Related papers: Volume-type functions for numerical cycle classes
For divisors over smooth projective varieties, we show that the volume can be characterized by the duality between pseudo-effective cone of divisors and movable cone of curves. Inspired by this result, we give and study a natural…
We prove that the partial derivative of the volume function of big classes along any real divisor in a compact Kaehler manifold is equal to the numerical restricted volume of that class to the divisor. A consequence of our main result is…
We show the existence of equivalence classes for large deviations. Stochastic dynamics within an equivalence class share the same large deviation properties.
The size function for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. It was conjectured to attain its maximum at the trivial class of Arakelov divisors. This conjecture was…
We elaborate on how two definitions of the volume of a big cohomology class are consistent. The first definition involves taking the absolutely continuous part of a closed positive current, and the second involves the non-pluripolar…
We define covering and separation numbers for functions. We investigate their properties, and show that for some classes of functions there is exact equality of separation and covering. We provide analogues for various geometric…
The level of a function f on an n-dimensional space encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the…
Given a space $Y$ in $X$, a cycle in $Y$ may be filled with a chain in two ways: either by restricting the chain to $Y$ or by allowing it to be anywhere in $X$. When the pair $(G,H)$ acts on $(X, Y)$, we define the $k$-volume distortion…
The volume of a Cartier divisor is an asymptotic invariant, which measures the rate of growth of sections of powers of the divisor. It extends to a continuous, homogeneous, and log-concave function on the whole N\'eron--Severi space, thus…
The vector space of all polynomial functions of degree $k$ on a box of dimension $n$ is of dimension ${n \choose k}$. A consequence of this fact is that a function can be approximated on vertices of the box using other vertices to higher…
This is the second part of our work on Zariski decomposition structures, where we compare two different volume type functions for curve classes. The first function is the polar transform of the volume for ample divisor classes. The second…
We give a relation between the existence of a Zariski decomposition and the behavior of the restricted volume of a big divisor on a smooth (complex) projective variety. Moreover, we give an analytic description of the restricted volume in…
A method of constructing an entire function with given zeros and estimates of growth is suggested. It gives a possibility to describe zero sets of certain classes of entire functions of one and several variables in terms of growth of volume…
Any permutation has a disjoint cycle decomposition and concept generates an equivalence class on the symmetry group called the cycle-type. The main focus of this work is on permutations of restricted cycle-types, with particular emphasis on…
If D is a big divisor and E is an effective divisor, then vol(D-E)<=vol(D)<=vol(D+E). We discuss when each inequality is an equality. Surprisingly, the answer is that the asymptotic equality vol(D-E)=vol(D) is equivalent to the equality of…
Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss…
We provide a formula to compute the volume of the intersection of a generalized cylinder with a hyperplane. Then we prove an integral inequality involving Bessel functions similar to Keith Ball's well-known inequality. Using this inequality…
In this paper, we show that the arithmetic volume function defined on the space of pairs of adelic R-Cartier divisors and base conditions is differentiable at a big pair, and that its derivative is given by an arithmetic restricted positive…
Given a reduced analytic space $Y$ we introduce a class of {\it nice} cycles, including all effective $\mathbb{Q}$-Cartier divisors. Equidimensional nice cycles that intersect properly allow for a natural intersection product. Using…
Let $M$ denote the space of complete conics. We compute the cone of effective and numerically effective $k$-cycles of $M$, $\mathrm{Eff}_k(M)$ and $\mathrm{Nef}_k(M)$, respectively. In addition, we compute the Bia\l{}ynicki-Birula…