Publications | Tsung-Ju Lee

Preprints

2024

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

Tsung-Ju Lee

Manuscript available upon request

Abs

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.

2020

Mirror symmetry for double cover Calabi–Yau varieties

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

to appear in J. Differential Geom.

Abs

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^∨) 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.
On a conjecture of Huang–Lian–Yau–Yu

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

to appear in Comm. Anal. Geom.

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.

Journal Articles

2024

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.

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}
}