# Homepage of  Tanya Kaushal Srivastava

IST Postdoctoral Fellow
Hausel’s group
IST Austria.

I am soon moving to CMI, India and this webpage has been moved to www.mathface.com

### Teaching

I am teaching a course on Moduli Space of Curves in Spring 2020 and you will find the details here

I took a (virtual) Quiz as a final exam for the course, you can find the details and quiz slides here

## Preprints

• Lifting Automorphisms on Abelian Varieties as Derived Autoequivalences (under minor revision for Archiv der Mathematik) (Preprint pdf)

I show that for Abelian Varieties, every automorphism lifts as a morphism if and only if it lifts as a derived autoequivalence to a characteristic zero lift. But even for the case of elliptic curves, there are more ways to lift an automorphism as an autoequivalence than just as a morphism, although in this case the autoequivalence we get are just twist of the lift of automorphism with the lift of the structure sheaf of the graph of the automorphism.

## In preparation

• Pathologies of Hilbert scheme of points on supersingular Enriques surface in characteristic 2 (Email for draft)

I show that Hilbert schemes of points on supersingular Enriques surface in characteristic 2, $Hilb^n(X)$, for $n \geq 2$ are simply connected, symplectic varieties but are not irreducible symplectic as Hodge number $h^{2,0} > 1$, even though a supersingular Enriques surface is an irreducible symplectic variety. These are new classes of varieties which appear only in characteristic 2 and show that the hodge number formula for Gottsche-Soergel does not hold over characteristic 2. It also gives examples of varieties with trivial canonical class which are neither irreducible symplectic nor Calabi-Yau, thereby showing that there are strictly more classes of simply connected varieties in positive characteristic than as given by Beauville-Bogolomov decomposition theorem over $\C$. Moreover, they give examples of varieties in each dimension 2n that admit liftings to characteristic zero to varieties which do not have trivial canonical bundle.

• Fully Faithfulness Criterion for Functors of (Twisted) Derived Categories, joint with Katrina Honigs

We give another proof for the Caldararu’s criterion of checking when a twisted Fourier-Mukai functor gives an equivalence and we are working on extending a modified version of this criterion to positive characteristic and even more generally to the setting of smooth proper Deligne Mumford stacks in positive characteristic, the coarse moduli space of such stacks can have at worst quotient singularities, which being in positive characteristic need not be even Cohen Macaulay, unlike in the case of characteristic zero.

• Counting Twisted Derived Equivalent Ordinary K3 Surfaces, joint with Sofia Tirabassi and Piotr Achinger.

We show that every Brauer class over an ordinary K3 surface has a preferred lift to the canonical lift of the underlying K3 surface and then we are working on a theory of moduli space of twisted K3 surfaces in characteristic p to be able to count the number of twisted derived Fourier-Mukai partner of an ordinary K3 surface.

## Notes/Slides

• Slides on Talk at IST Austria (19 June 2020) on Varieties with trivial canonical bundle in positive characteristic: pdf

This talk gives examples of Varieties with trivial canonical bundle in positive characteristic, which are irreducible symplectic varieties but don’t lift to Irreducible symplectic varieties (or Hyperkahlers) in char 0. Talk was given in the spirit of the remark that we don’t have a good definition of irreducible symplectic Varieties in  positive characteristic.

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