Publications | Tsung-Ju Lee

Preprints

2024

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

Tsung-Ju Lee

Abs arXiv

In this article, we study the finite distance problem with respect to the period-map metric on the moduli of non-Kähler Calabi–Yau \(∂\bar∂\)-threefolds via Hodge theory. We extended C.-L. Wang’s finite distance criterion for one-parameter degenerations to the present setting. As a byproduct, we also obtained a sufficient condition for a non-Kähler Calabi–Yau to support the \(∂\bar∂\)-lemma which generalizes the results by Friedman and Li. We also proved that the non-Kähler Calabi–Yau threefolds constructed by Hashimoto and Sano support the \(∂\bar∂\)-lemma.

2023

Twisted GKZ hypergeometric functions and relative cohomology

Tsung-Ju Lee, and Dingxin Zhang

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

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

Abs arXiv

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.

Journal Articles

2024

Mirror symmetry for double cover Calabi–Yau varieties

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

J. Differential Geom. 127 (2024) , No. 1, 409–431

Abs

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.
SYZ mirror symmetry for del Pezzo surfaces and affine structures

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

Adv. Math. 439 (2024) , 109488

Abs

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

A note on periods of Calabi–Yau fractional complete intersections

Tsung-Ju Lee

Math. Z. 304 (2023) , No. 4, 60

Abs

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.
Period domains for gravitational instantons

Tsung-Ju Lee, and Yu-Shen Lin

Trans. Amer. Math. Soc. 376 (2023) , 5461–5501

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.
On a conjecture of Huang–Lian–Yau–Yu

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

Comm. Anal. Geom. 31 (2023) , No. 6, 1341–1352

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

On Calabi–Yau fractional complete intersections

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

Pure Appl. Math. Q. 18 (2022) , No. 1, 317–342

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}
}

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 (2022) , No. 6, 921–955

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

A-hypergeometric systems and relative cohomology

Tsung-Ju Lee, and Dingxin Zhang

Internat. J. Math. 31 (2020) , No. 13, 2050113, 21

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

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

Tsung-Ju Lee, and Hui-Wen Lin

Asian J. Math. 23 (2019) , No. 3, 501–526

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

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

Tsung-Ju Lee

Math. Res. Lett. 25 (2018) , No. 2, 617–647

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},
  issn = {1073-2780},
  fjournal = {Mathematical Research Letters},
  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}
}