| 1. |
- Arone, Gregory Z., et al.
(författare)
-
The action of Young subgroups on the partition complex
- 2021
-
Ingår i: Publications mathématiques (Bures-sur-Yvette). - : Springer Science and Business Media LLC. - 0073-8301 .- 1618-1913. ; 133, s. 47-156
-
Tidskriftsartikel (refereegranskat)abstract
- We study the restrictions, the strict fixed points, and the strict quotients of the partition complex |Πn|, which is the Σn-space attached to the poset of proper nontrivial partitions of the set {1,…,n}.We express the space of fixed points |Πn|G in terms of subgroup posets for general G⊂Σn and prove a formula for the restriction of |Πn| to Young subgroups Σn1×⋯×Σnk. Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions.We uncover surprising links between strict Young quotients of |Πn|, commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients |Πn|⋄∧Σn(Sℓ)∧nand give a combinatorial proof of a splitting in derived algebraic geometry.Combining all our results, we decompose strict Young quotients of |Πn| in terms of “atoms” |Πd|⋄∧Σd(Sℓ)∧d for ℓ odd and compute their homology. We thereby also generalise Goerss’ computation of the algebraic André-Quillen homology of trivial square-zero extensions from F2 to Fp for p an odd prime.
|
|
| 2. |
- Berman, Robert, 1976, et al.
(författare)
-
A variational approach to complex Monge-Ampere equations
- 2013
-
Ingår i: Publications mathématiques. - : Springer Science and Business Media LLC. - 0073-8301. ; 117:1, s. 179-245
-
Tidskriftsartikel (refereegranskat)abstract
- We show that degenerate complex Monge-Ampère equations in a big cohomology class of a compact Kähler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kähler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kähler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kähler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.
|
|
| 3. |
|
|
