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most recent 30 from 2018-10-17T02:59:20Z

Probability estimate with a Lipschitz, weak* semicontinuous function on the $\ell^\infty$ unit ball

Sat, 10/13/2018 - 12:51

Suppose that $X_i$ for $i=0,1,\dots$ is an i.i.d. sequence of uniformly distributed random variables taking on values in $[-1,1]$. Fix a real number $L>0$ and suppose that $f_n:[-1,1]^n\rightarrow [-1,1]$ for $n=0,1,\dots$ is a sequence of functions that are $L$-Lipschitz with regards to the max metric on $[-1,1]^n$, i.e. $$|f_n(x_0,\dots,x_{n-1})-f_n(y_0,\dots,y_{n-1})|\leq L \max\{|x_0-y_0|,\dots,|x_{n-1}-y_{n-1}|\}$$ for any $n$ and $x_0,\dots,x_{n-1},y_0,\dots,y_{n-1}\in [-1,1]$. Note that each of these functions interpreted as a function on the $\ell^\infty$ unit ball is weak* continuous and therefore the function $f(x_0,x_1,\dots)=\sup_nf_n(x_0,\dots,x_{n-1})$ is weak* semicontinuous (and in particular measurable). Also note that $f$ is still $L$-Lipschitz with regards to the $\ell^\infty$ norm.

Let $Y=f(X_0,X_1,\dots)$ be the random variable given by applying $f$ to the sequence $X_i$. I want to be able to say that $Y$ cannot be too sharply bimodal with a bound given by $L$. To be more precise I want to say that there is some $r(L)<\frac{1}{2}$ such that $\min\{P(Y\leq\frac{1}{3}),P(Y\geq \frac{2}{3})\}\leq r(L)$ for any sequence of $L$-Lipschitz functions $f_n$.

Does such an estimate exist for every $L$?

Casimir operator of su(2) and relation with its matrix representation

Sat, 10/13/2018 - 12:47

I'm following Gilmore's recipe to compute the Casimir operator of a given algebra (in this example, I refer to algebra su(2)). My problem is that I obtain different results according to the specific matrix representation that I choose. To be more concrete, look at the difference between these two cases:

  1. Consider the defining commutators: $$ [J_1,J_2]=iJ_3, \qquad [J_2,J_3]=iJ_1, \qquad [J_3,J_1]=iJ_2 $$ The following $3\times 3$ matrices are a representation of this algebra: $$ J_1=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \\ \end{array} \right), \qquad J_2=\left( \begin{array}{ccc} 0 & 0 & i \\ 0 & 0 & 0 \\ -i & 0 & 0 \\ \end{array} \right), \qquad J_3= \left( \begin{array}{ccc} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right). $$ Applying Gilmore's method (see pag. 140), one can write matrix $$ X= \sum_{i=1}^3a_iJ_i, \qquad a_i\in\mathbb{R} $$ One therefore obtains: $$ X=\left( \begin{array}{ccc} 0 & -i a_3 & i a_2 \\ i a_3 & 0 & -i a_1 \\ -i a_2 & i a_1 & 0 \\ \end{array} \right) $$ At this point one computes the characteristic polynomial $$ P(\lambda)=\mathrm{det}(X-\lambda\mathbb{I})=-\lambda^3+\lambda(a_1^2+a_2^2+a_3^2) $$ Performing the substitution $a_i\to J_i$ in the coefficient of $\lambda$, as prescribed by the algorithm, one obtains that the algebra's Casimir is $$ C=\left( \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{array} \right) $$ which indeed commutes with $J_i, \quad \forall i$. This scheme therefore gives a correct result.

  2. Now consider the following alternative (but equivalent) definition of the algebra su(2):
    $$ [J_+,J_-]=2 J_3, \qquad [J_3,J_+]=+J_+, \qquad [J_3,J_-]=-J_-. $$ The following $3\times 3$ matrices are a representation of this algebra: $$ J_+= \left( \begin{array}{ccc} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 0 & 2 & 0 \\ \end{array} \right), \qquad J_-=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ -2 & 0 & 0 \\ \end{array} \right), \qquad J_3=\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \\ \end{array} \right) $$ Introducing three unknown coefficients $a_+,\,a_-,\,a_3$, one can computes matrix $X$ and the characteristic polynomial $P(\lambda)$ as seen before. One obtains that $$ P(\lambda)= -\lambda^3 +\lambda(a_3^2+4a_+a_-) $$ At this point, one has to perform the substitution $a_i\to J_i$. Even if one takes into accunt the need for symmetrization, i.e. $a_+a_-\to (J_+J_-+J_-J_+)/2$, the algorithm retrieves an incorrect result, i.e. the following matrix $$ \bar{C}=\left( \begin{array}{ccc} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \\ \end{array} \right) $$ which does not commute with matrices $J_+$ and $J_-$ and so constitutes a wrong result.

My question is: why does the first way give the correct result but the second scheme does not work? Did I miss any hypothesis which is needed by Gilmore's algorithm? I suspect that the problem might be either in the use of $J_\pm$ as basis elements of algebra su(2) or in the substitutions $a^i\to J_i$ (the book uses, in fact, upper and lower indices, a formalism which I am not familiar with). Please, take into account that my target is to understand how to compute the abstract Casimir operator of a certain Lie algebra in an algorithmic way. The matrix representation of linear and quadratic operators is just auxiliary to reach the target but it is not my core business.

Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$

Sat, 10/13/2018 - 12:23

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ The $\cup_1$ is a higher cup product. The $Sq^1 x= x \cup_1 x$. It shall be true that $$ \mathcal{P}(x) \mod 2= x \cup x. $$

Question 1: If $ x \cup x =0 \mod 2$, and if $ x \cup_1 Sq^1 x =0 \mod 2$, is it true that $$ \mathcal{P}(x) =0 \mod 4? $$

  • If not, please provide some counter examples.


Question 2: $\mathcal{P}(x)$ is a well-defined invariant for the cobordism $\Omega^4_{SO}(B^2 \mathbb{Z}_2)=\mathbb{Z}_4$. Is $$ \frac{1}{2}(\mathcal{P}(x) -x^2) \mod 2 = x \cup_1 Sq^1 x \mod 2 $$ a well-defined invariant of the cobordism $\Omega^4_{Spin}(B^2 \mathbb{Z}_2)=\mathbb{Z}_2$?

  • If not, please provide the correct way to write the cobordism generator of $\Omega^4_{Spin}(B^2 \mathbb{Z}_2)=\mathbb{Z}_2$.

Laplacian of an infinite graph and connected components

Sat, 10/13/2018 - 11:33

For a finite graph with undirected, unweighted edges, a well-known result is that the dimension of the null space of the Laplacian matrix gives the number of connected components. Does this result apply to infinite graphs as well?

The infinite graphs I'm interested are locally finite. That is, the degree of each node is finite. In my case, the number of nodes is countably infinite, there are no self-edges, and the edges are undirected.

At least according to the PDF from this course:

the connected components theorem does not assume anything about finite graphs. Can someone provide a reference (a paper or text) where this is explicitly discussed?

Thank You!

What are the difficulties involved in proving that the Kunen inconsistency holds in $NGB$

Sat, 10/13/2018 - 10:19

or (contrariwise) that $NGB$ + "There exists a Reinhardt cardinal" is consistent?

The question is partially in the title. $NGB$ is used for the reasons stated in the Hamkins, Kirmayer, and Perlmutter paper, "Generalizations of the Kunen inconsistency", Annals of Pure and Applied Logic, 163 (2012) pp. 1873-1876 (Section 1, "A few metamathematical preliminaries").

In order that this question should not be deemed overbroad, answers must be limited in scope to specific mathematical difficulties (eg. the Erdos-Hajnal theorem used in in proving the Kunen inconsistency in $NGBC$ or $NBC$ + $Choice$ has no known proof in $NGB$ or is known not to be provable without choice--however, those who would say that the Erdos-Hajnal theorem cannot be proven without choice must show how this fact allows for a non-trivial elementary embedding $j$: $V$ $\rightarrow$ $V$ and the existence of the requisite critical point $\kappa$ in $NGB$). Those who are so inclined might make mention of Rupert McCallum's recent attempt in showing that "the existence of a non-trivial elementary embedding $j$: $V_{\lambda + 2}$ $\rightarrow$ $V_{\lambda + 2}$ for some ordinal $\lambda$ not consistent with $ZF$" (quote from McCallum's withdrawn short preprint, The Choiceless Cardinals are Inconsistent, arXiv:1712.09768, which is discussed on Joel David Hamlin's blog — I would be especially interested in getting a nice explanation of the gap in this argument and the Laver paper which Gabriel Goldberg directed him to).

Though at first glance the answers to these questions might seem "primarily opinion-based", the mathoverflow community should realize the value of informed opinion and (yes) informed speculation regarding this topic, and use this question as an opportunity to provide these 'clues' (think of the proof of the 4-color theorem) to both the mathoverflow community and the mathematical community as a whole.

Improvement of Chernoff bound in Binomial case

Sat, 10/13/2018 - 01:44

We know from Chernoff bound $P\bigg(X \leq (\frac{1}{2}-\epsilon)N\bigg)\leq e^{-2\epsilon^2 N}$ where $X$ follows Binomial($N, \frac{1}{2}$).

If I take $N=1000, \epsilon=0.01$, the upper bound is 0.82. However, the actual value is 0.27. Can we improve this Chernoff bound?

Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$?

Fri, 10/12/2018 - 20:31

The name pretty much says it all: there's a well-defined operation on categories equivalent to modules over some ring: if $\mathcal{C}_1=A\operatorname{-mod}$ and $\mathcal{C}_2=B\operatorname{-mod}$, then $\mathcal{C}_1\boxtimes\mathcal{C}_2 =A\otimes B\operatorname{-mod}$ is independent of the choice of $A$ and $B$ (since Morita equivalence of one of the factors will induce Morita equivalence of the tensor product).

Is there some clever name for this operation?

Apologies if I should know this; it's not such an easy thing to Google for.

Lattice points in a square pairwise-separated by integer distances

Fri, 10/12/2018 - 18:08

Let $S_n$ be an $n \times n$ square of lattice points in $\mathbb{Z}^2$.

Q1. What is the largest subset $A(n)$ of lattice points in $S_n$ that have the property that every pair of points in $A(n)$ are separated by an integer Euclidean distance?

Is it simply that $|A(n)| = n$? And similarly in $\mathbb{Z}^d$ for $d>2$?

          $5 \times 5$ lattice square, $5$ collinear points.

Q2. What is the largest subset $B(n)$ of lattice points in $S_n$, not all collinear, that have the property that every pair of points in $B(n)$ are separated by an integer Euclidean distance?

          $5 \times 5$ lattice square, $4$ noncollinear points. A $9 \times 9$ example with $5$ noncollinear points, also based on $3{-}4{-}5$ right triangles, is illustrated in the Wikipedia article Erdős–Diophantine graph.

Behaviour of direct limit with matrices

Fri, 10/12/2018 - 02:02

I am trying to understand direct limits in the category of $C^*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts:

Let $(A_n,f_n)$ be a direct sequence of $C^*$-algebras. Does the direct limit behave well with matrices i.e. $$\lim_{\rightarrow} M_2(A_n)=M_2(\lim_{\rightarrow} A_n)$$ Where for the system $M_2(A_n)$ the connecting maps are the natural maps obtained using $f_n$ componentwise.

I do feel like the result should be true, but I don’t really have an argument. Any ideas?

Is transcendental Goldbach Conjecture true of the real numbers?

Fri, 10/12/2018 - 00:59

Let $x0$ be the real number $Pi$, Consider the below sequence of real numbers:

  • $s{0}$ = .1415926535897932384626433832795028841971...
  • $s{1}$ = .415926535897932384626433832795028841971...
  • $s{2}$ = .15926535897932384626433832795028841971...
  • $s{3}$ = .5926535897932384626433832795028841971...
  • ...

In other words, the real number $s{n+1}$ is formed by shifting the decimal expansion of $s{n}$ to the left, eliminating the first digit after the decimal point of $s{n}$. Naturally, this sequence has a $lub$ called, say, the $Major$ number of $x0$ and denoted by $Major(x0)$. Similarly, $minor(x0)$ can be defined with the $glb$ of the sequence.

In general, we can define $Major(x)$, $minor(x)$ for any real number $x$ since it has a decimal expansion. Similarly, $Major(c)$, $minor(c)$ where $c$ is a complex could also be defined.

Given Champernowne($x$) means $x$ is a Champernowne number, let's now state the, say, transcendental Goldbach Conjecture (akin to Goldbach Conjecture of the natural numbers):

$($transcendental($e$) $\land$ transcendental($Major(e))$$)$ $\rightarrow$ $\exists$ $p1,p2$ [(Champernowne($p1$) $\land$ Champernowne($p2$)) $\land$ $Major(e)$ = $Major(p1)$ + $Major(p2)$ ].

Is this transcendental Goldbach Conjecture true of the real numbers?

Absolute transcendence

All this also suggests a definition for the notion of absolute transcendence for real numbers:

$absoluteTranscendental$($at$) $\leftrightarrow$ $\exists t[(at = Major(t)) \land transcendental(t) \land transcendental(Major(t)) ]$.

Note: in the above, $minor(t)$ can be used in lieu of $Major(t)$.

The familiar transcendental numbers $Pi$ and $e$ aren't just mathematical constants: they do play some roles in physics. It'd be interesting if it turns out the $Major$ and $Minor$ numbers of $Pi$ and $e$ reflect some physics property and are absolutely transcendental.

lower bound the probability of at least L collisions

Wed, 10/10/2018 - 23:51

Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$.

If we now ask for the amount of collisions $X$ in $M$, where a collision is defined as a pair of indices $(i,j)$, $i\neq j$ with $M[i]=M[j]$ we obviously have $\mathbb{E}[X]=\frac{\binom{|M|}{2}}{N}=\frac{\binom{\sqrt{LN}}{2}}{N}\approx L$.

But unfortunately this is just the expected value and gives no information about the probability distribution. Is there anything we can say about the probability of $X$ deviating from its expectation here?

We could define indicator variables $X_{i,j}$ with $X_{i,j}= 1 \Leftrightarrow M[i]=M[j]$. Because the elements of M are drawn independently at random it holds that $X_{i,j}\sim \textrm{Ber}_\frac{1}{N}$. Unfortunately these indicator variables are not independent, otherwise the number of collisions would be binomial distributed and we could use a Chernoff like argument. Another approach to bound the probability that $X\geq \alpha\mathbb{E}[X]$ for some $\alpha<1$ could therefore be to somehow bound the variance of the sum of indicator variables and use the Chebyshev inequality. So far I did not find a non-trivial way to bound this variance.

This problem looks so common, that I can not imagine, that it hasn't been studied exhaustively in literature. Unfortunately, I was not able to find a solution somewhere so far. Help in form of own calculations as well as literature suggestions would be appreciated

How to maximize the total auction price for a set of bids subject to bidder constraints

Wed, 10/10/2018 - 19:50

I want to auction a set of ASSETS (A) and fetch the maximum total price. The bidding is simultaneous and works as follows.

Say I have a collection of BIDDERS (B) who, individually, bid to purchase a subset of the assets A. Each bidder is constrained by a maximum outlay which is typically less than the total price of their bids. I.e., Bidders can not generally purchase all the assets they bid on. They must settle on a subset as determined by the AUCTIONEER.

What method, algorithm or protocol can the auctioneer follow to ensure he fetches the MAXIMUM TOTAL PRICE? (Your ideal answer would include pseudocode.)

Fig 1. Matrix of assets and bids ASSETS -- (A) --------------------------------------------- A1 A2 A3 ... Ai ... An + -- -- -- --- -- --- -- BIDDERS B1 | A1B1 A2B1 A3B1 ... AiB1 ... AnB1 (B) B2 | A1B2 A2B2 A3B2 ... AiB2 ... AnB2 B3 | A1B3 A2B3 A3B3 ... AiB3 ... AnB3 ... | .... .... .... ... .... ... .... Bj | A1Bj A2Bj A3Bj ... AiBj ... AnBj ... | .... .... .... ... .... ... .... Bm | A1Bm A2Bm A3Bm ... AiBm ... AnBm

There is a similar assignment problem which the Hungarian Algorithm solves. Here is an online implementation. However, that doesn't exactly solve this problem because more than one asset can be assigned to each bidder subject to their total outlay constraints.

How should I think about the module of coinvariants of a $G$-module?

Wed, 10/10/2018 - 18:02

Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$.

The dual notion of $G$-invariants is very simple, and for most $G$-modules, it's easy to tell whether or not for example $M^G = 0$.

On the other hand, I don't have a good intuition for $M_G$.

I'd appreciate a list of results that say something about this module of coinvariants, and criteria which might help recognize when it vanishes. Even facts in the case where $G$ is a finite, abelian, or even finite abelian group would be welcome.

References would also be helpful.

Explicit description of the scheme obtained by relative gluing data over a base scheme

Wed, 10/10/2018 - 17:42

I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction given in the Stacks Project tag 01LH: (For those that are familiar with it, or just want to see the linked tag, it is the proof that a certain functor is representable, with representing object being the relatively glued scheme)

Let $S$ be a scheme. Let $\mathfrak{B}$ be a basis for the topology of $S$. Suppose given the following data:

1) For every $U \in \mathfrak{B}$ a scheme $f_{U}: X_{U} \rightarrow U$ over U.

2) For every pair $U,V \in \mathfrak{B}$ such that $ V \subset U$ a morphism $\rho^{U}_{V}:X_{V} \rightarrow X_{U}$.

Assume that

a) each $ρ^{U}_{V}$ induces an isomorphism $X_{V} \rightarrow f^{−1}_{U}(V)$ of schemes over $V$,

b) whenever $W,V,U \in \mathfrak{B}$, with $W \subset V \subset U$ we have $\rho^{U}_{W}= \rho^{U}_{V}\circ \rho^{V}_{W}$.

Then there exists a morphism $f:X \rightarrow S$ of schemes and isomorphisms $i_{U}:f^{−1}(U) \rightarrow X_{U}$ over $U \in \mathfrak{B}$ such that for $V,U \in \mathfrak{B} $ with $V\subset U$ the composition

$$ X_{V} \stackrel{i_{V}^{-1}}{\longrightarrow} f^{-1}(V) \stackrel{\text{inclusion}}{\longrightarrow} f^{-1}(U) \stackrel{i_{U}}{\longrightarrow} X_{U} $$ is the morphism $\rho^{U}_{V}$.

For the sake of simplicity let's just take the base $\mathfrak{B}$ to be the set of all affine opens, although this really shouldn't make a difference. I am convinced by the proof there that the scheme $X$ exists. But I really have no intuitive understanding of what this scheme is. When I read the gluing data in terms of a base of open sets, it reads much like the construction of a bundle of sections in topology. Indeed we are covering the "base space" by some basic open sets, then we have some projection maps down from the "total space".

How should one think about the scheme $X$ and morphism $f: X \rightarrow S$ more explicitly? What is it as a set? Is there some way to think of it as sections of a total space?

Is it possible to classify extensions $G$ of an abelian $A$ by an abelian $N$ such that the map $G\rightarrow A$ is abelianization?

Wed, 10/10/2018 - 16:10

Suppose we have a given action $\varphi : A\rightarrow\text{Aut}(N)$ with $A,N$ abelian groups. Is it possible describe the isomorphism classes of extensions $G$ of $A$ by $N$ realizing $\varphi$ such that the map $G\rightarrow A$ is abelianization (ie, such that $N = G'$)?

(Without the requirement $N = G'$ this is classified by the group cohomology $H^2(A,N)$)

Some remarks:

Nontrivial central extensions certainly give examples where classes in $H^2(A,N)$ do not necessarily satisfy the condition that $G' = N$.

I wonder if there is a condition on $\varphi$ which guarantees that every class in $H^2(A,N)$ represented by $G$ satisfies $G' = N$.

From relative semi-stable sheaf to semi-stability

Wed, 10/10/2018 - 15:55

Let $\pi:X \to \mathbb{P}^1_{\mathbb{C}}$ be a flat, projective morphism, so $\pi$ factors through a closed immersion into $\mathbb{P}^n_{\mathbb{P}^1}$ followed by the natural projection to $\mathbb{P}^1$. Let $F$ be a flat family of torsion-free semi-stable sheaf on $X$ i.e., $F$ is $\mathbb{P}^1$-flat and for every $t \in \mathbb{P}^1$, $F_t$ is a torsion-free, semi-stable sheaf on $X_t$. Consider now the Segree embedding of $\phi:\mathbb{P}^n \times \mathbb{P}^1 \to \mathbb{P}^N$ for $N=2n+1$. Is $F$ a semi-stable sheaf on $X$ with respect to the polarization induced by the Segre embedding?

How is Chern-Simons theory related to Floer homology?

Wed, 10/10/2018 - 14:42

Chern-Simons theory (say, with gauge group $G$) is the quantum theory of the Chern-Simons functional $$CS(A)=\frac{1}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$ while instanton Floer homology is roughly the Morse homology of the moduli space of $G$-connections on $M$ with the Chern-Simons functional $CS(A)$ as Morse function.

Other than the superficial fact that they both use the Chern-Simons functional $CS(A)$ in some way, is there any deeper connection between the Chern-Simons theory and the Floer homology?

Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence?

Wed, 10/10/2018 - 14:20

Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent subsequence, say $f_{n_k}$, converging to $f$. Can we say that $f_n\to f$ strongly?

Explicit algebraic constructions of Parshin covers

Wed, 10/10/2018 - 13:46

Let $X$ be an algebraic curve defined over a number field $K$ and has genus $g \geq 2$. Let $P$ be a $K$-point of $X$. We say that $X_P$ is a Parshin cover of $X$ if $X_P$ is defined over a finite field extension $K'$ of $K$ and such that there is a morphism $X_P \rightarrow X$ defined over $K$ which is only ramified at $P$.

Mazur gave a very short and easy-to-understand construction here, which is supposedly similar to Parshin's original construction. However the construction seems to pass through Riemann surface theory and so it is not clear to me how to produce explicit algebraic equations from it.

Are there any examples, say when $X$ is a fairly simple curve in $\mathbb{P}^2$ say with a straightforward defining equation, such that one can construct a Parshin cover explicitly? For example, take $X$ to be the Fermat curve $x^n + y^n = z^n$ with $n \geq 5$. Can one give an explicit Parshin cover for it?

Kahler manifolds and algebraic varieties

Wed, 10/10/2018 - 13:02

Let $X$ be a smooth complete algebraic variety over $\mathbb{C}$. Can it happen that the underlying complex manifold is not Kahler? If yes, are there explicit examples? If not - how to prove this?