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location National University of Singapore
age 29
visits member for 4 years, 10 months
seen Oct 3 at 14:43

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
comment 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
comment 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
comment 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
comment Homology of compact symmetric spaces
Could I ask what application do you have this homology information for?
Jul
13
comment 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
comment Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?
The fundamental group preserves products.
Jul
13
comment 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 $N(H\coprod_G K) \simeq NH\coprod_{NG} NK$.