bio | website | people.maths.ox.ac.uk/… |
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visits | member for | 4 years, 6 months |
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stats | profile views | 1,494 |
Apr 15 |
comment |
Do mutually dual finite vector spaces have the same orbit cardinalities under a linear group action?
One can also prove that the number of orbits are the same using the Fourier transform. It gives a canonical isomorphism $Fun(V) \to Fun(V^*)$ and hence $Fun_G(V) \to Fun_G(V^*)$. The dimensions of both sides is the number of $G$-orbits. |
Apr 12 |
awarded | Nice Answer |
Apr 8 |
comment |
Smooth mixed hodge modules - representations of fundamental group?
It is important to specify whether one considers an integral, real or complex mixed Hodge structure. For example, the integral lattice in a variation of pure Hodge structures of weight 1 is the same as a family of smooth curves which can be interesting (even over a disc). |
Mar 6 |
comment |
Schubert varieties which admit small resolutions of singularities
It is a result of Billey and Warrington that "321 hexagon avoiding" permutations in the symmetric group admit small resolutions. See: Billey and Warrington, Kazhdan-Lusztig Polynomials for 321-hexagon-avoiding permutations, J. of Algebraic Combinatorics, v. 13 (2001), no. 2, 111--136. |
Mar 2 |
revised |
Divisibility of all entries in an intersection form
added 4 characters in body |
Feb 28 |
awarded | Civic Duty |
Jan 13 |
revised |
Intersection homology for toric varieties
added 159 characters in body |
Dec 22 |
answered | Intersection homology for toric varieties |
Dec 17 |
comment |
Explicit examples presheaves associated to higher direct images which fail to be sheaves
Sorry, I misread what you were saying. I agree with (i) now. |
Dec 17 |
comment |
Explicit examples presheaves associated to higher direct images which fail to be sheaves
For (i), $f$ is a fibration with fibre $S^1$ and so $f^1_*(\underline{\mathbb{Z}})$ is a locally constant sheaf with fibre $\mathbb{Z}$. As $S^2$ is simply connected, we have $f_*^1$ is the constant sheaf, so I don't agree that (i) is an example. $f^3_*$ would however give an example for (ii). |
Oct 30 |
awarded | Necromancer |
Oct 30 |
answered | What to do now that Lusztig's and James' conjectures have been shown to be false? |
Oct 21 |
awarded | Yearling |
Oct 5 |
comment |
Higher degree generalizations of the Hard Lefschetz Theorem
Hard Lefschetz can be seen as "coming from $H^*(\mathbb{P}^n)$". Next on the scale of "easy" cohomology rings are Grassmannians. I wonder if the Chern classes of the tautological bundle give $2j$-HLP's. There are explicit formulas (Pieri rule) so it shouldn't be difficult to decide... |
Oct 5 |
comment |
Higher degree generalizations of the Hard Lefschetz Theorem
I like this question! One little fact which is relevant but doesn't answer your question: if one chooses any $d$ ample classes and uses them to define maps $H^0 \to H^2 \to H^4 \dots \to H^{2d}$ then hard Lefschetz and the Hodge-Riemann relations are still satisfied. Hence any product of $j$ ample classes will give you a $2j$-HLP. Cattani is the expert, e.g. arxiv.org/abs/0707.1352. |
Sep 22 |
revised |
Does anyone know this seemingly simple result in mixed Hodge theory?
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Sep 22 |
revised |
Does anyone know this seemingly simple result in mixed Hodge theory?
added 1636 characters in body |
Sep 22 |
comment |
Does anyone know this seemingly simple result in mixed Hodge theory?
You are right, the problem is "the above remarks imply that $K$ is of weight $\ge 0$", in fact one only gets $K$ of wt $\ge -1$. |
Sep 22 |
revised |
Does anyone know this seemingly simple result in mixed Hodge theory?
edited body |
Sep 22 |
answered | Does anyone know this seemingly simple result in mixed Hodge theory? |