Symplectic resolutions for Higgs moduli spaces

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

On the Voevodsky motive of the moduli space of Higgs bundles on a curve

Selecta Mathematica ◽

10.1007/s00029-020-00610-5 ◽

2021 ◽

Vol 27 (1) ◽

Author(s):

Victoria Hoskins ◽

Simon Pepin Lehalleur

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Characteristic Zero ◽

Rational Point ◽

Triangulated Category ◽

Higgs Bundles ◽

Projective Curve ◽

Smooth Projective Curve ◽

De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

Topology of the tropical moduli spaces $$\Delta _{2,n}$$

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry ◽

10.1007/s13366-021-00563-6 ◽

2021 ◽

Author(s):

Melody Chan

Keyword(s):

Moduli Spaces

EXTREMAL CASES OF RAPOPORT–ZINK SPACES

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748020000730 ◽

2021 ◽

pp. 1-56

Author(s):

Ulrich Görtz ◽

Xuhua He ◽

Michael Rapoport

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Complete List ◽

Qualitative Properties ◽

Model Case ◽

Divisible Groups

Abstract We investigate qualitative properties of the underlying scheme of Rapoport–Zink formal moduli spaces of p-divisible groups (resp., shtukas). We single out those cases where the dimension of this underlying scheme is zero (resp., those where the dimension is the maximal possible). The model case for the first alternative is the Lubin–Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.

Obstruction theory for moduli spaces of framed flags of sheaves on the projective plane

Journal of Geometry and Physics ◽

10.1016/j.geomphys.2021.104255 ◽

2021 ◽

pp. 104255

Author(s):

Rodrigo A. von Flach ◽

Marcos Jardim ◽

Valeriano Lanza

Keyword(s):

Projective Plane ◽

Moduli Spaces ◽

Obstruction Theory

Holomorphic anomaly equation for and the Nekrasov-Shatashvili limit of local

Forum of Mathematics Pi ◽

10.1017/fmp.2021.3 ◽

2021 ◽

Vol 9 ◽

Author(s):

Pierrick Bousseau ◽

Honglu Fan ◽

Shuai Guo ◽

Longting Wu

Keyword(s):

Elliptic Curve ◽

Moduli Spaces ◽

Topological String ◽

Holomorphic Anomaly ◽

Holomorphic Anomaly Equation ◽

Anomaly Equation ◽

Witten Theory ◽

Generating Series ◽

Maximal Contact ◽

Higher Genus

Abstract We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $\lambda _g$ -insertion is related to Gromov-Witten theory of the total space of ${\mathcal O}_X(-D)$ and local Gromov-Witten theory of D. Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${\mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to $S={\mathbb P}^2$ , we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${\mathbb P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${\mathbb P}^2$ , we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${\mathbb P}^2$ in the Nekrasov-Shatashvili limit.

Twisted moduli spaces and Duistermaat–Heckman measures

Journal of Geometry and Physics ◽

10.1016/j.geomphys.2020.104042 ◽

2021 ◽

Vol 161 ◽

pp. 104042

Author(s):

Ahmed J. Zerouali

Keyword(s):

Moduli Spaces

THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

Nagoya Mathematical Journal ◽

10.1017/nmj.2020.37 ◽

2020 ◽

pp. 1-23

Author(s):

MICHELE BOLOGNESI ◽

NÉSTOR FERNÁNDEZ VARGAS

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Vector Bundles ◽

Hyperelliptic Curve ◽

Geometric Description ◽

Birational Model ◽

Rank 2 ◽

Semistable Vector Bundles ◽

Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

On properness of K-moduli spaces and optimal degenerations of Fano varieties

Selecta Mathematica ◽

10.1007/s00029-021-00694-7 ◽

2021 ◽

Vol 27 (4) ◽

Author(s):

Harold Blum ◽

Daniel Halpern-Leistner ◽

Yuchen Liu ◽

Chenyang Xu

Keyword(s):

Moduli Spaces ◽

Fano Varieties