On the normally ordered tensor product and duality for Tate objects (joint with A. Heleodoro, O. Braunling and J. Wolfson)
We study external tensor product operations for Tate objects and duality. At the end of the paper we discuss applications to intersection theory and relate our formalism to various other generalisations of Pontryagin duality that can be found in the literature.
Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration (joint with D. Wyss and P. Ziegler)
In this article we prove a conjecture by Hausel-Thaddeus which asserts an equality of stringy Hodge number of appropriately defined moduli spaces of SL(n) and PGL(n)-Higgs bundles. While the original conjecture has a strong physics flavour we adopt an arithmetic perspective in our paper. The argument boils down to comparing two p-adic integrals, on the SL(n)-side the integral of the constant function with value 1 gets compared to a integral of the Hasse invariant of a certain gerbe (this gerbe is used to define the stringy Hodge numbers). Using Galois theory of local fields, and Tate duality for abelian varieties we can compare the two integrals over almost all points of the Hitchin base. The comparison of p-adic integrals implies the agreement of Hodge numbers using p-adic Hodge theory.
Rigid connections, F-isocrystals and integrality (joint with H. Esnault)
In this paper we prove that irreducible representations of fundamental groups of complex algebraic varieties, which are reduced and isolated points of the moduli space of all such representations, are in fact defined over a ring of algebraic integers. This statement has been conjectured by Simpson in the more general situation of irreducible and rigid representations (that is, isolated points in the moduli, which could be non-reduced). Our approach is entirely arithmetic, and passes through positive characteristic and p-adic fields. Curiously we get to use all known variants of non-abelian Hodge theory: the classical version over the complex numbers, Ogus-Vologodsky's characteristic p theory, Faltings's p-adic correspondence, and the theory of Lan-Sheng-Zuo. Besides that we prove that irreducible rigid flat connections give rise to F-isocrystals modulo p, and prove results towards the Grothendieck-Katz p-curvature conjecture for irreducible rigid flat connections: we prove that the monodromy is at least unitary (the expectation being finite monodromy).
Adelic Descent Theory, Compositio Mathematica
It's a well-known observation of Weil that vector bundles on a curve (up to isomorphism) can be described as a double quotient of the general linear group over the ring of adèles. In this paper I'm studying a generalisation of this picture to arbitrary dimensions. The paper's main result is a descent theorem for perfect complexes on Noetherian schemes. However we perform the glueing with respect to Beilinson's cosimplicial ring of adèles. Pairing this theorem with Tannakian reconstruction results by Bhatt we obtain that Noetherian schemes can be reconstructed from the cosimplicial ring of adèles.
Relative Tate Objects and Boundary Maps in the K-Theory of Coherent Sheaves (joint with O. Braunling, J. Wolfson), Homology, Homotopy & Applications
This paper is devoted to obtaining a more explicit understanding of the boundary maps arising in the localisation sequences of the K-theory of coherent sheaves. We approach this problem using relative versions of ind and pro-objects in exact categories. The K-theory of these categories is easily identified, and using them we can describe the boundary map in terms of concrete realisation functors, transforming coherent sheaves on an open variety into one of these relative ind or pro-objects. Curiously the realisation functor turns out to be closely related with Deligne's lower-shriek functor for coherent sheaves.
Geometric and analytic structures on the higher adèles (joint with O. Braunling, J. Wolfson), Res. Math. Sci.
We apply the canonical cubical structures on endomorphism rings of n-Tate vector spaces to prove a conjecture by Yekutieli which asserted a comparison of two cubical algebra structures on Beilinson-Parshin adèles.
Operator ideals in Tate objects (joint with O. Braunling, J. Wolfson), Math. Res. Lett.
Given an n-Tate vector space (think of something like k((t))((s)) as an example of a 2-Tate vector space), we study the the algebra of endomorphisms and show that it carries a canonical cubical structure in the sense of Beilinson. Cubical structures play a role in constructing abstract residue morphisms.
A Generalized Contou-Carrère Symbol and its Reciprocity Laws in Higher Dimensions (joint with O. Braunling, J. Wolfson)
This paper applies the index map defined below to study a higher-dimensional analogue of the Contou-Carrère symbol. The latter is a deformation of the tame symbol and has been studied by Contou-Carrère in the context of a geometric analogue of local class field theory. Interestingly there's a close connection to self-duality of the Jacobi variety of an algebraic curve. Contou-Carrère Symbol also arise in the theory of de Rham epsilon factors by Beilinson-Bloch-Esnault. All of these motivations are confined to the one-dimensional case of curves, so we were eager to get a glimpse at the higher-dimensional theory. We define a higher-dimensional Contou-Carrère symbol, and prove that it's essentially a composition of boundary maps in algebraic K-theory. Furthermore it satisfies a reciprocity law similar to Kato reciprocity for higher tame symbols. When we started working on this we drew a lot of inspiration from the two-dimensional case worked out by Osipov-Zhu. The work of Gorchinskiy-Osipov gives another treatment of higher-dimensional Contou-Carrère symbols.
The Index Map in Algebraic K-Theory (joint with O. Braunling, J. Wolfson), accepted in Res. Math. Sci.
Tate vector spaces (such as k((t)) for example) have a filtered poset of lattices. Assigning to each lattice a formal dimension (compatible with the relative dimension which are always well-defined) one obtains a torsor of so-called dimension theories. It's been observed by Kapranov that a similar construction allows one to define a torsor over graded lines, which gives a way to define the Kac-Moody extension of loop groups. Since graded lines are known to be the 1-truncation of the K-theory spectrum of a field, one would expect a homotopically refined version, producing a torsor over the K-theory spectrum. In this article we describe this torsor in terms of the classfying map (which we call the index map). This sheds new light on a description of the K-theory of Tate objects by Sho Saito.
Tate Objects in Exact Categories (joint with O. Braunling, J. Wolfson; appendix by J. Stovicek and J. Trlifaj), Mosc. Math. J.
Tate objects in exact categories generalise Lefschetz's idea of linearly locally compact vector spaces to other exact categories. It allows one to emulate properties of topological vector spaces, such as k((t)) within the realm of category theory. In particular this allows for iteration, that is, Tate objects in Tate objects (think of k((s))((t)) for example), etc. In this article we study the resulting exact categories of higher Tate objects, but also of ind and pro-objects. Our main result shows commensurability of lattices in Tate objects (this is tricky in this general context, as one cannot simply take the intersection of two lattices).
Moduli Problems in Abelian Categories and the Reconstruction Theorem (joint with J. Calabrese), Alg. Geom.
As graduate students John and I organised a reading group on stacks. At the time we were eager to put the functorial point of view on algebraic geometry to the test. We observed that the proof of Gabriel's reconstruction theorem (which allows one to recover schemes with certain technical properties, from the abelian category of quasi-coherent sheaves) does not directly apply to algebraic spaces. The reason for this is that Gabriel's strategy produces directly a locally ringed space. In this article we give a new proof of Gabriel's theorem which also works for separated (and quasi-compact) algebraic spaces. For every abelian category we define an abstract moduli problem which happens to recover the the functor of points for algebraic spaces satisfying our assumptions. The same approach also works for quasi-coherent sheaves twisted by a gerbe, and allows one to recover the space and the gerbe.
Hilbert schemes as moduli of Higgs bundles and local systems, IMRN
A chapter of my thesis which develops in detail a few toy examples for moduli spaces of (parabolic) Higgs bundles. All of the examples that appear are algebraic surfaces or Hilbert schemes of algebraic surfaces. A particularly important case is the cotangent bundle of an elliptic curve, which has been studied by Gorsky-Nekrasov-Rubtsov using gauge theory. The approach taken here is different, as we replace the use gauge theory by derived categories. In addition we show how equivariant Hilbert schemes of cotangent bundles of elliptic curves with complex multiplication give rise to further examples. I also describe the Hitchin map and use the McKay equivalence to extend the autoduality from the locus of smooth spectral curves to the entire Hitchin base.
Moduli stacks of maps for supermanifolds, (joint with T. Adamo) Adv. Theor. Math. Phys.
We wrote this paper while we were both graduate students at Oxford. We develop a theory of algebraic superstacks and show that there is an algebraic superstack of stable maps to a supervariety. We explain how this relates to high energy physics. Throughout the article we advocate dévissage strategies: instead of redeveloping the theory of super algebraic geometry from ground up we used existing concepts in algebraic geometry to describe the super-version of the theory.
Moduli of flat connections in positive characteristic, Math. Res. Lett.
This paper and the one above about Hilbert schemes are essentially my PhD thesis. Inspired by work of Bezrukavnikov-Braverman I show that the moduli stack of local systems in characteristic p is étale-locally equivalent to the moduli stack of Higgs bundles. In the second part of the paper I use this result (and work of Arinkin) to extend Bezrukavnikov-Braverman's geometric Langlands equivalence in positive characteristic, to the locus of integral spectral curves.
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