bio | website | |
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location | National University of Singapore | |
age | 29 | |
visits | member for | 4 years, 10 months |
seen | Oct 3 at 14:43 | |
stats | profile views | 3,890 |
I am a graduate student at the National University of Singapore studying complex geometry with To Wing Keung.
Sep 30 |
awarded | Explainer |
Sep 18 |
comment |
Stability of real polynomials with positive coefficients
In particular, as you noted, the condition that $|f(z)|<f(|z|)$ is equivalent to the associated Hermitian metric of $f$ satisfying the Sharp Global Cauchy Schwar inequality. The other condition that the first two and the last two coefficients of $f$ are strictly positive is equivalent to its associated globalizable Hermitian metric being negatively curved. |
Sep 18 |
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Stability of real polynomials with positive coefficients
can be isometrically embedded into complex projective space. Theorem A can be deduced from this result of Catlin and D'Angelo in the case where the complex compact manifold is the Riemann sphere and using the fact that every negative holomorphic line bundle over the Riemann surface is antiample. |
Sep 18 |
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Stability of real polynomials with positive coefficients
After reading the answers by you and Prof Handleman and having a discussion with my PhD supervisor, we realized that the affirmative answer to my question and in fact what is now Theorem A in your revised preprint is a corollary of an isometric embedding theorem of Catlin and D'Angelo, MRL 1999. There, Catlin and D'Angelo proved, in particular, that for a globalizable Hermitian metric on a holomorphic line bundle over a compact complex manifold, if this metric is negatively curved and satisfies the so called Sharp Global Cauchy Schwarz inequality, then sufficiently large powers of this metric |
Sep 11 |
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Proof by `universal receiver'
Perhaps Chow's theorem that a complex submanifold of complex projective space is a complex variety is also required before one is able to apply the machinery of algebraic geometry. |
Sep 11 |
accepted | Stability of real polynomials with positive coefficients |
Sep 10 |
awarded | Nice Question |
Sep 9 |
revised |
Stability of real polynomials with positive coefficients
added 10 characters in body |
Sep 9 |
comment |
Stability of real polynomials with positive coefficients
ah, thanks Emil and Douglas. I will at the monic condition in. |
Sep 9 |
asked | Stability of real polynomials with positive coefficients |
Sep 5 |
awarded | Good Question |
Aug 26 |
awarded | Popular Question |
Jul 22 |
accepted | Criterion for deloopable based map |
Jul 22 |
awarded | Famous Question |
Jul 20 |
asked | Criterion for deloopable based map |
Jul 20 |
comment |
Any non-constant surjective holomorphic map between connected compact complex manifolds of equal dimension is a ramified finite-sheet covering
The qualification that the covering is "ramified" is important here. Checking Wikpedia, the standard terminology is "branched covering" rather than "ramified", that is, a covering over a dense subspace of the base. With this qualification, your example is not yet a counterexample for your blowup is bijective outside the blowup point in the base projective plane. |
Jul 13 |
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Homology of compact symmetric spaces
Could I ask what application do you have this homology information for? |
Jul 13 |
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Is it true that the only interesting topologies are metric topologies and weak topologies?
Another interesting topology is the locally compact hausdorff topology, especially if equipped with a group structure. |
Jul 13 |
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Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?
The fundamental group preserves products. |
Jul 13 |
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Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?
The nerve functor preserves coproducts. By a result of Dwyer and Kan, the nerve functor preserves certain pushouts. Explicitly, for $G,H,K$ groupoids and for $f:G\to H$ and $g:G\to K$ groupoid homomorphisms, if (1) these groupoids have the same set of objects ($G_0=H_0=K_0$) and (2) both $f_0:G_0\to H_0$ and $g_0:G_0\to K_0$ is the identity morphism on objects, then $N(H\coprod_G K) \simeq NH\coprod_{NG} NK$. |