Welcome to the webpage of Tanya Kaushal Srivastava. I am an Algebraic Geometer.

Current Position:

Insfosys Postdoctoral Fellow

Chennai Mathematical Institute,

H1, SIPCOT IT Park, Siruseri
Kelambakkam 603103
India

Email: tanyak(at)cmi(dot)ac(dot)in

Till October 2020, I was a  postdoc in Hausel group  at IST Austria.  Before that I was a PhD student of Prof. Dr. Hélène Esnault at Freie Universität Berlin.  And even before that I was a BS-MS student at IISER Mohali, India.

##### Reserach Interests

The question that I am really curious about:

Which geometrical properties of an algebraic variety  can be determined by its derived category of coherent sheaves?

Attempts to answer this question has related a lot of fields of mathematics ranging from differential geometry, algebraic geometry, representation theory to number theory  and even to physics, notably string theory.

##### Teaching

I taught a course on Moduli Space of Curves in Spring 2020  at IST Austria 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.

## Accepted for Publication

• Pathologies of Hilbert scheme of points on supersingular Enriques surface in characteristic 2 (Arxiv Link) , accepted Bulletin des Sciences Mathematiques, 2021.

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 G\”ottsche-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 $\mathbb{C}$. 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.

## Under Preparation

• 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.

## Invited Talks

• Fourth Meeting for Young Women in Mathematics, University of Freiburg, May 1-3, 2019.
• University of Hannover at Guest Research Seminar in the group of Prof. Dr. Matthias Schuett, 13 December 2018.
• University of Bergen at Guest Research Seminar in the group of Prof. Dr. Sofia Tirabassi, 30 May 2016, Norway.

## Talks at seminars

• Talk on Varieties with trivial canonical bundle, IST Austria, August 2020.
• Talk on Lie algebra attached to smooth projective variety and hyperkahlers, IST Austria, March 2020.
• Talks on Virtual fundamental classes and dg scheme in Seminar on “Derived Algebraic Geometry”, IST Austria, Summer 2019.
• Mini Lecture series on Derived categories of Ordinary K3 surfaces in Working Seminar, IST Austria, Spring 2019.
• Talk on Formal Patching in Seminar on “Abhyankar’s Conjecture”, Freie Universitat Berlin, Summer 2018.
• Talk on Semistable Higgs Bundles in Seminar on “$p$-adic Simpson correspondence”, Freie Universit\”at Berlin, Winter 2017-18.
• Talk on Introduction to Deformation theory in Seminar on “Formal Geometry and Deformation Theory”, Freie Universit\”at Berlin, Winter 2017-18.
• Talk on Locally Noetherian Formal Schemes in Seminar on “Formal Geometry and Deformation Theory”, Freie Universit\”at Berlin, Winter 2017-18.
• Talk on The Comparison Theorem in Seminar on “Formal Geometry and Deformation Theory”, Freie Universitat Berlin, Winter 2017-18.
• Talk on Derived categories and semi-orthogonal decompositions in Seminar on “Derived categories and variation of GIT quotients”, Freie Universit\”at Berlin, Winter 2017-18.
• Talk on Crystalline Cohomology and crystalline sheaves in Seminar on “$p$-adic Hodge Theory”, Freie Universitat Berlin, Summer 2017.
• Talk on Almost Mathematics and Purity Theorem in Seminar on “$p$-adic Hodge Theory”, Freie Universitat Berlin, Summer 2017.
• Talk on Shtukas for GL2, II in Seminar on “Langlands correspondence for function fields”, Freie Universitat Berlin, Summer 2017.
• Talk on Koll\’ar’s paper on Nonrational hypersurfaces in Seminar on “Rationality of varieties and decomposition of the diagonal”, Humboldt Universitat, Winter 2016-17.
• Talk on $p$-adic integration and the Igusa zeta function in Seminar on “Berkovich spaces, birational geometry and motivic zeta functions”, Freie Universitat Berlin, Winter 2016-17.
• Talk on Comparison Isomorphisms and $p$-adic Hodge theory in Seminar on “Crystalline cohomology”, Freie Universitat Berlin, Winter 2016-17.
• Talk on Properties of crystalline Cohomology in Seminar on “Crystalline Cohomology”, Freie Universitat Berlin, Winter 2016-17.
• Talk on Introduction to Crystalline Cohomology in Seminar on“Crystalline cohomology”, Freie Universitat Berlin, Winter 2016-17.
• Talk on D-modules of K3 surfaces and Unirational varieties in Seminar on “D-modules”, Freie Universitat Berlin, Summer 2016
• Talk on Skeleta of curves in Seminar on “Berkovich Spaces”, Humboldt Universit\”at Berlin, Summer 2016
• Talk on Model Categories in Seminar on “Motivic Galois groups and periods”, Freie Universitat Berlin, Summer 2016
• Mini lecture series on Introduction to Derived Categories, Freie Universitat Berlin, Winter 2015-16.
• Talk on Supersingular K3 crystal in Seminar on “Supersingular K3 surfaces are Unirational”, Freie Universitat Berlin, Winter 2015-16.
• Talk on Functoriality, Affine stratification and Chow ring of Projective varieties in Seminar on “Intersection Theory”, Freie Universitat Berlin, Winter 2015-16.