The main aim of this paper is to develop general algebraic and cohomological tools for the study of the local geometry of moduli and parameter spaces in Algebraic Geometry, culminating in the so-called Hitchin (or KZ) (projective) connection over the Moduli of curves and the proof of its flatness. Among other things, we notably give a 'modular' (i.e. in terms of the modular problem) construction of the differential operators- of any order- acting on (general)modular vector bundles, and in particular for the Lie algebra of vector fields on a (general) Moduli space, with its natural action on functions. The methods involve mostly Lie algebras and their cohomology.
We compute the supports of the perverse cohomology sheaves of the Hitchin fibration for $GL_n$ over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields we find additional supports at the loci of reducible spectral curves. Their contribution to the global cohomology is governed by a finite twist of Hitchin fibrations for Levi subgroups. The corresponding summands give non-trivial contributions to the cohomology of the moduli spaces for every $n \geq 3$. A key ingredient is a restriction result for intersection cohomology sheaves that allows us to compare th…
For G = GL_2, PGL_2 and SL_2 we prove that the perverse filtration associated to the Hitchin map on the cohomology of the moduli space of twisted G-Higgs bundles on a Riemann surface C agrees with the weight filtration on the cohomology of the twisted G character variety of C, when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.
A conjectural recursive relation for the Poincaré polynomial of the Hitchin moduli space is derived from wallcrossing in the refined local Donaldson-Thomas theory of a curve. A doubly refined generalization of this theory is also conjectured and shown to similarly determine the Hodge polynomial of the same moduli space.
We calculate the monodromy action on the mod 2 cohomology for SL(2,C) Hitchin systems and give an application of our results in terms of the moduli space of semistable SL(2,R) Higgs bundles.
It is shown that the map from the Jacobian of the spectral curve to the moduli of stable bundles of rank 2 is generically simply branched along an irreducible divisor. This observation falsifies the key step in the "abelianization of the SU(2) WZW connection" presented in a recent paper [Yoshida, Annals 2006]
The main tool in Ngô Bao Châu's proof of the Langlands-Shelstad fundamental lemma, is a theorem on the support of the relative cohomology of the elliptic part of the Hitchin fibration. For GL(n) and a divisor of degree >2g-2, the theorem says that the relative cohomology is completely determined by its restriction to any dense open subset of the base of the Hitchin fibration. In this note we would like to present in this particular case, our extension of that theorem to the whole Hitchin fibration, including the global nilpotent cone.
In very rough terms, the main theorem is that the set, which consists of semistable vector bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projective variety, is a degeneration of a union of abelian varieties. More precisely, we consider the subset of the moduli space of Higgs bundles satisfying the analogous cohomological condition. We show that this set is Zariski closed and that if Sigma is the normalization of an irreducible component containing a stable point, then a connected component of a general fiber of the restriction of the Hitchin map …
Here we survey several results and conjectures on the cohomology of the total space of the Hitchin system: the moduli space of semi-stable rank n and degree d Higgs bundles on a complex algebraic curve C. The picture emerging is a dynamic mixture of ideas originating in theoretical physics such as gauge theory and mirror symmetry, Weil conjectures in arithmetic algebraic geometry, representation theory of finite groups of Lie type and Langlands duality in number theory.
We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new $p$-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of $p$. By coupling this $p$-periodicity in characteristic $p$ with lifting/specialization techniques in mixed characteristic, we find, in arbitr…
We give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is a substitute for the Narasimhan-Atiyah-Bott Kähler form that realizes the Chern class of the determinant-of-cohomology line bundle on the moduli space of bundles on a curve. As replacement we use an explicit realisation of the Atiyah class of this line bundle, based on the theory of the trace complex due to Beilinson-Schechtman and Bloch-Esnault.
We introduce a notion of Jacobi-Bernoulli cohomology associated to a semi-simplicial Lie algebra (SELA). For an algebraic scheme $X$ over $\C$, we construct a tangent SELA $\t_X$ and show that the Jacobi-Bernoulli cohomology of $\t_X$ is related to infinitesimal deformations of $X$.
We study the 0-th local cohomology module of the jacobian ring of a singular reduced complex projective hypersurface X, by relating it to the sheaf of logarithmic vector field along X. We investigate the analogies between the local cohomology and the well known properties of the jacobian ring of a nonsingular hypersurface. In particular we study self-duality, Hodge theoretic and Torelli type questions.
In this paper, we study the cohomology of the truncated Hitchin fibration, which was introduced in a previous paper. We extend Ngô's main theorems on the cohomology of the elliptic part of the Hitchin fibration. As a consequence, we get a proof of Arthur's weighted fundamental lemma
We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is generated by Picard-Lefschetz transformations associated to vanishing cycles of singular spectral curves. We construct such vanishing cycles explicitly and use this to show that the $SL(n,\mathbb{C})$ monodromy group is a {\em skew-symmetric vanishing lattice} in the sense of Janssen…
We prove that the $0$-th local cohomology of the jacobian ring of a projective hypersurface with isolated singularities has a nice interpretation it in the context of linkage theory. Roughly speaking, it represents a measure of the failure of Gherardelli's theorem for the corresponding graded modules. This leads us to a different and characteristic free proof of its self-duality, which turns out to be an easy consequence of Grothendieck's local duality theorem.
Let $X$ be a compact connected Riemann surface of genus $g \geq 2$, and let ${\mathcal M}_{\rm DH}$ be the rank one Deligne-Hitchin moduli space associated to $X$. It is known that ${\mathcal M}_{\rm DH}$ is the twistor space for the hyper-Kähler structure on the moduli space of rank one holomorphic connections on $X$. We investigate the group $\operatorname{Aut}({\mathcal M}_{\rm DH})$ of all holomorphic automorphisms of ${\mathcal M}_{\rm DH}$. The connected component of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ containing the identity automorphism is computed. There is a natural element o…
Any irreducible Dynkin diagram $Δ$ is obtained from an irreducible Dynkin diagram $Δ_h$ of type $\mathrm{ADE}$ by folding via graph automorphisms. For any simple complex Lie group $G$ with Dynkin diagram $Δ$ and compact Riemann surface $Σ$, we give a Lie-theoretic construction of families of quasi-projective Calabi-Yau threefolds together with an action of graph automorphisms over the Hitchin base associated to the pair $(Σ, G)$ . These give rise to Calabi-Yau orbifolds over the same base. Their intermediate Jacobian fibration, constructed in terms of equivariant cohomology, is isomorphic to t…
The Hitchin system is a completely integrable hamiltonian system (CIHS) on the cotangent space to the moduli space of semi-stable vector bundles over a curve. We consider the case of rank-two vector bundles with trivial determinant. Such a bundle $E$ defines a divisor $D_E$ in the Jacobian of the curve and for any smooth point of $D_E$ we define a cotangent vector (a Higgs field). The Hitchin map on these Higgs fields is then determined in terms of the Gauss map on the divisor $D_E$. We apply the results to the $g=2$ case and show how Hitchin's system is related to classical line geometry in $…
Let $f$ be an invertible polynomial and $G$ a group of diagonal symmetries of $f$. This note shows that the orbifold Jacobian algebra $\mathrm{Jac}(f,G)$ of $(f,G)$ defined by the authors and Elisabeth Werner in arXiv:1608.08962 is isomorphic as a $\mathbb{ZZ}/2\mathbb{ZZ}$-graded algebra to the Hochschild cohomology $\mathsf{HH}^*(\mathrm{MF}_G(f))$ of the dg-category $\mathrm{MF}_G(f)$ of $G$-equivariant matrix factorizations of $f$ by calculating the product formula of $\mathsf{HH}^*(\mathrm{MF}_G(f))$ given by Shklyarov in arXiv:1708.06030. We also discuss the relation of our previous resu…
In this article we give a new proof of Ngô's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_κ$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of end…
Let $X$ be a compact Riemann surface of genus $g \geq 2$, and let $D \subset X$ be a fixed finite subset. Let $\mathcal{M}(r,d,α)$ denote the moduli space of stable parabolic $G$-bundles (where $G$ is a complex orthogonal or symplectic group) of rank $r$, degree $d$ and weight type $α$ over $X$. Hitchin discovered that the cotangent bundle of the moduli space of stable bundles on an algebraic curve is an algebraically completely integrable system fibered, over a space of invariant polynomials, either by a Jacobian or a Prym variety of spectral curves. In this paper we study the Hitchin fibers …
We explicitly describe cycle-class maps c_H from motivic cohomology to absolute Hodge cohomology, for smooth quasi-projective and (some) proper singular varieties, and compute special cases of the latter. For smooth projective varieties, we also study Hodge-theoretically defined "higher Abel-Jacobi maps" on the kernel of c_H; this leads to new results on nontrivial indecomposable higher Chow cycles in the regulator kernel.
We express the kernel of Griffiths' Abel-Jacobi map by using the inductive limit of Deligne cohomology in the generalized sense (i.e. the absolute Hodge cohomology of A. Beilinson). This generalizes a result of L. Barbieri-Viale and V. Srinivas in the surface case. We then show that the Abel-Jacobi map for codimension 2 cycles and the Albanese map are bijective if a general hyperplane section is a surface for which Bloch's conjecture is proved. In certain cases we verify Nori's conjecture on the Griffiths group. We also prove a weak Lefschetz-type theorem for (higher) Chow groups, generalize a…
In this paper we consider the topological side of a problem which is the analogue of Sen's S-duality testing conjecture for Hitchin's moduli space of rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann surface. We prove that all intersection numbers in the compactly supported cohomology vanish, i.e. "there are no topological L^2 harmonic forms on Hitchin's space". This result generalizes the well known vanishing of the Euler characteristic of the moduli space of rank 2 stable bundles of fixed determinant of odd degree over the given Riemann surface. Our proof shows th…
Exploiting the description of rings of differential operators as Azumaya algebras on cotangent bundles, we show that the moduli stack of flat connections on a curve (allowed to acquire orbifold points) defined over an algebraically closed field of positive characteristic is étale locally equivalent to a moduli stack of Higgs bundles over the Hitchin base. We then study the interplay with stability and generalize a result of Laszlo-Pauly, concerning properness of the Hitchin map. Using Arinkin's autoduality of compactified Jacobians we extend the main result of Bezrukavnikov-Braverman, the Lang…
We study local theory of moduli schemes using the framework of the Ran space. With the help of the study of sheaves and complexes over the Ran space by Beilinson and Drinfeld in their theory of chiral algebras, we revisit Ran's works on the Jacobi complexes (the Chevalley complexes for sheaves of Lie algebras on the Ran space), the universal deformation rings of moduli problems, the higher Kodaira-Spencer maps, and construction of Hitchin-type flat connections. We give rigorous treatments in the algebraic setting, which seems to be new.
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 o…
We introduce a notion of Jacobi-Bernoulli cohomology associated to a semi-simplicial Lie algebra (SELA). For an algebraic scheme $X$ over $\C$, we construct a tangent SELA $\t_X$ and show that the Jacobi-Bernoulli cohomology of $\t_X$ is related to infinitesimal deformations of $X$.
The Gauss-Manin connection for nonabelian cohomology spaces is the isomonodromy flow. We write down explicitly the vector fields of the isomonodromy flow and calculate its induced vector fields on the associated graded space of the nonabelian Hogde filtration. The result turns out to be intimately related to the quadratic part of the Hitchin map.
In this paper, we define a certain Hodge-theoretic structure for an arbitrary variety X over the complex number field by using the theory of mixed Hodge module due to Morihiko Saito. We call it an arithmetic Hodge structure of X. It is shown that extension groups of arithmetic Hodge structure do not vanish even for degree $\geq2$. Moreover, we define higher Abel-Jacobi maps from Bloch's higher Chow groups of X to these extension groups. These maps essentially involve the classical Abel-Jacobi maps by Weil and Griffiths, and Mumford's infinitesimal invariants of 0-cycles on surfaces.
We show that the image of the Abel-Jacobi map admits functorially a model over the field of definition, with the property that the Abel-Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also rec…
This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and nearly free curves in terms of Tjurina numbers. Finally, we suggest a stronger form of Terao's conjecture on the freeness of a line arrangement being determined by its combinatorics.
The moduli spaces of flat $\mathrm{SL}_2$- and $\mathrm{PGL}_2$-connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tamás Hausel in Remark 3.30 of "Global topology of the Hitchin system".
We define Deligne-Beilinson cycle maps for Lichtenbaum cohomology $H_L^m(X, \mathbb Z(n))$ and that with compact supports $H_{c,L}^m(X, \mathbb Z(n))$ of an arbitrary complex algebraic variety $X.$ When $(m,n)=(2,1),$ the homological part of our cycle map with compact supports gives a generalization of the Abel-Jacobi theorem and its projection to the Betti cohomology yields that of the Lefschetz theorem on $(1,1)$-cycles for arbitrary complex algebraic varieties. In general degrees $(m,n),$ we show that the Deligne-Beilinson cycle maps are always surjective on torsion and have torsion-free co…
In this paper, we show some applications to algebraic cycles by using higher Abel-Jacobi maps which were defined in [the author: Motives and algebraic de Rham cohomology]. In particular, we prove that the Beilinson conjecture on algebraic cycles over number fields implies the Bloch conjecture on zero-cycles on surfaces. Moreover, we construct a zero-cycle on a product of curves whose Mumford invariant vanishes, but not higher Abel-Jacobi invariant.
We provide further evidence to the $P=W$ conjecture of de Cataldo, Hausel and Migliorini, by checking it in the Painlevé cases. Namely, we compare the perverse Leray filtration induced by the Hitchin map on the cohomology spaces of the Dolbeault moduli space and the weight filtration on the cohomology spaces of the irregular character variety corresponding to each of the Painlevé $I-VI$ systems. We find that the two filtrations agree. Along the way, we prove the Geometric $P=W$ conjecture of Katzarkov, Noll, Pandit and Simpson in the Painlevé cases, and show that in these cases the Geometric $…
We discuss an Abel-Jacobi invariant for algebraic cobordism cycles whose image in topological cobordism vanishes. The existence of this invariant follows by abstract arguments from the construction of Hodge filtered cohomology theories in joint work of Michael J. Hopkins and the author. In this paper, we give a concrete description of the new Abel-Jacobi map and Hodge filtered cohomology groups for projective smooth complex varieties.
The $P = W$ conjecture identifies the perverse filtration of the Hitchin system on the cohomology of the moduli space of Higgs bundles with the weight filtration of the corresponding character variety. In this paper, we introduce an enumerative approach to to this problem; our technique only uses the structure of the equivariant intersection numbers on the moduli space of Higgs bundles, and little information about the topology of the Hitchin map. In the rank $2$ case, starting from the known intersection numbers of the moduli of stable bundles, we derive the equivariant intersection numbers o…
We construct an infinitesimal invariant for cycles in a family with cohomology class in the total space lying in a given level of the Leray filtration. This infinitesimal invariant detects cycles modulo algebraic equivalence in the fibers. We apply this construction to the Ikeda family, which gives optimal results for the Beauville decomposition of the 1-cycle of a very general plane curve in its Jacobian.
On a threefold with trivial canonical bundle, Kuranishi theory gives an algebro-geometry construction of the (local analytic) Hilbert scheme of curves at a smooth holomorphic curve as a gradient scheme, that is, the zero-scheme of the exterior derivative of a holomorphic function on a (finite-dimensional) polydisk. (The corresponding fact in an infinite dimensional setting was long ago discovered by physicists.) An analogous algebro-geometric construction for the holomorphic Chern-Simons functional is presented giving the local analytic moduli scheme of a vector bundle. An analogous gradient s…
Dimer models provide a method of constructing noncommutative crepant resolutions of affine toric Gorenstein threefolds. In homological mirror symmetry, they can also be used to describe noncommutative Landau--Ginzburg models dual to punctured Riemann surfaces. For a zigzag consistent dimer embedded in a torus, we explicitly compute the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics. This includes a full characterization of the Batalin--Vilkovisky structure induced by the Calabi--Yau structure of the Jacobi algebra. We then compute the compactly supported Hochschild…
Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces it has to be redesigned when applied to other generalized cohomology theories such as the equivariant, the quantum cohomology, K-theory, and cobordism. All this cohomology theories are different deformations of the ordinary cohomology. In this note we show that there is in some sense the universal deformation of Schubert calculus which produces the above mentioned by specialization of the appropriate param…
It is well known that degree two Deligne cohomology groups can be identified with groups of isomorphism classes of holomorphic line bundles with connections. There is also a geometric description of degree three Deligne cohomology, due to J-L. Brylinski and P. Deligne, in terms of gerbes with connective structures and curvings. This paper gives a geometric interpretation of Deligne cohomology of all degrees, in terms of equivalence classes of higher line bundles with $k$-connections. It is also shown that the classical Abel-Jacobi isomorphism $Pic^0(X) \cong J(X)$ generalizes to the isomorphis…
We take another approach to Hitchin's strategy of computing the cohomology of moduli spaces of Higgs bundles by localization with respect to the circle-action. Our computation is done in the dimensional completion of the Grothendieck ring of varieties and starts by describing the classes of moduli stacks of chains rather than their coarse moduli spaces. As an application we show that the n-torsion of the Jacobian acts trivially on the middle dimensional cohomology of the moduli space of twisted SL_n-Higgs-bundles of degree coprime to n and we give an explicit formula for the motive of the modu…
We disprove Hitchin's conjecture to the effect that for a generic complex structure on a simply connected spin complex surface the square root of the canonical bundle has no more cohomology then is predicted by the Riemann--Roch theorem. (This paper will appear in Proc. Amer. Math. Soc.)
