LisMath Seminar 

13/07/2023, 14:00 — 14:30 — Online
Javier Orts, Instituto Superior Técnico, Universidade de Lisboa

Symmetric products of Galois-maximal varieties

A maximal space is a finite $C_2$-CW complex $X$ whose cohomology and the cohomology of its fixed points are related by the following equality:

$$\sum_{q=0}^{\dim X^{C_2}} \dim_{F_2} H^q(X^{C_2};F_2) = \sum_{q=0}^{\dim X} \dim_{F_2} H^q(X;F_2).$$

It is a fact that the action of $C_2$ on $H^*(X;F_2)$ is trivial when $X$ is a maximal space.

This class of spaces was generalised by Krasnov in [5] by allowing the action on cohomology to be non-trivial and modifying the above equality conveniently to take this into account. The new spaces are called Galois-maximal spaces. The equation that characterises them is the following:$$\sum_{q=0}^{\dim X^{C_2}} \dim_{F_2} H^q(X^{C_2};F_2) = \sum_{q=0}^{\dim X} \dim_{F_2} H^1(C_2, H^q(X;F_2)).
$$

Maximal and Galois-maximal spaces have always been of great interest in Real algebraic geometry [2], where the CW complex is now the complex locus of a variety over $\mathbb{R}$ and the $C_2$ action is given by complex conjugation, but have became especially relevant in the recent years due to the following result by Biswas and D'Mello [3]:

Theorem 1:
If $X_g$ is a maximal curve, then the symmetric product $SP_n X_g$ is a maximal variety for $n=2,3$ and $n\geq 2g-1$.

This problem was fully solved by Franz [4] for the case of maximal varieties, and by Baird [1] in the case of Galois-maximal curves.
The main result of this talk will address the general case for Galois-maximal varieties:

Theorem 2:
If $X$ is a Galois-maximal variety, then $SP_n X$ is a Galois-maximal variety for ever $n$.

References:

[1] Baird, T. J., “Symmetric products of a real curve and the moduli space of Higgs bundles”, J. Geom. Phys. 126, 7–21 (2018).

[2] Bertrand, B. “Asymptotically maximal families of hypersurfaces in toric varieties”, Geom. Dedicata. 118, 49–70 (2006).

[3] Biswas, I., D’Mello, S.,“M-curves and symmetric products”, Proc. In- dian Acad. Sci. Math. Sci. , 127, No. 4, 615–624 (2017).

[4] Franz, M., “Symmetric products of equivariantly formal spaces”, Canad. Math Bull. 61, No. 2,272–281 (2018).

[5] Krasnov, V. A., “Harnack–Thom inequalities for mappings of real al- gebraic varieties”, Izv. Akad. Nauk SSSR Ser Mat., 47, No. 2, 268–297 (1983).