The Poincare line bundle and autoduality of Hitchin fibers

Autoduality of compactified Jacobians for curves with plane singularities

Let C be an integral projective curve with planar singularities. Consider its Jacobian J and the compactified Jacobian J'. We construct a flat family P of Cohen-Macaulay sheaves on J' parametrized by J'; over J, the family P is the Poincare line bundle. We prove that the Fourier-Mukai transform given by P is an auto-equivalence of the derived category of J'.

具有平面奇点的曲线平展双覆盖的紧化Prym的自对偶性

Autoduality of compactified Pryms for étale double covers of curves with planar singularities

针对具有平面奇点的整曲线的平展双覆盖，构造了紧化Prym簇上的Poincaré层，并证明了相应的Fourier-Mukai变换是其导出范畴的自等价，进而应用于证明Corti-Hanamura的动机分解猜想并构造乘性动机反常滤过。

Fourier-Mukai and autoduality for compactified Jacobians II

To every reduced (projective) curve X with planar singularities one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, which are birational (possibly non-isomorphic) Calabi-Yau projective varieties with locally complete intersection singularities. We define a Poincare' sheaf on the product of any two (possibly equal) fine compactified Jacobians of X and show that the integral transform with kernel the Poincare' sheaf is an equivalence of their derived categories. In particular, any two fine compactified Jacobians are derived equivalent. When applied…

Moduli of flat connections in positive characteristic

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…

Complex K-theory of moduli spaces of Higgs bundles

We establish an isomorphism of complex $K$-theory of the moduli space $\check{\mathcal{M}}$ of $``SL_n"$-Higgs bundles of degree $d$ and rank $n$ (in the sense of Hausel--Thaddeus) and twisted complex $K$-theory of the orbifold $\hat{\mathcal{M}}$ of $PGL_n$-Higgs bundles of degree $e$, where $(n,d)=(n,e)=1$. Along the way we prove the vanishing of torsion for $H^*(\check{\mathcal{M}})$ and certain twisted complex $K$-theory groups of $\hat{\mathcal{M}}$. We also extend Arinkin's autoduality of compactified Jacobian to a derived equivalence between $SL_n$ and $PGL_n$-Hitchin systems over the e…

Construction of the Poincare sheaf for higher genus curves

Let $X$ be a smooth projective curve of genus $g$ and $L$ be a degree $l$ line bundle on $X$ with $l\geq 2g-1$. Denote the stack of rank two Higgs bundles on $X$ with value in $L$ by $\mathcal{H}iggs$ and the semistable part by $\mathcal{H}iggs_{ss}$. Let $H$ be the Hitchin base. In this paper we will construct the Poincare sheaf $\mathcal{P}$ on $\mathcal{H}iggs\times_{H}\mathcal{H}iggs_{ss}$ which is a maximal Cohen-Macaulay sheaf and flat over $\mathcal{H}iggs_{ss}$. In particular this includes the locus of nonreduced spectral curves.