The P=W conjecture in non-abelian Hodge theory.
• The P=W conjecture for GL_n (joint with D. Maulik). Ann. of Math. 200 (2024), Issue 2, 529-556.
This paper proves the P=W conjecture of de Cataldo-Hausel-Migliorini (2010) for GL_n.
More precisely, it proves that, for rank n and degree d coprime to n, the perverse filtration associated with the Hitchin system is matched with the weight filtration associated with the mixed Hodge structure of the character variety via Simpson's non-abelian Hodge correspondence.
• Hitchin fibrations, abelian surfaces, and the P=W conjecture (joint with M. A. de Cataldo and D. Maulik). J. Amer. Math. Soc. 35 (3), 2022, 911-953.
This paper connects the topology of Lagrangian fibrations associated with compact hyper-Kähler manifolds to the topology of Hitchin systems. As consequences, we prove the P=W conjecture for GL_n and genus 2.
• On te P=W conjecture for SL_n (joint with M. A. de Cataldo and D. Maulik). Selecta Math. 28 (2022), No.5. Paper No. 90.
This paper proves that for a prime number p, the P=W conjecture for GL_p and SL_p are equivalent. The proof relates the numerical version of the Hausel-Thaddeus mirror symmetry to the P=W conjecture for the variant cohomology.
• Cohomology of the moduli space of Higgs bundles via positive characteristic (joint with M. A. de Cataldo, D. Maulik, and Siqing Zhang). J. Eur. Math. Soc. (2023), pp. 1–21
This paper constructs symmetries of the cohomology of Hitchin moduli spaces using non-abelian Hodge theory in characteristic p. As applications, it provides direct/algebraic constructions on the Hitchin side of the Galois symmetry and the Hodge-Tate decomposition of the character variety.
The motivic decomposition conjecture for abelian fibrations.
• Perverse filtrations and Fourier transforms (joint with D. Maulik and Q. Yin). To appear at Acta Math.
This paper concerns a general framework studying the perverse filtration associated with an abelian fibration using a theory of Fourier transform. Over the locus of smooth fibers, such a Fourier theory is provided by the Beauville decomposition, therefore our work can be viewed as a partial extension of it. This framework can be used to prove the "multiplicativity" of the perverse filtration with respect to the cup-product and place the tautological classes in the correct place of the filtration via the Chern grading.
These techniques yield: (a) a proof of the motivic decomposition conjecture for certain abelian fibration, which generalizes Deninge-Murre's theorem for abelian schemes; (b) a new proof of the P=W conjecture for GL_r which does not rely on representation theory as in the previous proofs; (c) a proof of half of the P=C conjecture concerning refined BPS invariants for the local P2; (d) a proof of the multiplicativity of the perverse filtration for the compactified Jacobian associated with an integral locally planar curve, which generalizes a result of Oblomkov-Yun for homogeneous singularities.
• Algebraic cycles and Hitchin systems (joint with D. Maulik and Q.Yin). ArXiv 2407.05177, submitted.
This paper concerns some applications of the techniques developed in the last paper. Our main results involve a proof of the motivic decomposition conjecture of Corti-Hanamura for the Hitchin system associated with GL_n, and a motivic lifting of the \chi-indepdendence result.
• On generalized Beauville decompositions (joint with Y. Bae, D. Maulik and Q. Yin). ArXiv 2402.08861, submitted.
This paper concerns possible extension of the Beauville decomposition (i.e. a multiplicative and Fourier-stable splitting of the perverse filtration) from abelian schemes to compactified Jacobian fibrations. In a positive direction, we show that such an extension exists for a Beauville-Mukai system associated with an irreducible curve class on a K3 surface. In a negative direction, we show that such an extension is NOT possible in general even if we allow the simpliest (= nodal) singularities.
Our results suggest that the existence of a generalized Beauville decomposition may be a property for Lagrangian fibrations; we also discuss motivic lifting which is closed related to the Beauville-Voisin conjectures.
The Topological Mirror Symmetry Conjecture of Hausel-Thaddeus.
• Endoscopic decompositions and the Hausel-Thaddeus conjecture (joint with D. Maulik). ArXiv 2008.08520. Forum Math. Pi. 9 (2021), Paper No. e8, 49 pp.
This paper provides a new proof of the Hausel-Thaddeus conjecture connecting Hodge numbers for SL_n and PGL_n Hitchin moduli spaces; this conjecture was proved earlier by Groechenig-Wyss-Ziegler via p-adic integration. Our method is sheaf-theoretic, which combines support theorem (Ngô, Chauduouard-Laumon, de Cataldo) with vanishing cycle techniques. It provides a general strategy on using Ngô type support theorem to treat the (Lagrangian) Hitchin systems.
This idea was further used in the proof of the P=W conjecture of GL_n; see ArXiv 2209.02568.
• Cohomological χ-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles (joint with D. Maulik). Geom. Topol. 27:4 (2023), 1539-1586.
This paper provides a framework of using support theorem for singular moduli spaces and intersection cohomology complexes. As applications, it proves certain cases of \chi-independence conjectures by Bousseau and Toda rooted in enumerative geometry. This phenomenon can be viewed as a categorification of the strong rationality property of DT invariants.
• On the intersection cohomology of the moduli of SL_n-Higgs bundles on a curve (joint with D. Maulik). J. Topol. 15 (3) (2022), 1034-1057.
This paper starts to explore a formulation of the Hausel-Thaddeus mirror symmetry conjecture for any n and degree d without the coprime condition. Using the support theorem we established above, we prove that such a formulation holds for twisted (= meromorphic) Higgs bundles. We also discuss applications to intersection cohomology of the moduli of vector bundles when the rank is not coprime to the degree.
Geometry of hyper-Kähler varieties
• The D-equivalence conjecture for hyper-Kähler varieties via hyperholomorphic bundles, (joint with D. Maulik, Q. Yin, and R. Zhang). ArXiv 2408.14775v4. Submitted.
This paper proves the D-equivalence conjecture for any varieties with K3[n]-deformation type via projectively hyperholomorphic bundles.
• The period-index problem for hyper-Kähler varieties via hyperholomorphic bundles, (joint with J. Hotchkiss, D. Maulik, Q. Yin, and R. Zhang). ArXiv 2502.09774. Submitted.
This paper studies the period-index problem for hyper-Kähler varieties using Markman's hyperholomorphic bundles. The last paper showed that these bundles yield a full solution to the twisted D-equivalence conjecture for K3[n]-type, and this paper shows that these bundles also yield the best known bounds for the period-index problem for K3[n]-type.
• Topology of Lagrangian fibrations and Hodge theory of hyper-Kaehler manifolds, (joint with Q. Yin , an appendix by C. Voisin). Duke. Math. J. 171 (2022), no. 1, 209–241.
This paper connects the decomposition theorem of a Lagrangian fibration associated with a compact hyper-Kähler manifold to the Hodge theory of the ambient space. Applications involve a proof that the refined BPS invariants for K3 surfaces defined via perverse filtrations are matched with the "tranditional" calculations (GW/DT/KKV).
This is the first of a series of papers exploring "perverse-Hodge symmetry" for symplectic varieties; see below.
• P=W for Lagrangian fibrations and degenerations of hyper-Kaehler manifolds, (joint with A. Harder, Z. Li, and Q. Yin). Forum Math. Sigma. 9 (2021), Paper No. e50, 6 pp.
This paper proves the main results in the paper above from another persepective. In particular, it proves the P=W conjecture for compact hyper-Kähler manifolds by the first named author.
See this for a survey by Huybrechts and Mauri for the two papers above.
• Perverse-Hodge complexes for Lagrangian fibrations (joint with Q. Yin). ArXiv 2201.11283. EpiGA special volume in honor of C. Voisin, Article No. 6 (2023).
This paper proposes a categorification of the "perverse-Hodge symmetry" above using Saito's Hodge module. It also removes the compactness assumption in the cohomological version. This suggests that for any Lagrangian fibration, there is a mysterious symmetry concerning Hodge and perverse structures on the sheaf-theoretic level.
The main conjecture of this paper is now proved in this paper by Christian Schnell.
• Fourier-Mukai transform and the decomposition theorem for integrable systems. (joint with D. Maulik and Q. Yin). ArXiv 2301.05825. To appear at Algebr. Geom.
For an integrable system (roughly, a Lagrangian fibration), this paper proposes a correspondence between (1) Fourier-Mukai transform of the k-th exterier power of the cotangent bundle, and (2) the Hodge module arises from the k-th piece of the decomposition theorem. This specializes to a version of the categorification of the "perverse-Hodge symmetry" discussed in the last two papers above, and further suggests that the vector bundle of Kaehler differentials is "dual" to the Hodge module given by the decomposition theorem. We proof this correspondence for smooth case and certain 2-dimensional cases.
• Derived categories of K3 surfaces, O'Grady's filtration, and zero-cycles on holomorphic symplectic varieties, (joint with Q. Yin and X. Zhao ). Compos. Math. 156 (2020), no. 1, 179–197.
This paper proves (1) O'Grady's conjecture on the sheaf/cycle correspondence for K3 surfaces, and (2) Voisin's conjecture for certain K3[n]-type varieties on the existence of coisotropic subvarieties swept out by constant cycle subvarieties.
• K3 categories, one-cycles on cubic fourfolds, and the Beauville-Voisin filtration, (joint with Q. Yin ). J. Inst. Math. Jussieu.19 (2020), no. 5, 1601-1627.
This paper proposes a non-commutative version of O'Grady's conjecture using the Kuznetsov category associated with a cubic 4-fold. We also discuss the nature of the existence of such a correspondence, and suggest that this phenomenon is rooted in Calabi-Yau 2 categories.
• Rational curves in holomorphic symplectic varieties and Gromov-Witten invariants, (joint with G. Oberdieck and Q. Yin ). Adv. Math. 357 (2019), 106829, 28 pp.
We use Gromov-Witten invariants to study rational curves in hyper-Kähler varieties of K3[n]-type. Among others, we provide a complete criterion for the existence of a uniruled divisor swepted out by primitive rational curves for K3[n]-type, and we classify all primitive rational curves for the Fano variety of lines on a general cubic 4-fold.
Our results illustrate that the structures for rational curves in K3 and its higher dimensional analogs (i.e. K3[n]-type symplectic varieties) are very different.
• On O'Grady's generalized Franchetta conjecture, (joint with N. Pavic and Q. Yin), Int. Math. Res. Not. (2017), 16, 4971-4983.
This paper proves O'Grady's generalized Franchetta conjecture (that universally defined 0-cycles on K3 surfaces are proportional to the second Chern class) for Mukai models.
Enumerative geometry.
• Generator for the cohomology ring of the moduli of one-dimensional sheaves on P2. (joint with W. Pi). ArXiv 2204. 0586. Algebr. Geom.10 (4) (2023) 504–520.
This paper explores cohomology of Le Potier's moduli of 1-dimensional stable sheaves on P2 in terms of the tautological classes. It proves that the tautological classes in degrees < =2d-4 (i) do not admit any relation in degrees <= 2d-4, and (ii) generate the entire cohomology as a Q-algebra.
The connection to the Gopakumar-Vafa theory of local P2 is explained in the following two paper.
• Perverse fitrations, Chern filtrations, and refined BPS invariants for local P2 (joint with Y. Kononov and W. Pi). ArXiv 2211.06991. Adv. Math. 433 (2023). 1209294.
This paper formulates the P=C conjecture for P2 which can be viewed an analog of the P=W conjecture for Hitchin systems. Using the generation results in the paper above, P=C specializes to a product formula for the refined BPS invariants of local P2 obtained by refined Pandharipande-Thomas invariants/Nekrasov partition functions. The main conjectures of this paper has been proven in the following paper:
• Cohomological stabilization, perverse filtrations, and refined BPS invariants for del Pezzo surfaces (joint with F. Si, W. Pi, and F. Zhang). ArXiv 2406.10004
This paper proves an asymptotic product formula for the refined BPS invariants of a local del Pezzo surface; consequently this proves the asymptotic P=C match. In the case of P2, this resolves conjectures of Kononov-Pi-Shen.
• Curve counting on elliptic Calabi-Yau 3-folds via derived categories, (joint with G. Oberdieck). J. Eur. Math. Soc. 22 (2020), no. 3, 967-1002.
We build connections between (1) relative Fourier-Mukai transforms of elliptic Calabi-Yau 3-folds, and (2) the elliptic transformation law of Jacobi forms for the counting invariants as conjectured by Huang-Katz-Klemm from string theoretic considerations.
• Reduced Donaldson-Thomas invariants and the ring of dual numbers, (joint with G. Oberdieck), Proc. Lond. Math. Soc. (3) 118 (2019), no. 1, 191-220.
We develop motivic Hall algebra techniques using the ring of dual numbers, to handle reduced invariants for abelian 3-folds and K3xE. In particular this proves some formulas conjectured earlier by Bryan-Oberdieck-Pandharipande-Yin, and Oberdieck-Pandharipande.
• Cobordism invariants of the moduli space of stable pairs, J. Lond. Math. Soc (2) 94 (2016), no.2, 427-446.
We define virtual classes in algebraic cobordism, computing their Chern numbers after pushing forward to a point, and study cobordism invariants for Pandharipande-Thomas moduli spaces of stable pairs. We show that over a point, cobordism invariants are expressed in terms of descendent invariants