In his proof of the fundamental lemma of the Langlands program, Ngô initiated the study of the decomposition theorem for abelian fibrations. When an abelian fibration admits a duality structure, the decomposition theorem and the perverse filtration on cohomology exhibit rich structures. The purpose of these notes is to describe a framework for dualizable abelian fibrations and to discuss some recent progress and applications.

We prove the conjectural correspondence between logarithmic Gromov-Witten theory and logarithmic Donaldson/Pandharipande-Thomas theory for pairs (Y|∂ Y) consisting of a toric threefold Y and any torus invariant divisor ∂ Y, with primary insertions. The results are the first verifications of this conjecture when ∂ Y is singular, i.e. the “fully logarithmic” setting. When ∂ Y is empty, we get a new proof of the known toric correspondence, but our methods also lead to stronger conclusions. In particular, we show the PT series is a Laurent polynomial in the presence of sufficient positivity and prove a 2008 conjecture of Oblomkov, Okounkov, Pandharipande, and the first author stating the capped vertex is a Laurent polynomial. The methods also verify the logarithmic DT/PT conjecture for toric threefold pairs.

Motivated by the Beauville decomposition of an abelian scheme and the “Perverse = Chern” phenomenon for a compactified Jacobian fibration, we study in this paper splittings of the perverse filtration for compactified Jacobian fibrations. On the one hand, we prove for the Beauville–Mukai system associated with an irreducible curve class on a K3 surface the existence of a Fourier-stable multiplicative splitting of the perverse filtration, which extends the Beauville decomposition for the nonsingular fibers. Our approach is to construct a Lefschetz decomposition associated with a Fourier-conjugate 𝔰𝔩_2 -triple, which relies heavily on recent work concerning the interaction between derived equivalences and LLV algebras for hyper-Kähler varieties. Motivic lifting and connections to the Beauville–Voisin conjectures are also discussed. On the other hand, we construct for any g≥ 2 a compactified Jacobian fibration of genus g curves such that each curve is integral with at worst simple nodes and the (multiplicative) perverse filtration does not admit a multiplicative splitting. Our argument relies on the recently established universal double ramification cycle relations. This shows that in general an extension of the Beauville decomposition cannot exist for compactified Jacobian fibrations even when the simplest singular point appears.

We study Fourier transforms induced by Markman's projectively hyperholomorphic bundles on products of hyper-Kähler varieties of K3^[n]-type. As applications, we prove the following. (a) Derived equivalent hyper-Kähler varieties of K3^[n]-type have isomorphic homological motives preserving the cup-product. (b) All smooth projective moduli spaces of stable sheaves on a given K3 surface have isomorphic homological motives preserving the cup-product. (c) Assuming the Franchetta properties for the self-products of polarized K3 surfaces, the isomorphisms in (b) can be lifted to Chow motives for K3 surfaces of Picard rank 1. These results provide evidence for the Orlov conjecture and a conjecture of Fu-Vial.

We study the interplay between the Fourier-Mukai transform and the decomposition theorem for an integrable system π: M → B. Our main conjecture is that the Fourier-Mukai transform of sheaves of Kähler differentials, after restriction to the formal neighborhood of the zero section, are quantized by the Hodge modules arising in the decomposition theorem for π. For an integrable system, our formulation unifies the Fourier-Mukai calculation of the structure sheaf by Arinkin-Fedorov, the theorem of the higher direct images by Matsushita, and the "perverse = Hodge" identity by the second and the third authors. As evidence, we show that these Fourier-Mukai images are Cohen-Macaulay sheaves with middle-dimensional support on the relative Picard space, with support governed by the higher discriminants of the integrable system. We also prove the conjecture for smooth integrable systems and certain 2-dimensional families with nodal singular fibers. Finally, we sketch the proof when cuspidal fibers appear.

The purpose of this paper is to study motivic aspects of the Hitchin system for GL_n. Our results include the following. (a) We prove the motivic decomposition conjecture of Corti-Hanamura for the Hitchin system; in particular, the decomposition theorem associated with the Hitchin system is induced by algebraic cycles. This yields an unconditional construction of the motivic perverse filtration for the Hitchin system, which lifts the cohomological/sheaf-theoretic perverse filtration. (b) We prove that the inverse of the relative Hard Lefschetz symmetry is induced by a relative algebraic correspondence, confirming the relative Lefschetz standard conjecture for the Hitchin system. (c) We show a strong perversity bound for the normalized Chern classes of a universal bundle with respect to the motivic perverse filtration; this specializes to the sheaf-theoretic result obtained earlier by Maulik-Shen. (d) We prove a χ-independence result for the relative Chow motives associated with Hitchin systems. Our methods combine Fourier transforms for compactified Jacobian fibrations associated with integral locally planar curves, nearby and vanishing cycle techniques, and a Springer-theoretic interpretation of parabolic Hitchin moduli spaces.

Gromov-Witten (GW) theory produces Chow and cohomology classes on the moduli of curves, and there are several conjectures/speculations about their relation to the tautological ring. We develop new degeneration techniques to address these. In Chow, we show that GW cycles of complete intersections in products of projective spaces (and more generally a broad class of toric varieties) with restricted insertions are tautological. This gives significant evidence for a 2010 speculation of Pandharipande that GW cycles of varieties over the algebraic numbers are tautological. In particular, the 0-cycle for curves on the quintic threefold is proportional to a zero stratum in the moduli space of stable curves. In cohomology, we show that in normal crossings degenerations, GW classes of the general fiber lie in the span of absolute GW classes of the special fiber strata. This confirms a 2006 conjecture of Levine-Pandharipande for targets that degenerate into elementary pieces, including complete intersections in products of projective spaces and many toric varieties. Our proofs rely on several reconstruction theorems in logarithmic GW theory, which make the logarithmic degeneration formula an inductive tool to compute GW cycles via snc degenerations. We prove a folklore conjecture that logarithmic GW cycles of a pair are determined by absolute invariants of the strata. We prove a conjecture of Urundolil Kumaran and the second author that GW cycles of toric pairs are tautological, and analogous results for broken toric bundles. We also develop tools to study GW cycles with vanishing cohomology and strengthen the logarithmic degeneration formula to allow iteration.

We study the interaction between Fourier-Mukai transforms and perverse filtrations for a certain class of dualizable abelian fibrations. Multiplicativity of the perverse filtration and the "Perverse $\supset$ Chern" phenomenon for these abelian fibrations are immediate consequences of our theory. We also show that our class of fibrations include families of compactified Jacobians of integral locally planar curves. Applications include the following: (a) we prove the motivic decomposition conjecture for this class (including compactified Jacobian fibrations), which generalizes Deninger-Murre's theorem for abelian schemes; (b) we provide a new proof of the P=W conjecture for $\mathrm{GL}_r$; (c) we prove half of the P=C conjecture concerning refined BPS invariants for the local $\mathbb{P}^2$; (d) we show that the perverse filtration for the compactified Jacobian associated with an integral locally planar curve is multiplicative, which generalizes a result of Oblomkov-Yun for homogeneous singularities. Our techniques combine Arinkin's autoduality for coherent categories, Ng\^o's support theorem for the decomposition theorem, Adams operations in operational K-theory, and Corti-Hanamura's theory of relative Chow motives.

The purpose of this paper is to study the cohomology rings of universal compactified Jacobians. Over the moduli space ℳ_g of nonsingular curves, we show that the cohomology ring of the universal Jacobian is independent of the degree. Over the moduli space ℳ_g,n of Deligne-Mumford stable marked curves with n≥ 1, on the one hand we show that the cohomology ring of a universal fine compactified Jacobian is sensitive to the choice of a nondegenerate stability condition which answers a question of Pandharipande; on the other hand, we prove that the cohomology ring admits a degeneration via the perverse filtration which is independent of the (nondegenerate) stability condition. The latter defines the intrinsic cohomology ring of the universal compactified Jacobian which only relies on g,n. Our main tools include the support theorems, the recently developed Fourier theory for dualizable abelian fibrations, and the universal double ramification cycle relations associated with the universal Picard stack.

For any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by GSp(2g) between the cohomology of the moduli spaces of stable Higgs bundles which preserve the perverse filtrations. As consequences, we prove two structural results concerning the cohomology of Higgs moduli which are predicted by the P=W conjecture in non-abelian Hodge theory: (1) Galois conjugation for character varieties preserves the perverse filtrations for the corresponding Higgs moduli spaces. (2) The restriction of the Hodge-Tate decomposition for a character variety to each piece of the perverse filtration for the corresponding Higgs moduli space gives also a decomposition. Our proof uses reduction to positive characteristic and relies on the non-abelian Hodge correspondence in characteristic p between Dolbeault and de Rham moduli spaces.

We show that birational hyper-Kähler varieties of K3^[n] -type are derived equivalent, establishing the D -equivalence conjecture in these cases. The Fourier–Mukai kernels of our derived equivalences are constructed from projectively hyperholomorphic bundles, following ideas of Markman. Our method also proves a stronger version of the D -equivalence conjecture for hyper-Kähler varieties of K3^[n] -type with Brauer classes.

Given a Brauer class on a K3 surface defined over a number field, we prove that there exists infinitely many specializations where the Brauer class vanishes, under certain technical hypotheses, answering a question of Frei--Hassett--V\'arilly-Alvarado.

We prove new bounds for the period-index problem for hyper-Kähler varieties of K3^[n]-type using projectively hyperholomorphic bundles constructed by Markman. We show that dim(X) is a bound for any X of K3^[n]-type. We also show that 1/2dim(X) is a bound for most Brauer classes when the Picard rank of X is at least two, providing evidence for a conjecture of Huybrechts.

We propose a logarithmic enhancement of the Gromov-Witten/Donaldson-Thomas correspondence, with descendants, and study the behaviour of the correspondence under simple normal crossings degenerations. The formulation of the logarithmic correspondence requires a matching of tangency conditions and relative insertions. This is achieved via a version of the Nakajima basis for the cohomology of the Hilbert schemes of points on logarithmic surfaces. We then establish a strong form of the degeneration formula in logarithmic DT theory - the numerical DT invariants of the general fiber of a degeneration are determined by the numerical DT invariants attached to strata of the special fiber. The GW version of this result, which we also prove here, is a strengthening of the currently known formulas. A key role is played by a certain exotic class of insertions, introduced here, and can be thought of as non-local incidence conditions coupled across multiple boundary strata of the target geometry. Finally, we prove compatiblity of the new logarithmic GW/DT correspondence with degenerations, and in particular, knowledge of the conjecture on the strata of the special fiber of a degeneration implies it on the general fiber. Several examples are included to illustrate the nature and utility of the formula.

We prove the P = W conjecture for GL(n) for all ranks n and curves of arbitrary genus g >= 2. The proof combines a strong perversity result on tautological classes with the curious Hard Lefschetz theorem of Mellit. For the perversity statement, we apply the vanishing cycles constructions in our earlier work to global Springer theory in the sense of Yun, and prove a parabolic support theorem.

Given a smooth projective curve C , nonabelian Hodge theory gives a diffeomorphism between two different moduli spaces associated to C . The first is the moduli space of Higgs bundles on C of rank n , which is equipped with the structure of an algebraic completely integrable Hamiltonian system. The second is the character variety of representations of the fundamental group of C into \mathrm{GL}(n) . In 2012, de Cataldo, Hausel, and Migliorini [1] proposed the P=W conjecture which identifies the perverse filtration on the cohomology of the Higgs moduli space with the weight filtration on the cohomology of the character variety. Recently, in 2022, two independent proofs of the P=W Conjecture appeared, in work of Maulik &Shen [2] and Hausel, Mellit, Minets &Schiffmann [6]. The aim of the Arbeitsgemeinschaft was to understand the P=W Conjecture and these two recent proofs.

Let X be a smooth threefold with a simple normal crossings divisor D. We construct the Donaldson-Thomas theory of the pair (X|D) enumerating ideal sheaves on X relative to D. These moduli spaces are compactified by studying subschemes in expansions of the target geometry, and the moduli space carries a virtual fundamental class leading to numerical invariants with expected properties. We formulate punctual evaluation, rationality and wall-crossing conjectures, in parallel with the standard theory. Our formalism specializes to the Li-Wu theory of relative ideal sheaves when the divisor is smooth, and is parallel to recent work on logarithmic Gromov-Witten theory with expansions.

We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent of the Euler characteristic of the sheaves. We also prove an analogous result for the moduli space of semistable Higgs bundles with respect to an effective divisor $D$ of degree $\mathrm{deg}(D)>2g-2$. Our results confirm the cohomological $\chi$-independence conjecture by Bousseau for $\mathbb{P}^2$, and verify Toda's conjecture for Gopakumar-Vafa invariants for certain local curves and local surfaces. For the proof, we combine a generalized version of Ngo's support theorem, a dimension estimate for the stacky Hilbert-Chow morphism, and a splitting theorem for the morphism from the moduli stack to the good GIT quotient.

Let p be a prime number. We prove that the $$P=W$$ conjecture for $$\mathrm {SL}_p$$ is equivalent to the $$P=W$$ conjecture for $$\mathrm {GL}_p$$ . As a consequence, we verify the $$P=W$$ conjecture for genus 2 and $$\mathrm {SL}_p$$ . For the proof, we compute the perverse filtration and the weight filtration for the variant cohomology associated with the $$\mathrm {SL}_p$$ -Hitchin moduli space and the $$\mathrm {SL}_p$$ -twisted character variety, relying on Gröchenig–Wyss–Ziegler’s recent proof of the topological mirror conjecture by Hausel–Thaddeus. Finally we discuss obstructions of studying the cohomology of the $$\mathrm {SL}_n$$ -Hitchin moduli space via compact hyper-Kähler manifolds.

Let $A$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$ . We prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves.