Fix $n$, $m\ge 1$, and let $d=\binom{m+n}{m}$ and $N=mn+m+n$. Consider the Segre embedding $\sigma:\mathbb{P}^m\times \mathbb{P}^n \hookrightarrow \mathbb{P}^{N}$, which has degree $d$. I'm trying to understand the pushforward of the external tensor product of sheaves. Namely:
What is $\sigma_*({\mathcal{O}}_{\mathbb{P}^m}(a)\boxtimes{\mathcal{O}}_{\mathbb{P}^n}(b))$ in terms of (various) ${\mathcal{O}}_{\mathbb{P}^N}(l)$?
It is well-known that the K-theory of $\mathbb{P}^N$ is generated by $[{\mathcal{O}}_{\mathbb{P}^N}(l)]$ for $l=0,...,N$. So my follow up would be:
What is $[\sigma_*({\mathcal{O}}_{\mathbb{P}^m}(a)\boxtimes{\mathcal{O}}_{\mathbb{P}^n}(b))]$ in K-theory, in terms of $[{\mathcal{O}}_{\mathbb{P}^N}(l)]$ for $l=0,...,N$?
A potential approach:
The pushforward of the Segre embedding in K-theory can be computed using the Segre classes of the sheaves involved. The Segre embedding $\sigma: \mathbb{P}^m \times \mathbb{P}^n \hookrightarrow \mathbb{P}^N$ sends a pair of homogeneous coordinates $([x_0:\cdots:x_m],[y_0:\cdots:y_n])$ to the point $[z_{ij}]$ in $\mathbb{P}^N$, where $z_{ij} = x_i y_j$.
Let $E$ be the rank $d$ vector bundle on $\mathbb{P}^m \times \mathbb{P}^n$ given by $E = \mathcal{O}{\mathbb{P}^m}(a) \boxtimes \mathcal{O}{\mathbb{P}^n}(b)$. Then the pushforward $\sigma_*(E)$ is the vector bundle on $\mathbb{P}^N$ whose fiber at a point $[z_{ij}]$ is the direct sum of the fibers of $E$ over all pairs of points in $\mathbb{P}^m \times \mathbb{P}^n$ that map to $[z_{ij}]$ under the Segre embedding. This direct sum decomposes as a direct sum of line bundles $\mathcal{O}_{\mathbb{P}^N}(l)$, where $l$ varies between $0$ and $N$.
To compute the pushforward in K-theory, we need to find the multiplicity of each line bundle $\mathcal{O}{\mathbb{P}^N}(l)$ in the decomposition of $\sigma(E)$. This can be done using the Segre classes of $E$. Recall that the Segre class of a vector bundle $E$ on a variety $X$ is the class $s(E) \in A^(X)$ given by $s(E) = \sum_{i=0}^{\infty} s_i(E)$, where $s_i(E)$ is the $i$-th Segre class of $E$. The pushforward of a vector bundle $E$ in K-theory can be expressed as $$\sigma_*(E) = \sum_{l=0}^N c_l s_l(E),$$ where $c_l$ are coefficients that depend on the Segre classes of $E$ and the degrees of the line bundles $\mathcal{O}_{\mathbb{P}^N}(l)$.
In our case, since $E = \mathcal{O}{\mathbb{P}^m}(a) \boxtimes \mathcal{O}{\mathbb{P}^n}(b)$, the Segre classes of $E$ can be computed as the products of the Segre classes of $\mathcal{O}{\mathbb{P}^m}(a)$ and $\mathcal{O}{\mathbb{P}^n}(b)$. Using these Segre classes, we can compute the coefficients $c_l$ and express the pushforward $\sigma_*(E)$ in K-theory as a linear combination of the classes $[\mathcal{O}_{\mathbb{P}^N}(l)]$ for $l = 0, \dots, N$.
To express the pushforward $\sigma_*(E)$ in K-theory as a linear combination of the classes $[\mathcal{O}_{\mathbb{P}^N}(l)]$ for $l = 0, \dots, N$, we need to compute the Segre classes of $E = \mathcal{O}_{\mathbb{P}^m}(a) \boxtimes \mathcal{O}_{\mathbb{P}^n}(b)$. Since $E$ is an external tensor product, its Segre classes are given by the products of the Segre classes of $\mathcal{O}_{\mathbb{P}^m}(a)$ and $\mathcal{O}_{\mathbb{P}^n}(b)$:
$$ s(E) = s(\mathcal{O}_{\mathbb{P}^m}(a)) \cdot s(\mathcal{O}_{\mathbb{P}^n}(b)). $$
The Segre classes of a line bundle $\mathcal{O}_X(d)$ on a projective space $X = \mathbb{P}^k$ are given by
$$ s(\mathcal{O}_X(d)) = (1 + dH)^{k+1}, $$
where $H$ is the hyperplane class on $X$. Then, the Segre classes of $E$ can be computed as
$$ s(E) = (1 + aH_m)^{m+1} \cdot (1 + bH_n)^{n+1}, $$
where $H_m$ and $H_n$ are the hyperplane classes on $\mathbb{P}^m$ and $\mathbb{P}^n$, respectively.
Now, we can use the well-known formula for the pushforward of the Segre classes under the Segre embedding:
$$ \sigma_* s(E) = \sigma_* (s(\mathcal{O}_{\mathbb{P}^m}(a)) \cdot s(\mathcal{O}_{\mathbb{P}^n}(b))) = \sum_{l=0}^N c_l [\mathcal{O}_{\mathbb{P}^N}(l)], $$
where the coefficients $c_l$ depend on the Segre classes of $E$ and the degrees of the line bundles $\mathcal{O}_{\mathbb{P}^N}(l)$.
To compute the coefficients $c_l$, we can expand the product $(1 + aH_m)^{m+1} \cdot (1 + bH_n)^{n+1}$ and match the terms with the same degrees in the hyperplane classes $H_m$ and $H_n$. This expansion gives us the Segre classes of $E$ in terms of the Chern classes of $\mathcal{O}_{\mathbb{P}^m}(a)$ and $\mathcal{O}_{\mathbb{P}^n}(b)$. Then, using the pushforward formula for the Segre classes, we can compute the coefficients $c_l$ and express the pushforward $\sigma_*(E)$ in K-theory as a linear combination of the classes $[\mathcal{O}_{\mathbb{P}^N}(l)]$ for $l = 0, \dots, N$.
Note that this computation can be quite involved and might require the use of a computer algebra system to carry out the expansions and match the terms with the same degrees. Once the coefficients $c_l$ are obtained, the desired expression for the pushforward in K-theory is
$$ [\sigma_*(\mathcal{O}_{\mathbb{P}^m}(a) \boxtimes \mathcal{O}_{\mathbb{P}^n}(b))] = \sum_{l=0}^N c_l [\mathcal{O}_{\mathbb{P}^N}(l)]. $$
The coefficients $c_l$ capture the information about the interaction between the Segre classes of $E$ and the degrees of the line bundles $\mathcal{O}_{\mathbb{P}^N}(l)$. They provide a way to describe the pushforward $\sigma_*(\mathcal{O}_{\mathbb{P}^m}(a) \boxtimes \mathcal{O}_{\mathbb{P}^n}(b))$ as a combination of line bundles on $\mathbb{P}^N$.
In summary, the problem of expressing the pushforward of the external tensor product of line bundles on a product of projective spaces under the Segre embedding in K-theory can be solved by computing the coefficients $c_l$ that result from the expansion of the product of Segre classes and the pushforward formula. This computation can be complex and may require the aid of computer algebra systems to perform the expansions and match the terms with the same degrees.
Corrections
$E$ has rank 1, and the pushforward $\sigma_*(E)$ is a coherent sheaf, not a vector bundle.
Example
Now, let's consider a small example to illustrate the computation of $c_l$.
Let $m = n = 1$. In this case, the Segre embedding $\sigma: \mathbb{P}^1 \times \mathbb{P}^1 \hookrightarrow \mathbb{P}^3$ is given by $(x_0:x_1, y_0:y_1) \mapsto (x_0y_0:x_0y_1:x_1y_0:x_1y_1)$, and the degree of the Segre embedding is $d = 2$.
Now, let $E = \mathcal{O}_{\mathbb{P}^1}(a) \boxtimes \mathcal{O}_{\mathbb{P}^1}(b)$. We want to compute the pushforward $\sigma_*(E)$.
First, we need to find the Segre class $s(E)$. For line bundles on projective spaces, the Segre class is simply $1 + c_1(E)$. In our case,
$$ s(E) = (1 + a[\mathbb{P}^1]) \otimes (1 + b[\mathbb{P}^1]). $$
Now, we apply the pushforward formula:
$$ \sigma_*(s(E)) = (1 + a[\mathbb{P}^1]) \cdot (1 + b[\mathbb{P}^1]) \cdot s(\mathcal{O}_{\mathbb{P}^3}(2)). $$
Since $s(\mathcal{O}_{\mathbb{P}^3}(2)) = 1 - 2[\mathbb{P}^3]$, the formula becomes:
$$ \sigma_*(s(E)) = (1 + a[\mathbb{P}^1])(1 + b[\mathbb{P}^1])(1 - 2[\mathbb{P}^3]). $$
Expanding the product, we obtain:
$$ \sigma_*(s(E)) = 1 - 2[\mathbb{P}^3] + (a+b)[\mathbb{P}^1] - 2ab[\mathbb{P}^1][\mathbb{P}^3]. $$
Now, we can match the terms with the same degrees in the expansion. In this case, $c_0 = 1$, $c_1 = 0$, $c_2 = a+b$, and $c_3 = -2ab$. Therefore, the pushforward of $E$ in K-theory is given by:
$$ [\sigma_*(E)] = 1[\mathcal{O}_{\mathbb{P}^3}(0)] + 0[\mathcal{O}_{\mathbb{P}^3}(1)] + (a+b)[\mathcal{O}_{\mathbb{P}^3}(2)] - 2ab[\mathcal{O}_{\mathbb{P}^3}(3)]. $$
Now that we have computed the pushforward of $E$ in K-theory, we can use this result to analyze the behavior of the pushforward for different values of $a$ and $b$. The expression for the pushforward in K-theory is given by:
$$ [\sigma_*(E)] = 1[\mathcal{O}_{\mathbb{P}^3}(0)] + 0[\mathcal{O}_{\mathbb{P}^3}(1)] + (a+b)[\mathcal{O}_{\mathbb{P}^3}(2)] - 2ab[\mathcal{O}_{\mathbb{P}^3}(3)]. $$
Notice that the coefficients $c_l$ of the pushforward depend on the values of $a$ and $b$, and they determine the linear combination of the line bundles $\mathcal{O}_{\mathbb{P}^3}(l)$ that forms the pushforward.
For example, if $a = b = 1$, then the pushforward is given by:
$$ [\sigma_*(E)] = 1[\mathcal{O}_{\mathbb{P}^3}(0)] + 0[\mathcal{O}_{\mathbb{P}^3}(1)] + 2[\mathcal{O}_{\mathbb{P}^3}(2)] - 2[\mathcal{O}_{\mathbb{P}^3}(3)]. $$
In this case, the pushforward is a linear combination of the trivial line bundle $\mathcal{O}_{\mathbb{P}^3}(0)$, the line bundle $\mathcal{O}_{\mathbb{P}^3}(2)$ with coefficient 2, and the line bundle $\mathcal{O}_{\mathbb{P}^3}(3)$ with coefficient -2.
As another example, if $a = 1$ and $b = -1$, then the pushforward is given by:
$$ [\sigma_*(E)] = 1[\mathcal{O}_{\mathbb{P}^3}(0)] + 0[\mathcal{O}_{\mathbb{P}^3}(1)] + 0[\mathcal{O}_{\mathbb{P}^3}(2)] - 2[\mathcal{O}_{\mathbb{P}^3}(3)]. $$
In this case, the pushforward is a linear combination of the trivial line bundle $\mathcal{O}_{\mathbb{P}^3}(0)$ and the line bundle $\mathcal{O}_{\mathbb{P}^3}(3)$ with coefficient -2.
These examples illustrate how the pushforward of $E$ in K-theory can be expressed as a linear combination of line bundles $\mathcal{O}_{\mathbb{P}^3}(l)$, with coefficients that depend on the values of $a$ and $b$. This information can be useful for understanding the geometry of the Segre embedding and the behavior of the pushforward under different choices of line bundles on the product $\mathbb{P}^1 \times \mathbb{P}^1$.