Deformations of algebras in noncommutative geometry

These are significantly expanded lecture notes for the author's minicourse at MSRI in June 2012, as published in the MSRI lecture note series, with some minor additional corrections. In these notes, we give an example-motivated review of the deformation theory of associative algebras in terms of the Hochschild cochain complex as well as quantization of Poisson structures, and Kontsevich's formality theorem in the smooth setting. We then discuss quantization and deformation via Calabi-Yau algebras and potentials. Examples discussed include Weyl algebras, enveloping algebras of Lie algebras, symplectic reflection algebras, quasihomogeneous isolated hypersurface singularities (including du Val singularities), and Calabi-Yau algebras.Comment: A few minor corrections after publication (e.g., continuity of star products

Character Varieties

We study properties of irreducible and completely reducible representations of finitely generated groups Gamma into reductive algebraic groups G in in the context of the geometric invariant theory of the G-action on Hom(Gamma,G) by conjugation. In particular, we study properties of character varieties, X_G(Gamma)=Hom(Gamma,G)//G. We describe the tangent spaces to X_G(Gamma) in terms of first cohomology groups of Gamma with twisted coefficients, generalizing the well known formula. Let M be an orientable 3-manifold with a connected boundary F of genus > 1 and let X_G^g(F) be the subset of the G -character variety of F composed of conjugacy classes of good representations. By a theorem of Goldman, X_G^g(F) is a holomorphic symplectic manifold. We prove that the set of good G-representations of pi_1(F) which extend to representations of pi_1(M) is an isotropic submanifold of X_G^g(F). If these representations correspond to reduced points of the G-character variety of M then this submanifold is Lagrangian.Comment: 34 pages, to appear in Transactions of AM

The Strominger-Yau-Zaslow conjecture and its impact

This article surveys the development of the SYZ conjecture since it was proposed by Strominger, Yau and Zaslow in their famous 1996 paper, and discusses how it has been leading us to a thorough understanding of the geometry underlying mirror symmetry.Comment: v2: minor corrections in Section 8; 22 pages. appears in "Selected Expository Works of Shing-Tung Yau with Commentary. Vol. II", 1183-1208, Adv. Lect. Math. (ALM) 29, Int. Press, Somerville, MA, 201

Representation Growth and Rational Singularities of the Moduli Space of Local Systems

We relate the asymptotic representation theory of $SL(d,\mathbb{Z}_p)$ and the singularities of the moduli space of $SL(d)$-local systems on a smooth projective curve, proving new theorems about both. Regarding the former, we prove that, for every d, the number of n-dimensional representations of $SL(d,\mathbb{Z}_p)$ grows slower than $n^{22}$, confirming a conjecture of Larsen and Lubotzky. Regarding the latter, we prove that the moduli space of $SL(d)$-local systems on a smooth projective curve of genus at least 12 has rational singularities. Most of our results apply more generally to semi-simple algebraic groups. For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities.Comment: preliminary version, comments are welcome. v2. Revised version, now covering all semi simple group

Classification of real Bott manifolds and acyclic digraphs

We completely characterize real Bott manifolds up to affine diffeomorphism in terms of three simple matrix operations on square binary matrices obtained from strictly upper triangular matrices by permuting rows and columns simultaneously. We also prove that any graded ring isomorphism between the cohomology rings of real Bott manifolds with $\mathbb Z/2$ coefficients is induced by an affine diffeomorphism between the real Bott manifolds. Our characterization can also be described in terms of graph operations on directed acyclic graphs. Using this combinatorial interpretation, we prove that the decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors. Finally, we produce some numerical invariants of real Bott manifolds from the viewpoint of graph theory and discuss their topological meaning. As a by-product, we prove that the toral rank conjecture holds for real Bott manifolds.Comment: 27 pages, 5 figures. It is a combination of arXiv:0809.2178 and arXiv:1002.4704, including some new result

Geometric quantization via SYZ transforms

The so-called quantization problem in geometric quantization is asking whether the space of wave functions is independent of the choice of polarization. In this paper, we apply SYZ transforms to solve the quantization problem in two cases: (1) semi-flat Lagrangian torus fibrations over complete compact integral affine manifolds, and (2) projective toric manifolds. More precisely, we prove that the space of wave functions associated to the real polarization is canonically isomorphic to that associated to a complex polarization via SYZ transforms in both cases.Comment: Final versio

Wild Character Varieties, points on the Riemann sphere and Calabi's examples

We will give several descriptions of some basic examples of wild character varieties, including a discussion of links to work of Sibuya, Calabi and Euler, amongst others.Comment: 23 pages. v5: terminology regarding Stokes filtrations adjusted, better reflecting the analogy with Hodge structures, as in arXiv:1903.12612 (v4 is closest to the published version

Affine structures and non-archimedean analytic spaces

In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is motivated by the junction of Homological Mirror conjecture and geometric Strominger-Yau-Zaslow conjecture. In particular, we glue from "flat pieces" an analytic K3 surface. As a byproduct of our approach we obtain an action of an arithmetic subgroup of the group $SO(1,18)$ by piecewise-linear transformations on the 2-dimensional sphere $S^2$ equipped with naturally defined singular affine structure.Comment: 80 pages, 8 figure

The reduced spaces of a symplectic Lie group action

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, and Molino. In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.Comment: 37 page

The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations

This survey article begins with a discussion of the original form of the Strominger-Yau-Zaslow conjecture, surveys the state of knowledge concering this conjecture, and explains how thinking about this conjecture naturally leads to the program initiated by the author and Bernd Siebert to study mirror symmetry via degenerations of Calabi-Yau manifolds and log structures.Comment: 44 pages, to appear in the Proceedings of the 2005 AMS Symposium on Algebraic Geometry, Seattl