The positivity tag has no wiki summary.

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### Weak Fano and Log fano varieties

A projective smooth variety $X$ is weak Fano if $-K_X$ is nef and big. We say that $X$ is log Fano is there exists a divisor $D$ such that $-(K_X+D)$ is ample and $(X,D)$ is Kawamata log terminal.
Is ...

**4**

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**3**answers

157 views

### Exponential of a specific hypergeometric series

This is motivated by this question.
Let $f$ be the hypergeometric series
$ f(x) = 2 x \, _{4}F_3([1, 1, 4/3, 5/3], [2, 2, 2], 27 x) $
which is explictly given by
$ f(x) = \sum_{n \geq 1} ...

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**2**answers

182 views

### Push-forward of a nef bundle

Let $f:X\rightarrow Y$ be a finite morphism between normal varieties. Let $E$ be a vector bundle on $X$ and let us consider its pushforwad $f_{*}E$.
Does anyone know an example where $E$ is nef but ...

**8**

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289 views

### Higher degree generalizations of the Hard Lefschetz Theorem

Let $M$ be a $2d$-dimensional manifold. We say that $\omega \in H^2(M)$ has the Hard Lefschetz Property (HLP) if multiplication with $\omega^j$ is an isomorphism $H^{d-j} \to H^{d+j}$. This holds for ...

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131 views

### (Relative) ampleness on algebraic spaces

This is a follow-up (of sorts) to this question.
Let $f : X \to T$ be a proper morphism of schemes. Then the notion of a relative ample (or $f$-ample) line bundle can be defined in several ...

**1**

vote

**1**answer

58 views

### Does every $\alpha$-normal ordered Banach space have minimal upper bounds?

Let $\alpha>0$ and $X$ be an $\alpha$-normal (meaning, for $x,y\in X$,
$0\leq x\leq y$ implies $\|x\|\leq\alpha\|y\|$) ordered Banach space
with closed generating cone $X_{+}$. If $X$ is reflexive, ...

**4**

votes

**1**answer

266 views

### If $L$ is positive, $E\otimes L^k$ is Nakano positive for some $k$

I'm trying to prove the following:
Let $L$ be a positive holomorphic line bundle on a compact complex manifold $X$. For any hermitian holomorphic vector bundle $E$ on $X$, there is $k \in ...

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votes

**1**answer

96 views

### Degree principles for non-symmetric polynomials

A theorem of Timofte says that a symmetric polynomial inequality of degree $d$ holds on $\mathbb{R}^{n}_{+}$ if and only if it holds for all vectors in $\mathbb{R}^{n}_{+}$ with at most $\max\{\lfloor ...

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votes

**1**answer

158 views

### Sommese's theorem (generalized Weak Lefschetz) in arbitrary characteristic?

Sommese's theorem is a natural generalization of the Weak Lefschetz; for a smooth projective (connected) $X$, an ample vector bundle $E/X$ of rank $e$, and a section $s:X\to E$ it states that the ...

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111 views

### Quadratic forms over symmetric $3\times3$ matrices

This question is motivated by a technique called compensated compactness (CC), which was elaborated by L. Tartar and F. Murat for the analysis of PDEs. The foreword of CC is a problem about quadratic ...

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**1**answer

150 views

### Positive definite functions on G from Hilbert space vectors?

Let $G$ be a countable discrete group. Given a vector $\xi \in l^{2}(G)$, is there any way to naturally construct a positive definite function on $G$ using $\xi$?
This question is rather vague and ...

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**1**answer

116 views

### Mapping a subset of semi-definite matrices through arcsinus

Hi
I am meeting a problem concerning semi-definite positive matrices, and I have no clue concerning them, the classical approaches I know have not given any result, maybe people used to manipulating ...

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319 views

### What is the physical difference between states and unital completely positive maps?

Mathematically, completely positive maps on C*-algebras generalize positive linear functionals in that every positive linear functional on a C*-algebra $A$ is a completely positive map of $A$ into ...

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91 views

### Proving non-negativity of a hypergeometric type sum

I am trying to prove the following inequality:
$$
\sum\limits_{k=0}^{m}\frac{(a)_k(a+\mu)_{m-k}}{(c)_k(c+\mu)_{m-k}}\binom{m}{k}(m-2k+\mu)\geq{0},
$$
where $(a)_0=1$, $(a)_k=a(a+1)\cdots(a+k-1)$ is ...

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votes

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890 views

### Matrices whose inverse is positive

I have recently come across some examples of matrices with a special structure.
I will describe these matrices here and I hope that somebody will be able
to point out a source where I can find more ...

**4**

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**1**answer

360 views

### Numerically equivalent effective divisors and semiampleness

Recall that a divisor $M$ on a variety $X$ is said to be semiample if $kM$ is base point free for a certain $k > 0$.
Being semiample is not a numerical property (take for example torsion and a ...

**4**

votes

**1**answer

403 views

### When is the norm of all positive operators on an ordered Banach space determined by their values on the positive cone?

I'm trying to investigate the interplay between the norm and cone of positive elements in ordered Banach spaces. In particular, I would like a nice characterization of when the norm of a positive ...

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261 views

### Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?

As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on ...

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**1**answer

324 views

### Which (reducible) projective varieties could be presented as 'relatively smooth' hyperplane sections of irreducible (normal) ones?

Are there any restrictions known on a (complex reducible) projective variety $Y$ that can be presented as $Z\cap H$, where $Z$ is a normal (or just irreducible) closed subvariety of a smooth ...

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491 views

### Norm of Cesaro matrix

EDIT (28.11.2013).
This math.SE question about Cesaro operators should also be of interest (it discusses upper bounds; Noam has produced lower bounds below)
Federico's recent question Norm of upper ...

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529 views

### Positive operators - norm equality

I hope that somebody can help me with the following problem:
Let $A$ be a positive operator on $\mathbf{B}(\mathcal{H})$, ( $\mathcal{H}$ is a Hilbert space) with its spectral measure $E$. Show that ...

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349 views

### Signed factors of harmonic polynomials

Let ${\rm Harm}_n^d$ be the space of real harmonic polynomials in $n$ variables, homogeneous of degree $d$. If $P\in{\rm Harm}_n^d$, then
$$\left(\frac{\partial^2}{\partial ...

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226 views

### Positivity of polynomial sequences via generating series

In this question I address
the problem of proving the nonnegativity of a numerical sequence
$a_0,a_1,a_2,\dots$ via generating series technique. In the notation
$A(x)=\sum_{n=0}^\infty a_nx^n\ge0$ ...

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426 views

### Effective algorithm to test positivity

Let $f(x_1,\ldots, x_n)$ be a real polynomial in several variables. Is there an effective algorithm to test whether $f$ is positive (or nonnegative) on the whole of ${\mathbb{R}}^n$?

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317 views

### volume of big line bundles under finite morphisms

Let $X$, $Y$ be complex projective varieties of dimension $n$, let $f:X \rightarrow Y$ be a surjective finite morphism of degree $d$ and let $B$ be a big line bundle on $Y$.
Is that true that ...

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603 views

### Polynomials with presumably positive coefficients

After seeing that some positivity problems get their solutions on MO,
I am quite enthusiastic of posing my (and not only) problem of positive flavour.
In order to state it, I have to introduce the ...

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vote

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465 views

### Nakano semipositivity

Let $X$ be a compact Kaehler manifold.
What is a good, possibly algebraic-geometric, way to think to Nakano semipositivity of holomorphic vector bundles on $X$?
Is the trivial line bundle ...

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845 views

### Nef divisors with few global sections

Are there nef divisors D on a complex projective manifold X such that $h^0(X,D)$ is less than or equal to $\dim X$?
Edit: In fact I'm interested in nef line bundles D, not just divisors.

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### sum of squares in ring of integers

Lagrange proved that every (positive) rational integer is a sum of 4 squares.
Are there general results like this for ring of integers of a number field? Is this class field theory?
Explicity, ...

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2k views

### When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix?

Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the ...