I am interested in algebraic geometry and geometric representation theory, especially in topics related to derived categories and derived algebraic geometry.
Derived projectivizations of complexes. Mem. Amer. Math. Soc. (Accepted). (arXiv: 2202.11636; handwritten slides.)
This paper develops the counterparts of Grothendieck’s theory of projectivizations in the realm of derived algebraic geometry (DAG). The main results include generalized Serre’s theorem, a derived version of Beilinson’s relations, and semiorthogonal decompositions. Applications: classical situations such as reducible schemes, stabilization maps of prestable curves, certain singular threefolds, and moduli situations such as Hecke correspondences.
Lascoux-type resolutions, derived categories, and flips. J. Algebra. (2025), DOI.
This paper is the first part of the preprint “Derived categories of Quot schemes of locally free quotients,” now divided into several parts for publication. It focuses on the Key Lemma and its geometric implications. We introduce Lascoux-type complexes that generalize the classical Lascoux complexes for resolving generic determinantal ideals. These complexes naturally arise when comparing two types of resolutions of determinantal varieties, with applications to blowups, standard flips, virtual flips, and projectivizations.
On the Chow theory of Quot schemes of locally free quotients. J. Pure Appl. Algebra, 229 (2025), no. 1, Paper No. 107782, 40 pp. DOI. (arXiv:2010.10734.)
This paper establishes the Chow-theoretical Quot formula. Applications include the Chow-theoretical blowup formula for blowups of determinantal ideals, decompositions of Chow groups for moduli of linear series on curves, and for generalized nested Hilbert schemes.
Brill—Noether theory for Hilbert schemes of points on surfaces (joint with Arend Bayer and Huachen Chen). Int. Math. Res. Not. (IMRN) (2024), no. 10, 8403–8416. DOI. (arXiv 2304.12016);
This paper presents two proofs that the Brill--Noether loci in the Hilbert scheme of points on a smooth connected surface are Cohen--Macaulay, irreducible, and have expected dimensions, and that they are non-empty whenever their expected dimensions are positive.
On the Chow theory of projectivizations. J. Inst. Math. Jussieu, 22 (2023), no. 3, 1465–1508, DOI. (arXiv: 1910.06730).
This paper establishes the Chow-theoretical projectivization formula. Applications: symmetric powers of curves, nested Hilbert schemes, and varieties resolving Voisin’s maps.
Derived category of projectivization and flops (joint with Conan Leung). Adv. Math. 396 (2022), Paper No. 108169, 44 pp, DOI. (arXiv:1811.12525).
This paper establishes (i) the projectivization formula for derived categories and (ii) ``flop--flop=twist" results for flops from Springer-type resolutions for determinantal hypersurfaces. Applications: a blowup formula for blowup along Cohen--Macaulay codimension-$2$ subschemes, symmetric powers of curves, $\Theta$-flops, and nested Hilbert schemes on surfaces.
Categorical Pl\"ucker formula and homological projective duality (joint with Conan Leung and Ying Xie). J. Eur. Math. Soc. (JEMS) 23 (2021), no. 6, 1859--1898.
This paper generalizes the fundamental theorem of homological projective duality (HPD) of Kuznetsov from linear sections to general nonlinear intersections.
For a visual overview of the proof strategy, see the diagrams for fully faithfulness and generation..
Abel maps for integral curves via a derived perspective. 34pages. Submitted. (arXiv: 2508.11786)
This paper develops a general framework for Abel maps associated with families of integral curves, using derived algebraic geometry. We construct relative quasi-smooth derived enhancements of Abel maps for compactified Picard schemes studied by Altman–Kleiman, and naturally extend these results to higher-rank torsion-free sheaves and their coherent systems. Our approach yields unified semiorthogonal decompositions that significantly extend previous results for symmetric powers, varieties of linear series, and Thaddeus pairs.
Continuum envelops on Fargues--Fontaine curves and elliptic curves (joint with Heng Du and Yucheng Liu). 79 pages, 17 figures. (arXiv: 2404.04551)
This paper explores the applications of Bridgeland stability conditions in studying Fargues-Fontaine curves. We introduce the continuum envelopes on these curves and analyze their homological properties. Additionally, we investigate SL(2,Z) variants of Colmez--Fontaine's division algebra. Fargues--Fontaine curves exhibit strong similarities with elliptic curves and noncommutative tori from these perspectives.
Derived categories of derived Grassmannians. 38 pages. Submitted. (arXiv: 2307.02456)
Poster for Imperial Conference on July 7, 2023, and here for a simple worked-out example. 2024 Oberwolfach extended abstract.
This paper establishes semiorthogonal decompositions for derived Grassmannians of [0,1]-perfect complexes. These semiorthogonal decompositions serve as a categorified version of the BBDG decomposition in this context. This verifies the author's Quot formula conjecture, generalizing and strengthening Toda's result in [Tod23]. Applications: blowups of determinantal ideals, Fulton–MacPherson degeneration schemes, and varieties of linear series on curves. Our approach uses DAG, allowing us to work over arbitrary base over Q, and providing concrete descriptions of Fourier-Mukai kernels in terms of derived Schur functors.
Derived Grassmannians and derived Schur functors. 99 pages. Submitted. (arXiv: 2212.10488.)
This paper develops two theories, the geometric theory of derived Grassmannians (and flag schemes) and the algebraic theory of derived Schur (and Weyl) functors, and establishes their connection, a derived generalization of the Borel--Weil--Bott theorem.
Derived categories of Quot schemes of locally free quotients. 102 pages. (arXiv: 2107.09193.)
(Note: This paper will be split into several parts. We will maintain this original, comprehensive manuscript here for reference, to illustrate how the various sections interconnect. Please cite the new versions for specific results. We apologize for any inconvenience this may cause.)
This paper systematically studies the Grothendieck Quot schemes of locally free quotients, proposes a conjecture on their structures (the Quot formula), and verifies it in various cases. Our approach unifies various known formulae, such as formulae for blowups, Grassmannian bundles, standard flips/flops, and projectivizations, into a simple framework, and removes smoothness assumptions. It also leads to new results such as formulae for virtual flips, $\mathrm{Quot}_2$-formula, blowup formulae for determinantal ideals of codimension $\le 4$, etc.
Derived category of projectivization and generalized linear duality. 10 pages. (arXiv: 1812.05685.)
This paper establishes the duality between the projectivization formula and the blowup formula, in the sense of HPD.
Categorical duality between joins and intersections (joint with Conan Leung). 39 pages. (arXiv:1811.05135.)
This paper establishes the categorical duality between joins and intersections, in the sense of homological projective duality (HPD).
Blowing up linear categories, Refinements, and Homological Projective Duality with base locus (joint with Conan Leung). 39 pages. (arXiv: 1811.05132.)
This paper defines blowups of linear categories and their refinements in the sense of Lefschetz category theory and establishes its behavior under homological projective duality (HPD).
Instantons on blow-ups and the affine vertex algebra (joint with Weiping Li and Yu Zhao). In preparation.
This paper provides a solution to the question of Nakajima and Vafa–Witten on the coincidence between the blow-up formula for instanton moduli and the characters of affine $\mathrm{gl}_r$ and WZW models, by constructing affine $\mathrm{gl}_r$ actions on various cohomology theories. A key ingredient is a new representation-theoretic perspective on Grassmannians of $[0,1]$-perfect complexes, explored through Clifford algebra representations and connected to the original question via the Boson–Fermion correspondence.
Galois descent and ascent for Fargues--Fontaine curves. (title subject to changes; joint with Heng Du and Yucheng Liu). In preparation.
This paper series continues our investigation into the applications of Bridgeland stability conditions in the study of Fargues-Fontaine curves. We primarily examine their equivariant categories, explore the relations between their stability conditions and tilted hearts, investigate generalizations of Banach-Comez spaces, and establish Galois descent and ascent theories for these curves.
Triangle varieties from LLSVS eigthfolds. (joint with Arend Bayer and Huachen Chen). In preparation.
This paper constructs two different triangle varieties in the sense of Voisin for LLSvS eightfolds associated with very general cubic fourfolds
Hecke correspondences for surfaces revisited. In preparation.
This paper revisits the Hecke correspondences for surfaces, studied by Andrei Neguts, from the perspective of derived projectivizations. We will establish the relation between the algebra from Hecke correspondences and the elliptic Hall algebra for any surface.