More Like this

1. Endoscopic decompositions and the Hausel–Thaddeus conjecture

   https://doi.org/10.1017/fmp.2021.7  

   Maulik, Davesh; Shen, Junliang (January 2021, Forum of Mathematics, Pi)

   Abstract We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and Thaddeus concerning $$\mathrm {SL}_n$$ - and $$\mathrm {PGL}_n$$ -Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable $$\mathrm {SL}_n$$ -Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss and Ziegler via p -adic integration. Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration, whose supports are simpler. 

   more » « less
2. A Parabolic Analog of a Theorem of Beilinson and Schechtman

   https://doi.org/10.1093/imrn/rnae085  

   Biswas, Indranil; Mukhopadhyay, Swarnava; Wentworth, Richard (May 2024, International Mathematics Research Notices)

   For a simple, simply connected, complex group G, we prove an explicit formula to compute the Atiyah class of parabolic determinant of cohomology line bundle on the moduli space of parabolic G-bundles. This generalizes an earlier result of Beilinson-Schechtman. 

   more » « less
3. Cohomology of the Moduli Space of Cubic Threefolds and Its Smooth Models

   https://doi.org/10.1090/memo/1395  

   Casalaina-Martin, Sebastian; Grushevsky, Samuel; Hulek, Klaus; Laza, Radu (February 2023, Memoirs of the American Mathematical Society)

   We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily–Borel and toroidal compactifications of the ball quotient model, due to Allcock–Carlson–Toledo. Our starting point is Kirwan’s method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli space of cubic surfaces is discussed in an appendix. 

   more » « less
4. Hitchin fibrations, abelian surfaces, and the P=W conjecture

   https://doi.org/10.1090/jams/989  

   de Cataldo, Mark; Maulik, Davesh; Shen, Junliang (November 2021, Journal of the American Mathematical Society)

   We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus 2 2 curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role. 

   more » « less
5. Positively curved Riemannian metrics with logarithmic symmetry rank bounds

   https://doi.org/10.4171/CMH/340  

   Kennard, Lee (November 2014, Commentarii Mathematici Helvetici)

   We prove an obstruction at the level of rational cohomology to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As one application, we provide evidence for a generalized conjecture of H. Hopf, which states that no symmetric space of rank at least two admits a metric with positive curvature. Other applications concern product manifolds, connected sums, and manifolds with nontrivial fundamental group. 

   more » « less