# Browsing by Subject "math.AG"

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Item Open Access Applications to A1-enumerative geometry of the A1-degreePauli, Sabrina; Wickelgren, KirstenThese are lecture notes from the conference Arithmetic Topology at the Pacific Institute of Mathematical Sciences on applications of Morel's A1-degree to questions in enumerative geometry. Additionally, we give a new dynamic interpretation of the A1-Milnor number inspired by the first named author's enrichment of dynamic intersection numbers.Item Open Access Compactly supported $\mathbb{A}^{1}$-Euler characteristic and the Hochschild complex(2020-03-20) Arcila-Maya, Niny; Bethea, Candace; Opie, Morgan; Wickelgren, Kirsten; Zakharevich, InnaWe show the $\mathbb{A}^{1}$-Euler characteristic of a smooth, projective scheme over a characteristic $0$ field is represented by its Hochschild complex together with a canonical bilinear form, and give an exposition of the compactly supported $\mathbb{A}^{1}$-Euler characteristic $\chi^{c}_{\mathbb{A}^{1}}: K_0(\mathbf{Var}_{k}) \to \text{GW}(k)$ from the Grothendieck group of varieties to the Grothendieck--Witt group of bilinear forms. We also provide example computations.Item Open Access Completion of two-parameter period maps by nilpotent orbits(2023-12-01) Deng, Haohua; Robles, ColleenWe show that every two-parameter period map admits a Kato--Nakayama--Usui completion to a morphism of log manifolds.Item Open Access Hodge theory of the Goldman bracket(Geometry and Topology, 2020-01-01) Hain, RichardIn this paper we show that, after completion in the I-adic topology, the Goldman bracket on the space spanned by homotopy classes of loops on a smooth, complex algebraic curve is a morphism of mixed Hodge structure. We prove similar statements for the natural action (defined by Kawazumi and Kuno) of the loops in X on paths from one "boundary component" to another. These results are used to construct torsors of isomorphisms of the the completed Goldman Lie algebra with the completion of its associated graded Lie algebra. Such splittings give torsors of partial solutions to the Kashiwara--Vergne problem (arXiv:1611.05581) in all genera. Compatibility of the cobracket with Hodge theory is established in arXiv:1807.09209.Item Open Access Hodge Theory of the Turaev Cobracket and the Kashiwara--Vergne Problem(Journal of the European Mathematical Society, 2021-01-01) Hain, RichardIn this paper we show that, after completing in the $I$-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve $X$ with an algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper (arXiv:1710.06053) on the Goldman bracket to construct torsors of solutions of the Kashiwara--Vergne problem in all genera. The solutions so constructed form a torsor under a prounipotent group that depends only on the topology of the framed surface. We give a partial presentation of these groups. Along the way, we give a homological description of the Turaev cobracket.Item Open Access Mirror Symmetry and DiscriminantsAspinwall, Paul S; Plesser, M Ronen; Wang, KangkangWe analyze the locus, together with multiplicities, of "bad" conformal field theories in the compactified moduli space of N=(2,2) superconformal field theories in the context of the generalization of the Batyrev mirror construction using the gauged linear sigma-model. We find this discriminant of singular theories is described beautifully by the GKZ "A-determinant" but only if we use a noncompact toric Calabi-Yau variety on the A-model side and logarithmic coordinates on the B-model side. The two are related by "local" mirror symmetry. The corresponding statement for the compact case requires changing multiplicities in the GKZ determinant. We then describe a natural structure for monodromies around components of this discriminant in terms of spherical functors. This can be considered a categorification of the GKZ A-determinant. Each component of the discriminant is naturally associated with a category of massless D-branes.Item Open Access Notes on Projective, Contact, and Null CurvesBryant, RobertThese are notes on some algebraic geometry of complex projective curves, together with an application to studying the contact curves in CP^3 and the null curves in the complex quadric Q^3 in CP^4, related by the well-known Klein correspondence. Most of this note consists of recounting the classical background. The main application is the explicit classification of rational null curves of low degree in Q^3. I have recently received a number of requests for these notes, so I am posting them to make them generally available.Item Open Access Notes on the Universal Elliptic KZB Equation(Pure and Applied Mathematics Quarterly, 2020-01-30) Hain, RThe universal elliptic KZB equation is the integrable connection on the pro-vector bundle over M_{1,2} whose fiber over the point corresponding to the elliptic curve E and a non-zero point x of E is the unipotent completion of \pi_1(E-{0},x). This was written down independently by Calaque, Enriquez and Etingof (arXiv:math/0702670), and by Levin and Racinet (arXiv:math/0703237). It generalizes the KZ-equation in genus 0. These notes are in four parts. The first two parts provide a detailed exposition of this connection (following Levin-Racinet); the third is a leisurely exploration of the connection in which, for example, we compute the limit mixed Hodge structure on the unipotent fundamental group of the Tate curve minus its identity. In the fourth part we elaborate on ideas of Levin and Racinet and explicitly compute the connection over the moduli space of elliptic curves with a non-zero abelian differential, showing that it is defined over Q.Item Open Access Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces(2001-03-05) Bryant, RLI use local differential geometric techniques to prove that the algebraic cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly speaking, foliated by rigid subvarieties in a nontrivial way). These rigidity results have a number of applications: First, they prove that many subvarieties in Grassmannians and other Hermitian symmetric spaces cannot be smoothed (i.e., are not homologous to a smooth subvariety). Second, they provide characterizations of holomorphic bundles over compact Kahler manifolds that are generated by their global sections but that have certain polynomials in their Chern classes vanish (for example, c_2 = 0, c_1c_2 - c_3 = 0, c_3 = 0, etc.).