6/5/2023 0 Comments Algebraic geometryLet X be a projective algebraic variety over the field of complex numbers.Algebraic geometry is a branch of mathematics which classically studies zeros of multivariate polynomials. Lurie, Deligne-Illusie from the Prismatic Perspective (slides). Moreover, in certain examples the algebraic Frobenius structures for different primes fit together in a manner that as yet lacks a general explanation. When the connection is rigid (recovering a result of Esnault-Groechenig) or superrigid (extending work of Klevdal-Patrikis). In particular, when the connection is of geometric origin, we obtain an isomorphism of vector bundles with connection which constitutes an “algebraic Frobenius structure”.Ĭonversely, in certain cases one can establish this isomorphism directly to show that E is an isocrystal, e.g., (in a sense which we will make precise in the talk). We construct another vector bundle with logarithmic connection which interpolates the pullbacks along all analytic Frobenius lifts on $X$ Then for any vector bundle $E$ on $X$ equipped with a logarithmic (for $Z$) integrable connection, for almost all primes $p$ Let $Z$ be a strict normal crossings divisor on $X$. Let $X$ be a smooth projective variety over $Q$. To the obstruction to lifting Frobenius of $X_0$ to $X_1$ and a certain cohomology class arising from the discrepancy between $S^p$ and $\Gamma^p$. I will state Petrov’s basic formula relating the first possibly non-trivial extension class in the canonical filtration of the de Rham complex of $X_0/k$ In particular, for $X_0/k$ smooth, lifted to $X_1$ over $W_2(k)$, I will present Petrov’s example, and give an idea of the strategy of its proof. The purpose of this and the next lecture is to give an idea of the main points and techniques in the proofs,Īs well as to discuss a number of related results pertaining to de Rham cohomology of smooth schemes over $k$ and their liftings over $W_2(k)$ or $W(k)$. In addition, Petrov showed that the Sen operator of Bhatt-Lurie on $X_0$ is not semisimple. Thus solving a question raised by Deligne-Illusie in 1987. Such that the Hodge to de Rham spectral sequence of $X_0 = X \otimes k$ does not degenerate at $E_1$, Petrov has recently constructed a projective, smooth scheme $X/W(k)$, of relative dimension $p 1$, Let $k$ be a perfect field of characteristic $p>0$.Ī. Illusie, Non-decomposability of the de Rham complex and non-semisimplicity of the Sen operator, after A. Then, I will formulate some conjectures about canonical flows on the space of stability conditions that imply several important conjectures on the structure of derived categories, such as the D-equivalence conjecture and Dubrovin’s conjecture. I will present a new unifying framework for studying these semiorthogonal decompositions using Bridgeland stability conditions. There are many situations in which the derived category of coherent sheaves on a smooth projective variety admits a semiorthogonal decomposition that reflects something interesting about its geometry. The noncommutative minimal model program (slides). This has natural consequences for derived equivalences between Kummer fourfolds. We do this by giving an explicit description of the symplectic involutions on the fourfolds. We have extended the results of Hassett and Tschinkel and characterized the Galois action on the even cohomology. In recent joint work with Katrina Honigs, we study Kummer-type fourfolds over arbitrary fields which arise as moduli spaces of stable sheaves on an abelian surface. The middle cohomology of hyperkahler fourfolds of Kummer type was studied by Hassett and Tschinkel, who showed that a large portion is generated by cycle classes of fixed loci of symplectic involutions. This is work in progress and we shall discuss the projective case during the lecture.įrei, Symplectic involutions of Kummer-type fourfolds. While the earlier proof was via characteristic $p$, the new one is purely $p$-adic and uses $p$-adic topology. This improves our earlier work where, if $X$ was not projective, we assumed a strong cohomological condition (which is fulfilled for Shimura varieties of real rank $\geq 2$),Īnd we obtained only infinitely many $\mathbb F_q$ of growing characteristic.
0 Comments
Leave a Reply. |