equivalent to the A-model of another (the untwisted part of the genus zero orbifold Gromov–Witten invariants). The technique can be generalized and applied to the case when the nef-partition is trivial. As a byproduct, we also verify Morrison’s conjecture in certain circumstances.
Math. Ann.

Finite distance problem on the moduli of non-Kähler Calabi–Yau $∂\bar∂$-threefolds

Tsung-Ju Lee

Math. Ann. 392, no. 2, pp. 1541–1583, Mar 2025

DOI
arXiv

A Supplement to the anticanonical Volumes of weak \mathbbQ-Fano threefolds of Picard rank two

Ching-Jui Lai and Tsung-Ju Lee

Feb 2025

Abs

We show that for a weak \mathbbQ-Fano threefold X (\mathbbQ-factorial with terminal singularities and -K_X is nef and big) of Picard rank ρ(X)≤2, either -K_X^3≤64 or -K_X^3=72 and X=\mathbbP_\mathbbP^2(\mathcalO_\mathbbP^2⊕\mathcalO_\mathbbP^2(3)). This is supplementary to the previous work in arXiv:2501.12555.
Math. Ann.

Twisted GKZ hypergeometric functions and relative homology

Tsung-Ju Lee and Dingxin Zhang

Math. Ann. 391, pp. 5553–5580, 2025

Abs DOI arXiv

We investigate the GKZ A-hypergeometric \mathscrD-modules, introduced by Gel’fand, Kapranov, and Zelevinskii, arising from cyclic covers of toric varieties and find its Riemann–Hilbert partner. This extends our earlier results in arXiv:1902.01536.

2024

J. Differential Geom.

Mirror symmetry for double cover Calabi–Yau varieties

Shinobu Hosono, Tsung-Ju Lee, Bong H. Lian, and 1 more author

J. Differential Geom. 127, no. 1, pp. 409–431, 2024

Abs DOI

The presented paper is a continuation of the series of papers by the first, third, fourth authors, and Tagaki. In this paper, utilizing Batyrev and Borisov’s duality construction on nef‑partitions, we generalize their recipe to construct a pair of singular double cover Calabi–Yau varieties (Y,Y^∨) over toric manifolds and compute their topological Euler characteristics and Hodge numbers. In the 3-dimensional cases, we show that (Y, Y^∨) forms a topological mirror pair, i.e., h^p,q (Y ) = h^3-p,q (Y^∨) for all p, q.
Adv. Math.

SYZ mirror symmetry for del Pezzo surfaces and affine structures

Siu-Cheong Lau, Tsung-Ju Lee, and Yu-Shen Lin

Adv. Math. 439, pp. 109488, 2024

Abs DOI

We prove that the Landau–Ginzburg superpotential of del Pezzo surfaces can be realized as a limit of their hyperKähler rotation toward the large complex structure limit point. As a corollary, we compute the limit of the complex affine structure of the special Lagrangian fibrations constructed by Collins–Jacob–Lin in P1×P1 [16] and compare it with the integral affine structures used in the work of Carl–Pumperla–Siebert [9]. We also construct the Floer-theoretical Landau–Ginzburg mirrors of smoothing of An-singularities and monotone del Pezzo surfaces, by using the gluing method of Cho–Hong–Lau [13] and Hong–Kim–Lau [41]. They agree with the result of limit of hyperKähler rotations.

2023

arXiv

Non-commutative resolutions as mirrors of singular Calabi–Yau varieties

Tsung-Ju Lee, Bong H. Lian, and Mauricio Romo

Jul 2023

Abs

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.
Math. Z.

A note on periods of Calabi–Yau fractional complete intersections

Tsung-Ju Lee

Math. Z. 304, no. 4, 2023

Abs DOI

We prove that the GKZ }}\backslashmathscr {D}}}-module }}{\backslashmathcal {M}}_{A}\^{\backslashbeta }}}arising from Calabi–Yau fractional complete intersections in toric varieties is complete, i.e., all the solutions to }}{\backslashmathcal {M}}_{A}\^{\backslashbeta }}}are period integrals. This particularly implies that }}{\backslashmathcal {M}}_{A}\^{\backslashbeta }}}is equivalent to the Picard–Fuchs system. As an application, we give explicit formulae of the period integrals of Calabi–Yau threefolds coming from double covers of }}\backslashtextbf{P}\^{3}}}branched over eight hyperplanes in general position.
Trans. of the AMS

Period domains for gravitational instantons

Tsung-Ju Lee and Yu-Shen Lin

Trans. Amer. Math. Soc. 376, pp. 5461–5501, 2023

Abs arXiv

Based on the uniformization theorems of gravitation instantons by Chen–Chen arXiv:1505.01790, Chen–Viaclovsky arXiv:2110.06498, Collins–Jacob–Lin arXiv:2111.09260, and Hein–Sun–Viaclovsky–Zhang arXiv:2111.09287, we prove that the period maps for the ALH*, ALG, and ALG* gravitational instantons are surjective.
Comm. Anal. Geom.

On a conjecture of Huang–Lian–Yau–Yu

Tsung-Ju Lee, Bong H. Lian, and Dingxin Zhang

Comm. Anal. Geom. 31, no. 6, pp. 1341–1352, 2023

Abs arXiv

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.

2022

Pure Appl. Math. Q.

On Calabi–Yau fractional complete intersections

Tsung-Ju Lee, Bong H. Lian, and Shing-Tung Yau

Pure Appl. Math. Q. 18, no. 1, pp. 317–342, 2022

Bib

@article{2022-Lee-Lian-Yau-on-calabi-yau-fractional-complete-intersections,
  author = {Lee, Tsung-Ju and Lian, Bong H. and Yau, Shing-Tung},
  date-added = {2022-04-15 20:35:16 -0400},
  date-modified = {2022-04-15 20:35:41 -0400},
  fjournal = {Pure and Applied Mathematics Quarterly},
  issn = {1558-8599},
  journal = {Pure Appl. Math. Q.},
  mrclass = {14D07 (14J32 32G20 32Q25)},
  mrnumber = {4381855},
  number = {1},
  pages = {317--342},
  title = {On {C}alabi--{Y}au fractional complete intersections},
  url = {https://mathscinet-ams-org.ezp-prod1.hul.harvard.edu/mathscinet-getitem?mr=4381855},
  volume = {18},
  year = {2022},
  bdsk-url-1 = {https://mathscinet-ams-org.ezp-prod1.hul.harvard.edu/mathscinet-getitem?mr=4381855}
}

ATMP

On the complex affine structures of SYZ fibration of del Pezzo surfaces

Siu-Cheong Lau, Tsung-Ju Lee, and Yu-Shen Lin

Adv. Theor. Math. Phys. 26, no. 6, pp. 921–955, 2022

Bib

@article{2022-Lau-Lee-Lin-on-the-complex-affine-structures-of-syz-fibration-of-del-pezzo-surfaces,
  author = {Lau, Siu-Cheong and Lee, Tsung-Ju and Lin, Yu-Shen},
  date-added = {2023-07-25 08:39:40 +0800},
  date-modified = {2023-07-25 08:40:18 +0800},
  fjournal = {Advances in Theoretical and Mathematical Physics},
  issn = {1095-0761},
  journal = {Adv. Theor. Math. Phys.},
  mrclass = {53D37 (14J33)},
  mrnumber = {4557613},
  number = {6},
  pages = {921--955},
  title = {On the complex affine structures of {SYZ} fibration of del {P}ezzo surfaces},
  url = {https://mathscinet-ams-org.ezp-prod1.hul.harvard.edu/mathscinet-getitem?mr=4557613},
  volume = {26},
  year = {2022},
  bdsk-url-1 = {https://mathscinet-ams-org.ezp-prod1.hul.harvard.edu/mathscinet-getitem?mr=4557613}
}

2020

Internat. J. Math.

A-hypergeometric systems and relative cohomology

Tsung-Ju Lee and Dingxin Zhang

Internat. J. Math. 31, no. 13, pp. 2050113, 21, 2020

DOI Bib

@article{2020-Lee-Zhang-a-hypergeometric-systems-and-relative-cohomology,
  author = {Lee, Tsung-Ju and Zhang, Dingxin},
  date-added = {2021-12-06 09:03:33 -0500},
  date-modified = {2021-12-06 09:03:33 -0500},
  doi = {10.1142/S0129167X2050113X},
  fjournal = {International Journal of Mathematics},
  issn = {0129-167X},
  journal = {Internat. J. Math.},
  keywords = {14F10 (14M25)},
  mrnumber = {4192455},
  number = {13},
  pages = {2050113, 21},
  title = {{$A$}-hypergeometric systems and relative cohomology},
  volume = {31},
  year = {2020},
  bdsk-url-1 = {https://doi.org/10.1142/S0129167X2050113X}
}

2019

Asian J. Math.

Tautological systems under the conifold transition on G(2, 4)

Tsung-Ju Lee and Hui-Wen Lin

Asian J. Math. 23, no. 3, pp. 501–526, 2019

DOI Bib

@article{2019-Lee-Lin-tautological-systems-under-conifold-transitions-on-G24,
  author = {Lee, Tsung-Ju and Lin, Hui-Wen},
  date-added = {2021-12-06 09:08:02 -0500},
  date-modified = {2021-12-06 09:08:02 -0500},
  doi = {10.4310/AJM.2019.v23.n3.a7},
  fjournal = {Asian Journal of Mathematics},
  issn = {1093-6106},
  journal = {Asian J. Math.},
  keywords = {14J33 (14M15 14M25)},
  mrnumber = {3983757},
  number = {3},
  pages = {501--526},
  title = {Tautological systems under the conifold transition on {$G(2, 4)$}},
  volume = {23},
  year = {2019},
  bdsk-url-1 = {https://doi.org/10.4310/AJM.2019.v23.n3.a7}
}

2018

Math. Res. Lett.

A Hodge theoretic criterion for finite Weil–Petersson degenerations over a higher dimensional base

Tsung-Ju Lee

Math. Res. Lett. 25, no. 2, pp. 617–647, 2018

DOI Bib

@article{2018-Lee-a-Hodge-theoretic-criterion-for-finite-WP-degenerations-over-a-higher-dimensional-base,
  author = {Lee, Tsung-Ju},
  date-added = {2021-12-06 09:03:33 -0500},
  date-modified = {2021-12-06 09:03:33 -0500},
  doi = {10.4310/MRL.2018.v25.n2.a14},
  fjournal = {Mathematical Research Letters},
  issn = {1073-2780},
  journal = {Math. Res. Lett.},
  keywords = {32G20 (14D07 14J32)},
  mrnumber = {3826838},
  number = {2},
  pages = {617--647},
  title = {A {H}odge theoretic criterion for finite {W}eil--{P}etersson degenerations over a higher dimensional base},
  volume = {25},
  year = {2018},
  bdsk-url-1 = {https://doi.org/10.4310/MRL.2018.v25.n2.a14}
}