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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
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Feb
28
awarded  Civic Duty
Jan
13
revised Intersection homology for toric varieties
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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?
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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?
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Sep
22
answered Does anyone know this seemingly simple result in mixed Hodge theory?