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I was wondering,

What could be a geometric/intuitive meaning of taking the (tensor) product of two line bundles on a smooth variety. Could you depict this to me with some example?

Computationally it is very clear in terms of transition functions: we are just taking the product of the two. What I miss is a mental picture of $L_1\otimes L_2$.

These thoughts originated from considering the Adjunction Formula

$K_V = (K_X \otimes [V])_V$

Here $X$ is a compact manifold and $K_V$ is the canonical bundle of a smooth hypersurface $V\subset X$, expressed in terms of the canonical bundle of $X$ times $[V]$, the line bundle of $X$ associated to $V$.

So a second and more specific question is about the geometric meaning of the adjoint bundle $K_X \otimes [V]$ on $X$. Could you depict this to me with examples? Thanks!

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1 Answer 1

up vote 5 down vote accepted

About your first question:

Since you're variety is smooth, you can think of (isomorphism classes of) line bundles as (equivalence classes of) Weil divisors. The tensor product then corresponds to the sum of Weil divisors, which is very geometric.

Intuitively this correspondence also makes a lot of sense, since you would certainly expect the zeroes of a product to be the union of the zeroes of the two individual sections.

As an example i would take lines in the projective plane $\mathbb{P}^2$. A bundle with degree 2, normally denoted $\mathcal{O}_{\mathbb{P}^2}(2)$ corresponds to a Weil divisor given by a conic. Similarly a bundle of degree 1 $\mathcal{O}_{\mathbb{P}^2}(1)$ corresponds to a degree 1 curve: a line.

The tensor product is $\mathcal{O}_{\mathbb{P}^2}(3)$, the sum of the Weil divisors is a conic plus a line, which is a (degenerate, but nonetheless a) cubic curve. This makes total sense: the tensor product has degree three so we know a priori that a corresponding Weil divisor must have degree 3.

As for the second question, a nice proof is given in Georges' answer to a question of mine posted earlier: Adjunction for varieties with higher codimension I don't think you're interested in the question, but you might be in the answer, it is relevant for your question.

Finally, i don't know any specific geometric interpretation of the adjoint bundle. Maybe someone else knows. I can however tell you two places where it pops up a lot.

First, it is part of the Riemann-Roch formula on surfaces which is an immensely useful theorem in the theory of algebraic surfaces. Possbily it is also used in other Riemann-Rochs, i don't know.

Second, if the dimension of $X$ is $n$, Serre duality gives roughly an isomorphism $$ H^i(X, \mathcal{L}) \cong H^{n-i}(X, K_X \otimes \mathcal{L}^{-1}) $$ again an occurence of this adjoint bundle as you call it.

Note that i wrote roughly: the precise statement of Serre duality is different and i am simplyfing a bit (in a harmless way for the mental picture).

I hope this helps.

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