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Algebraicity of Hodge loci

Let $X$ be a smooth complex projective variety. If $Y$ is a subvariety of codimension $k$, then it gives rise to a cycle class $[Y]$ in $H^{2k}(X)$, and it is reasonable to ask which cohomology classes are “algebraic” in the sense of arising from (rational sums of) cycle classes. The Hodge conjecture gives a proposed answer: The algebraic classes are exactly the rational cohomology classes which are also contained in the appropriate component of the Hodge decomposition. We call such a rational cohomology class a Hodge class. Now, consider a family $X \to B$ of smooth projective varieties. Then as $t \in B$ varies, how does the dimension of the space of Hodge classes $\mathrm{Hdg}^{2k}(X_t)$ change? We define the Hodge locus to be the set of $t \in B$ for which this dimension is not minimal. If the Hodge conjecture is true, then one can show that the Hodge locus is algebraic, in the sense that it is a countable union of algebraic sets.

Cattani–Deligne–Kaplan showed that this consequence holds unconditionally, without having to invoke the Hodge conjecture. More recently, Bakker–Klingler–Tsimerman gave a considerably simpler proof of the same result which is based on o-minimal geometry. The goal of the seminar is to learn this proof. If there is extra time, we will cover other topics related to Hodge loci depending on interest.

References#

  • C. Voisin, Hodge loci, link
  • B. Bakker, B. Klingler, J. Tsimerman, Tame topology of arithmetic quotients and algebraicity of Hodge loci, link

Schedule#

  1. Introduction (September 25th, Spencer)
  2. Variations of Hodge structures (October 2nd, Shurui): Voisin section 2
  3. Hodge loci (October 9th, Ronnie): Voisin section 3, review of algebraic groups, define Siegel sets if time permits
  4. Introduction to o-minimality (October 16th, Spencer): up through the definable Chow theorem
  5. Borel–Serre compactifications (October 23rd, Vaughan): BKT sections 2.2–2.3, more on Siegel sets and Borel–Serre compactifications
  6. Definability of arithmetic quotients (October 30th, Daniel): BKT, section 2.1 and section 3
  7. Definability of the period map, part 1 (November 6th, Jiahao): BKT sections 4.1–4.2
  8. Definability of the period map, part 2 (November 13th, Stepan): prove theorem 1.5, following sections 4.3–4.5