For a cup of coffee more

This post is long overdue, but being in the middle of nowhere helps finally progressing a little.

In the last post, I discussed a lemma of Kronecker. Let me discuss another version

Lemma (Kronecker, second version) Consider A, B linear maps from a vectorspace X to another one Y over any infinite field. Assume that im(A) and B ker(A) intersect trivially. Then the generic linear combination of A and B has kernel equal to the intersection of kernels of A and B, in other words, the kernel is as small as possible.

This is a very powerful tool to construct high-rank maps in vector spaces, in particular in the case when X and Y are dual in a Poincaré duality algebra. Then im A and ker A are orthogonal complements, and linear algebra tells us that orthogonal complements intersect trivially if and only if the Poincaré pairing does not degenerate on either space. This is exploited in my proof of the g-conjecture to construct Lefschetz elements, see here, and also in this survey (which I was invited to for winning the EMS prize, which I still cannot quite believe).

I call this property, that a pairing does not degenerate on certain subspaces, the biased pairing property. The relevance of biased pairings to the Lefschetz property was then rediscovered by two Greek mathematicians, Stavros Papadakis and Vasiliki Petrotou, who gave a marvellous second proof of the g-conjecture, though initially only in characteristic two.

They proved stronger that in a certain field extension (and characteristic two), the pairing does not degenerate at ANY ideal (let’s call this total anisotropy). This is quite marvellous. In essence, they proved that if the Artinian reduction of the equivariant cohomology ring of the variety is chosen along indendent transcendental variables, then the fundamental class deg satisfies

where is a partial differential after the transcental variables in question, specifically in direction of vertices of a simplex . Since the right-hand has to be non-zero for SOME simplex . Hence is nonzero, and the biased pairing property holds for every principal ideal, and in particular any ideal.

We later extended this partially to general characteristic, providing a second complete proof of the g-conjecture, see here. So far, so good. You can see a detailed explanation in the Hadamard Lectures I was honored to give, here.

Anisotropy and biased pairings seems to be of great importance in generic algebraic geometry going forward, and I have been wondering how far this extends, in particular the identity above and biased pairings. In any case, you can count on some exciting news in this direction soon.