In this talk, I will first introduce the background of Bridgeland stability condition. Then I will mention some existence result of Bridgeland stability. Next I will prove the Bogomolov-Gieseker type inequality of X(2,4), Calabi-Yau threefold of complete intersection of quadratic and quartic hypersufaces, by proving the Clifford type inequality of the curve X(2,2,2,4). Then this will provide the existence of Bridgeland stability condition of X(2,4).
I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a 'mirror P=W' conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting.
This is a report on work in progress with my student Cristian Rodriguez. The mirror of a ℚ-Fano 3-fold with b2 = 1 is a rigid K3 fibration over ℙ1 such that Hodge bundle is degree 1 and some power of the monodromy at infinity is maximally unipotent. Although prior work focused on the maximally unipotent case (without base change), perhaps a classification of such Picard-Fuchs equations is possible.
In the smooth case these fibrations were described explicitly by Przyjalkowski, and Doran-Harder-Novoseltsev-Thompson showed that they are given by etale covers of the (1-dimensional) moduli of rank 19 K3 surfaces. In the case of a single 1/2(1,1,1) singularity they are given by rigid rational curves on the (2-dimensional) moduli of rank 18 K3 surfaces, and examples suggest they are Teichmuller curves in A2 (via the Shioda-Inose correspondence relating rank 18 K3s and abelian surfaces), as studied by McMullen.
If a Calabi-Yau threefold varies in a one-parameter family and aquires some double points, a small resolution will produce a rigid space. The local monodromy at such a 'conifold transition' is of infinite order. In the talk I report on some work done with S. Cynk (Krakow), which shows similar transitions to rigid Calabi-Yaus are possible with monodromy of finite order, in sharp distinction to what can happen for K3 surfaces.
In the recent breakthrough on the uniform Mordell-Lang problem by Dimitrov-Gao-Habegger and Kuhne, their key result is a uniform Bogomolov type of theorem for curves over number fields. In this talk, we introduce a refinement and generalization of the uniform Bogomolov conjecture over global fields, as a consequence of bigness of some adelic line bundles in the setting of Arakelov geometry. The treatment is based on the new theory of adelic line bundles of Yuan-Zhang and the admissible pairing over curves of Zhang.
Fano varieties of (derived) K3 type are Fano varieties whose Hodge structure (derived category) contains a K3-type sub-Hodge structure (subcategory). Many examples of such varieties are known, arising as zeroes of homogeneous bundles on Grassmannians, in dimensions that grow up to 19. In this talk, I will first present joint work with Fatighenti and Manivel showing that many of these examples can be related by geometric correspondences and have actually the same K3-type Hodge structure. I will also present an ongoing project with Fatighenti, Manivel and Tanturri, whose aim is to show that in the case of Fano fourfolds, the only possible K3-type structures which are not actual K3 can arise from Gushel-Mukai, cubics and Küchle c5 fourfolds.
You must be logged in to post a comment.